Lecture No 1 Q1. What is a function and give some examples?

Transcription

1 Lecture No 1 Q1. What is a function and give some examples? Ans: We can define a function informally as Function is a machine which assigns every element of one set a unique element of other set". It means that we need two sets and function is a relation which relate every element of one set a unique element of other set. Suppose that we have two sets A, B and f is a function which assign a unique element of B to every element of A, that is f(a)=b for all a in the set A. here we say that image of a is b if this is the case and for every a we have unique b then we will say that f is function from A to B and write f:a Β. Example: Suppose that we have A= as set of real numbers and B= + be the set of + 2 positive real numbers. Then f : defined by f( x) = x is a function. The Set is + known as Domain of the function and is known as Co domain of the function. In general the set of elements from which your function takes the elements is known as Domain of the function and the set of elements which your function assign the elements of Domain is known as Co domain. Finally the Set { f( a) = b a A/ b B} B of Co domain is known as the Range of the function. In Calculus I and II we will discus the function which has domain and co domain as real numbers. We discuss many functions in Calculus I and for more examples you should go to LMS. Q2. What are octants in 3d space? Ans: As you know that in 2d our axis divide the plane into for quadrants and the sign of any point in the plane depends on the fact that,in which quadrant the point lies and you know that in the first quadrant both x and y -coordinates have positive sign in the second quadrant x-coordinate has negative sign and y-coordinate has positive sign, in the third quadrant both x and y-coordinates are negative and in the fourth quadrant x-coordinate is positive while y-coordinate is negative. As shown in figure below. Prepared By: Irfan Khan Page 1

2 Similarly in 3d space we have eight different combinations of signs of the x,y and z- coordinates and this divide the space into eight regions and these regions are known as octants. So the conclusion is that there is no difference between quadrants and octants but these are the terminologies for the similar thing in 2d space and 3d space. As shown in figure below. Q3 what are reference axes and why we need them? Also explain the 8 octants in 3d. Ans: Eight octants are explained above and reference axes are the axis by which you can locate every point in the space. Here locate means that you can assign its position with respect to a fixed point known as origin and by three mutually perpendicular axis known as x,y and z axis. Q4. What is Cartesian co-ordinate system? Ans: Cartesian coordinate system is another name of Rectangular coordinates and these two are the same concepts. In which we locate the position of a point in the space with respect to the axis namely x, y and z-axis which are perpendicular to each other. Lecture no 1 Prepared By: Irfan Khan Page 2

3 What are three dimensional co-ordinates? A point in the three dimensional space has three co-ordinates these co-ordinates are called three dimensional co-ordinates. Q5. How can we define the position of a point P in a three dimensional coordinate system, if it's not static? Ans: If you are dealing with the particle which is continuously changes its position then definitely you can locate its position at particular time and at that time you will have fixed position of the particle. And if you are not given a particular time then the x y and z coordinate of particle will change continuously but we can define the co-ordinates of such a point as a function of t and then you are also given a time interval for that particle now if you want to find out the location of the particle at any time simply put the value of t in the given functions then you will get the location of the point. I hope that you will get the point. Example: Consider a particle is moving along a circle as shown in the figure. Then for any value of t the coordinate of the position of P(x,y) is given by x=a cost and y=asint where a is the radius of the circular path. Q6. Can there be any other Co-ordinate system containing axis more than x,y,and z.if kindly inform? Ans: There are many axis which can also locate the position of the particle in the space For example Spherical coordinates and Polar coordinates and you will study these coordinates in your course. If you want to study these coordinate system you can find them in your book of calculus, for this see the index of the your book. Q7. Is there any possibility of placing the "TIME" in 3d environment. Also time and space can be placed in any axis system? How? Ans: Yes people are working in four dimensional and even more dimensions but you can't views this reference Systems as you can view the things in 2d or 3d. For example when we talk about Theory of relativity then we discuss the motion of the planets and Prepared By: Irfan Khan Page 3

4 stars in four dimensions. In fact there are two major branches of Mathematics (i) Applied Mathematics (ii) Pure Mathematics and in Pure Mathematics we study n dimensional vector Spaces and even Infinite Dimensional spaces. Q8. What are complex numbers? What is the difference between real numbers and complex numbers? Why complex numbers are in Mathematics. Ans: There are certain equations which does not have real number as their solution for 2 example x + 1= 0 this equation does not have any real number as solution because there is no real number we have whose square is -1. It means that real numbers are not sufficient for our real life problems. And from here we get the motivation of the complex numbers, complex numbers are the set bigger then the set of rel numbers, we have every real number as complex numbers or we can say Mathematically set of real numbers is subset of set of complex numbers. We define the set of complex numbers as C = {x + i y where x and y are real numbers} and "i" denote square root of (-1) and is called iota. Complex numbers are usually denoted by z = x + i y where x is known as the real part of the complex number and y is known as imaginary part of complex number z. In the figure below we have shown that the set of Complex numbers is the super set of all the sets. Q9. Explain the 3 dimensional planes? Ans: We have three dimensional space and there are three mutually perpendicular lines to locate any point in the three dimensional space. Now a plane is always two dimensional but it is may be placed in 3d. Consider the equations x = 0, y =0 Now if you are discussion is in 2d then these equations will represent the equation of lines namely y- axis and x-axis respectively. And if you discuss these equations in 3d space then these equations represents the planes and together with z = 0 divide the 3d space into eight Prepared By: Irfan Khan Page 4

5 octants in which each octant has particular sign of x,y and z-coordinates for every point lie in it. Note that in the 2d we have quadrants instead of octants. Q10. Why we takeπ = 3.14? Where as in geometry its value is 180 degree. Ans: Well the number π has very prominent place in Mathematics and usually define as Ratio of circumference to the diameter of a circle. Also note that the number π is irrational number it means that it has non-recurring decimal expression. So being irrational number nobody knows the exact value of theπ and since in many real life problems π occurs hence we use the approximate values of the number π and some people take it as Now as you know that an arc of length equal to radius subtends an angle of one radian. Also one radian= 57degree38 minutes and also there are 2π radian in a complete revolution also we know that there are 360 degrees in a complete revolution of circle, so360degree=2π radian and180=π or π radian. So the conclusion is that when the angles are measured in radians π =3.14 and when the angles are measured in degrees then π =180. Q11. What is onto and one to one function? Ans: You should know what is a function and if you don't know what is a function then see the answers of the questions above. Suppose that you have a function f:a Β. Then we will say that this function is onto if for every element of the set B say it "b" we have a unique element in set A say "a" such that f(a)=b holds. In other words we can say that a function is onto if every element of its co domain is the image of some element of its domain. Now we will say that the function is one to one if f(x 1 )=f(x 2 ) implies x 1 =x 2 where you have x 1 and x 2 in the set A. In other words we will say a function "f" is one to one if distinct elements in the domain have distinct images. Example: Identity function is both one to one and onto on any set A. And you can check every function, whether it is one to one or not. Similarly you can check every function whether it is onto or not. Q12. Explain why we need the z axis? Ans: When you are doing geometry or discuss a particle in plane then you need only two perpendicular lines to locate the position of the particle because it lies in the plane. Now if a body is moving in the 3d space then the position of the particle at a particular instant can not be expressed as you deal with bodies in plane. For example if you are interested in the location of the particle on the cube then this may have length breadth and height. So for the location of any point in the 3d we need three perpendicular lines, you are already know in plane we have two perpendicular lines namely x-axis and y axis, you also know that you can find out a third line which is perpendicular to both by cross product of the two vectors corresponding to these lines the direction of that vector which is perpendicular to both the x and y axis is the direction of Prepared By: Irfan Khan Page 5

6 z -axis and this direction can be obtained by the Right hand rule explained in the Lecture. Now the problem is that we are unable to show the three perpendicular line on the plane so we made the z axis on the plane and consider it is coming out of the page. I hope that you will understand why we need z-axis and how to draw it on the plane. Q13. What is mean by one to one correspondence? Ans: When you are saying that there is a one to one correspondence between between two sets it means that you can define at least a function from one set to the other set which is both one to one (Injective) and onto (Surjective) or more compactly we can say that the function is bijective.if between any two sets we have a function (f say) like that then we say that there is one to one correspondence between these sets. Hence whenever we will talk about one to one correspondence it means that we there is a bijective function between the two sets. Q14. What is the difference between Calculus and Discrete Mathematics? Ans: Discrete Mathematics concerns with the process which consist of individual steps unlike continuous steps as in calculus. This distinguishes it from calculus, which studies continuously changing processes. While the ideas of calculus were fundamental to the science and technology of the industrial revolution, the ideas of discrete mathematics underline the science and technology specific to the computer age. Lecture # 2 What is the formula to determine the value for the zeros of parabola (where the x will zero)? To find the zeros of any parabola, equate the equation to zero. for example: x^2+8x+9 = 0 (x-1)(x+9)=0 Now the equation will be zero when either x = 1 or x = -9. These value of x are called zeros of the equation. Domain and Range? Domain and range of the function: WHEN ONE THING DEPENDS ON ANOTHER, as, for example, the area of a circle depends on the radius, or the temperature on the mountain Prepared By: Irfan Khan Page 6

7 depends on the height, then we say that the first is a "function" of the other. The area of a circle is a function of -- it depends on -- the radius. More precisely: A function is a relationship between two variables, typically called x and y. For example, y = 2x + 3. The values that x may assume are called the domain of the function, and the values ofy (that correspond to each value of x) are called the range. Thus if 5 is a value in the domain of that function, then = 13 is the corresponding value in the range. y is called a function of x if and only if there is a rule that assigns to every value of xin the domain one -- and only one -- value of y. A "function" is therefore single-valued ("one and only one"). The rule is typically in the form of an equation, e.g. y = 2x + 3. It is customary to call x the independent variable, because we are given, or we must choose, the value of x first. y is then called the dependent variable, because its value will depend on the value of x. By the value of the function, we mean the value of y. The range, then, is composed of the values of the function. What are real life examples of parabolas, hyperbolas, and ellipses? Parabola's real world application includes satellite dishes, which engineers take into account into their design for data reception. Hyperbola's example is Focus lenses designs which will determine focal points depending on your curve design. And Ellipses probably race track designs that help designers take into account top speeds and such depending on shape. Why circles ellipse parabolas and hyperbolas are called conic sections? Prepared By: Irfan Khan Page 7

8 Circles, parabolas, ellipses, and hyperbolas are called conic sections because you can get those shapes by placing two cones - one on top of the other - with only the tip touching, and then you cut those cones by a plane. When you move that plane around you get different shapes. If you want to see an illustration of these properties, click on the link below on the related links section What is Cone and Circular Cylinder,Right Circular Cylinder? Cone Generally a cone is a pyramid having circular base. A right circular cone is cone with its vertex vertically above the center of the base (circle). Circular cylinder A circular cylinder is a cylinder whose centers of the bases forms a line perpendicular the bases. Right Circular cylinder Right circular cylinder is cylinder whose bases are circles and are aligned directly one above the other Q15. Explain Hyperbola? And what are its real life applications? Ans: As you know that when defining the Ellipse we have two fixed points, similarly we have two fixed points for the Hyperbola all the points in the plane are included in the Hyperbola if the difference of distance from the fixed points is a given positive number and that positive number is less then the distance between the fixed points. The plane curves line, circle, Ellipse,Parabola and Hyperbola are known as the conic Sections because these all curves are obtained by the intersection of a cone and a plane. The fixed points are said to be foci and there are many terminologies about the Hyperbola for more detail see page 606 Calculus by Howard Anton or go to the following URL Q16. What is a Number how can we define term Number? Is it a defined term or undefined like terms set and element? Ans: Well the question which you asked is very interesting, the numbers we know naturally now what does it mean, well no body teach us about one or zero. When we grow up then we naturally know that Allaha is one by our belief. The famous Muslim Mathematician Mohammed ben Musa al-khowarizmi said about the numbers When I considered what people generally want in calculating, I found that it always is a number. Prepared By: Irfan Khan Page 8

9 Most older books on the history of math and numbers say that at some point (they are vague on when) somebody made a connection between the number of fingers on a hand and the number of cows or sheep, and figured out the concept of "number." Not enough attention has been paid to the digit zero in the course of history. It was the last of the decimal digits to be established in Western systems, but it has so many strong effects on our lives. Where would we be without zeros? In big trouble... In our paychecks, zeros are critical. We all like to see lots of zeros lined up on a check. Speaking of lines, there would be a big hole in the number line if there was no zero, you could fall through and get hurt. In fact, all of the other numbers would just slip though and then what would we be left with? The digit 1 has been called "The loneliest number," yet many organizations, including sports teams, companies and countries take pride in calling themselves "Number 1." Similarly other numbers are also have history from the real life, actually we know these numbers and Nature learn us about these numbers. Q.17 what is an Asymptote to a curve? Ans: If you draw a curve of a function then any equation of straight line is said to be asymptote of the curve if you approaches towards infinity as an input of the function it means that we draw the curve for larger and larger values, then the distance between the straight line and curve approaches to zero whose graph you are discussing. Then the straight line is known as asymptote to the curve. For example you can see that in the case of Hyperbola the straight line y = x is an asymptote because the distance between line y=x and one branch of the parabola approaches towards zero. Similarly you can see that the straight line y= - x is also an asymptote to the other branch of Hyperbola. Q18. What is vertex (maximum & minimum) and Axis of symmetry? What is difference between two dimensional geometry three dimensional geometry? 2 Ans: Since you know that any quadratic equation ax + bx + c = 0 with a 0 represents a parabola. Now the parabola may opens upward or downward depending on the sign of a". We say the maximum or minimum point on the parabola as vertex if it opens down wards and upwards respectively. And that maximum or minimum point of a vertex is known as the vertex of the parabola. Where as any line passing through the vertex is b x = known as axis of symmetry of the parabola. And that line is always at 2a. In two dimensional space we discuss the particles in the plane while in the 3dimensional Geometry we consider the particle in 3d, and also note that in 2d we have two components of any point in the plane namely x and y regarding to the x and y axis. While in 3d we have three position component of every point and these three are refer to the x,y and z axis. We can say that 3d is the extension of 2d. Q19. How we will find the distance between two points a and b where a lies on the Parabola and b is outside and not static? Prepared By: Irfan Khan Page 9

10 Ans: If you have a point on the parabola then you can find out the coordinates of that point now if a particle is not static it means that it changes its position time by time then its position vector will be given in terms of some function of "t"(note that if the particle is in plane then both of its coordinates are given as function of "t") and you can find out the distance between the non static point and the point on the parabola by distance formula, also note that the formula which you will find out for the distance will be a function of "t".because "t" is involved in the coordinates of non static point. If you are given a particular time and asked to find out the distance between the points then you can find out the distance simply by putting the value of t in the distance formula. Q20. Why square root of a number gives us two values but cube root doesn't give 3 values could you please tell me that why this is so? q Ans: Suppose that you want to find out the value of where q is positive real number. We are interesting in finding out the values such that x= q in other words we will find out the solution of that equation. We can write 2 2 x = q x q = 0 ( x q)( x+ q) = 0 which shows that x= q and x= q are both are the solution of the above equation in other words we can say that the quadratic equation has two real roots that is square root of a positive real number may be with negative sign or with positive sign. So we can write ± x= q. It means that While evaluating square root of a number we write + and sign with the answer. You will get the point by an example suppose that you want find out the square root of "9" then you write the answer ± 3, because both +3 and -3 have the same square 9. It means that we don't know the 9 is obtained by the square of +3 or -3, that s why we have two values when we are taking the square root. On the other hand when you take the cube 1 1 root of a number say ( q) 3 then let x= ( q) 3 taking cube of both sides we get x q = 0 ( x q)( x + xq + q ) = 0 Now this equation only have one real solution and two imaginary solutions that s why we take only one value while taking square root of a real number. Q21. What is the Difference between hyperbola and parabola? Ans: In Parabola we have a fixed line known as Directrix and a fixed point known as focus and all the points in the plane are the points on the parabola which satisfy the condition that the distance from the fixed point and the fixed line is the same. And by using this definition we can draw the parabola. For Hyperbola we have two fixed points known as the foci of the Hyperbola and all the points on the plane are in Hyperbola such that the difference of the distances from the fixed points is a constant positive number which is less then the distance between the distance of the foci. It means that in the definition of the Hyperbola we need two fixed points and a number "c" such that c is less then the distance between the fixed points. And all the points in the Prepared By: Irfan Khan Page 10

11 plane which satisfy this condition are the points on the Hyperbola. Where as in the Parabola we have only one fix point and a fixed line. Q22. I have some problems in understanding parabola and circles and what is difference among parabola, hyperbola and ellipse is. Ans: For a circle we have a fixed point known as the center of the circle and a positive real number known as the radius of the circle and all the points whose distance from the fixed point is equal to the positive real number (that is radius) are the points of the circle. In definition of Parabola we have a fixed line and a fixed point and all the points in the plane are the points of the parabola which satisfy the condition that the distance from the fixed point and the fixed line is the same. In the definition of Ellipse we have two fixed points and a positive real number which is greater then the distance between the fixed points and all the points in the plane whose sum of the distances from the fixed points is equal to that positive real number. The fixed points are known as the foci and fixed positive real number is usually denoted by "c". Where as for Hyperbola we have two fixed points known as the foci of the Hyperbola and all the points whose difference of the distances from the fixed points is a constant positive number which is less then the distance between the distances of the foci Also circle is the special case of Ellipse. Q23. What is parametric representation of surface? Ans: When you define surface in the space then you give the equation of that surface in rectangular coordinates. (Rectangular coordinates means you have x, y and z-axis as Reference axis). But there is another method for defining the surface which is known as the parametric representation in this method you define the coordinates of every point on the surface by another parameter, that is you define the x,y and z - coordinates of every point in terms of the parameter. For example parametric equation of the parabola are x= at 2 and y= 2at and you can check these equations satisfy the equation of Parabola and define the every point on the parabola for certain value of t.. Q24. I can t understand example of " g (x,y,z)=zsin xy " explain. Ans: The idea is that first we define four functions of three variables namely g, u,v and w (these are the functions) and the variables are x, y and z. Then we use u,v and w as input to the function g(x,y,z) So we write with this input instead of x,y,z and get the result g(u,v,w). Now since u,v and w are the functions of x,y and z and also defined in the example by the formulas, so we will replace them by u(x,y,z), v(x,y,z) and w(x,y,z).so we get the expression as g(u(x,y,z), v(x,y,z), w(x,y,z)). But we are also given in the example that g (x,y,z)=zsin xy and in terms of new input it becomes g (u,v,w)=wsin uv but in the examples formulas for w,u and v are also given, hence we put these formulas here and get the value of the function whose input is u,v and w. I hope that you will understand now the example. Prepared By: Irfan Khan Page 11

12 Q25. What is difference between vertex (maximum) and Axis of symmetry? Ans: We say the maximum or minimum point on the parabola as vertex if it opens down wards and upwards respectively. Vertex is a point on the parabola where as any line passing through the vertex is known as axis of symmetry of the parabola. Q26. Define conjugate axis. Ans: In the definition of Hyperbola you have two fixed points these two points are knwon as the foci of the parabola and any line which passes through these foci is known as Major axis and any line which is perpendicular to Major axis and passes through the mid point of the line segment joining the foci is known as Conjugate axis of Hyperbola. Lecture # 3 How can we defined a general equation of plane? General equation of plane is ax+by+cz+d=0 Where n=(a, b, c) is non-zero normal vector. how the general Equation of plane drive? We define a plane in a three dimensional space by specifying a point and a normal vector to the plane. Let r be the position vector of any point P in the plane from the origin O, and let n be a nonzero vector normal to the plane. Now D is the perpendicular distance from the origin to the plane. r.n = r n cos A. But n = 1 so we have r.n = r cos A = D. we split both r and n into their components. We write r = xi + yj + zk and n = n1i + n2j + n3k Therefore r.n = (xi + yj + zk). (n1i + n2j + n3k) = D so r.n = xn1 + yn2 + zn3 = D n1, n2 and n3 (the components of the unit surface normal vector) give us the A, B and C in the equation Ax + By + Cz = D which is the general equation of the plane. What are direction Cosines? Prepared By: Irfan Khan Page 12

13 The direction cosines of a vector are merely the cosines of the angles that the vector makes with the x, y, and z axes, respectively. We label these angles (angle with the x axis), (angle with the y axis), and (angle with the z axis); and we define The cosines of the angles can be found by taking dot products. Thus, if we have a unit vector given by, the direction cosines are Q27. Explain surfaces in 3D Space in detail? Ans: When you have a function of two variable say f(x,y), then you have two independent variables as input, it means that you need two lines for the location of these real numbers which are independent and a third line for the value of the function say z=f(x,y) and we plot this to the z -axis in the 3d space and the resulting figure is also in 3d and is known as Surface. Example: Consider the function f(x,y)= x 2 + y 2 where 2 x 1and 2 y 1.Then we have a surface in 3d as shown in the figure below. Prepared By: Irfan Khan Page 13

14 Note that the above 3d surface, you can t draw with your own hands and the above figure is generated by Matematica. You can easily see the three axis two correspond to the values of x and y and since we give the input as 2 x 1and 2 y 1 and the line perpendicular to both these lines is the z-axis along which we plot the values of the function. Similarly if we consider f(x,y)=x 2-2xz+z 3 where 3 x 3and 3 y 3 Then the 3d surface is shown below Q28. What do your mean by FOCI? Ans: In the definitions of Parabola, Ellipse and Hyperbola we have one fix point for parabola and two fixed points for Ellipse and Hyperbola, these fixed points are known as Foci for the Ellipse and Hyperbola and is just plural of focus which is in the case Parabola. It means that if we have only one fixed point then we will say that it is focus as in the case of Parabola and if there are more then two fixed points then we will say them foci as in the case of Ellipse and Hyperbola. Prepared By: Irfan Khan Page 14

15 What is Plane? The orientation of a plane is defined by means of the direction of a perpendicular vector. The position of the plane is added by means of some initial conditions. Thus, a plane is given by the equation Usually, the right side of the equation is brought to the left-hand side to obtain Q29. What are Direction ratios and Direction Cosines of a line? Ans: If you have any line in 3d it will made some angles with x,y and z- axis these angles are known as Direction Angles say "a","b","c",are the angles which line make with x,y and z-axis respectively. Now if you take the cosine of these three angles that is Cos(a), Cos(b) and Cos(c) these are known as Direction Cosines and numbers which are proportional to it are known as Direction ratios. Or the numbers a 1, b 1 and c 1 are cos( a) cos( b) cos( c) = = = k Direction ratios if a1 b1 c1 (Where k is constant of proportionality) And we give the formulas for both Direction Cosines as well as Direction Ratios in our Lecture and also available at LMS. Q30. Would you please elaborate multidimensional surfaces and coordinates with result of curves? Ans: You are already familiar with the 2d Geometry, now since you have two perpendicular lines as reference axis usually known as x and y-axis. Now if a point lies in the space then its position cannot be obtained by the two axis which we use for 2d. So we need another axis which can be obtained by the cross product of the two axis which we have in 2d and thus we have three perpendicular lines for the location of the point in space. Any function of two variables say f(x,y) can be drawn into 3d space and is known as surface in 3d. Now if any plane and a surface intersect each other then we get a curve which is placed in 3d space. And you can visualize this fact by taking example in the real life. how do we get the zeros of parabola x=0 and x = 4 for the equation -x² + 4x = 0? The zeros of the parabola (i.e. the point where the parabola meets x-axis) are the solutions to -x 2 +4x = 0 so x = 0 and x = 4. Prepared By: Irfan Khan Page 15

16 As it touches with x-axis only, so the value of y-axis is zero and we get the points (0, 0) and (4,0) Q31. Is there exist any four dimensional system or more than four? Ans: Yes there are four dimensional and more then four dimensional coordinate systems but you can't see the figures as you did in 2d. When we discuss the bodies in Theory of relativity then we consider the four dimensions and three of them are position coordinates and the fourth is time. Similarly there are more then four dimensional considerations but these ideas are abstract. Leture 4,5,6 Q32. What is Domain of a function? Ans: When you define a function by some formula then you have an independent variable if your function is of one variable, now the domain of the function will be all those values of the independent variable for which the formula of the function gives you finite and real values. Example: 2 (i) Consider the function defined by the formula f( x) = x Then your independent variable is x and all the values of x which gives you finite real value as out put are the values from your domain and the set containing all such values is known as domain of that function and here domain of the function is whole real line which we write as. (ii) Now consider the function defined by the formula f( x) = x Now as you know that if your independent variable is negative then ve # is a complex number. But remember that in calculus we always study the real valued functions. So we will give as input for that formula all the non-negative numbers which is the domain of that function. Also note that if our function f is of one variable then our domain will be a subset of real numbers. Similarly if we have a function of two variables then the the domain of the function will be some subset of the plane because we have two variables and we will find out the two subsets corresponds to these two variables and together these two sets in from of Cartesian product of these sets will form the domain of the function which will be subset of plane because we know that there is one to one correspondence between the set of points of plane and.and you also know that we define the 2d space as. Example: Prepared By: Irfan Khan Page 16

17 Consider the function which is because x + y 2 2 f( xy, ) = x + y 2 2 then its domain is the whole 2d space gives you a real number for all the ordered pairs (x, y). 2 2 Now if you have the function f( xy, ) = 1 x y then as you know that the square root of a positive real number is a real number, so the natural condition on the domain of the function is 1 x y 0 1 x + y which is domain of the function and is the region inside the circle of radius 1 including boundary and you can also draw this domain in plane. Q33. What is the basic difference between rectangular Polar and spherical Coordinates? Ans: In rectangular coordinate system we get the position of the particle in space by using three mutually perpendicular lines while in the spherical coordinates we get the position of the particle by "r" which is distance of the particle from the origin also known as pole and "θ " which is the angle made by the line from origin to that point with x-axis in the positive orientation then the third coordinate is " ϕ " which again an angle made by the line from origin to the point but measure with the z-axis. So in Spherical coordinates a point will have coordinates Prθϕ (,, ).Where as in rectangular coordinate system we get the position of the particle by getting the length of the perpendicular lines form the point to the axis. In polar coordinates we get the position of the particle in space by "r" which is distance from the origin to the point,"θ " which is the angle in the positive orientation with the line passing through origin and the third and last coordinate is "z" which is the distance from origin to the foot of perpendicular, drawn from the point to the z -axis. Any point in Polar coordinates will have coordinates as Pr (, θ, z). What is "Horizontal Elliptic Cylinder " equation.? An elliptic cylinder is a cylinder whose cross-sections are ellipses. General elliptic cylinder equation is (x 2 /a 2 )+(y 2 /b 2 )=1 with Z=constant For horizontal elliptic cylinder, y=constant and (x 2 /a 2 )+(z 2 /b 2 )=1 e.g. (x 2 /4)+(z 2 /1)=1 or x 2 +4z 2 =4 Q34. What is NATURAL DOMAIN? Prepared By: Irfan Khan Page 17

18 Ans: Natural domain of a function is the set of those values of the independent variables for which the function has finite real values. For example consider the function 2 f( x) = x then its natural domain is whole real line and we write it as (, ) because function gives finite values for every real number. Now we define the function by f( x) = x its natural domain is all nonnegative real numbers because we know that square root of negative real numbers are complex numbers which is not real, hence we exclude these real numbers from the natural domain of the function. 1 f( x) = Again consider another function defined by x its natural domain is whole of the real numbers except "0" because the function does not have finite value at x=0,so we exclude this point from the natural domain of the function. Q35. Any plane parallel to coordinate axis possess one coordinate of every point in that plane as constant, In particular if plane passes through origin then one coordinate is zero. Why? Ans: Any plane parallel to one of the coordinate axis must have one of the coordinate constant from the three coordinates of any point in that plane. But it is not necessary that the every point in that plane has one zero coordinate. Every point in the plane parallel to the one of the coordinate axis will have one coordinate as zero if that plane passes through origin. Because when a plane passes through origin and is parallel to two axis then the coordinate corresponding to this axis varies while the third coordinate will be zero. And if the plane does not passes through origin and and it is parallel to the two axis from the three axis then the plane must intersect the remaining axis to which it is not parallel and that number remain constant for all the points in the plane. For example any plane parallel to the x and y axis must have a constant z coordinate for all the points in the that plane. As we did in our lecture x=0 is plane in which every point has x coordinate as 0. But x=x 0 is also an equation of the plane parallel to the same axis (that is y and z axis) but in this case every point in the plane has fixed x coordinate which is x 0. Where x 0 is constant. What is abscissa and where it is used? Let suppose any point on the plane is repressed by the order pair (x, y). Here the value of x is know as abscissa and the value of y is know ordinate Any point on the x-axis will be denoted by an order pair.let suppose ( 4,0) whose first element (that is 4 )which is also known as abscissa is a real number and other element of the order pair which is also known as ordinate will has 0 values (because on y-axis, the point has no value on y-axis ) Prepared By: Irfan Khan Page 18

19 Q36. Can we draw 3D graphs or surfaces of function of two variables? Ans: You can t draw 3d graph on the page but there are soft wares which show the 3d graphs when we give them the functions and also the ranges of the independent variables numbers. So don t bother about the sketching of the 3d graphs. The 3d surfaces and graphs of functions of two variables drawn in your book are sketched by the software Mathematica,Mat lab, Maple etc. But remember that you can draw the domain of the function in 2d as you have questions in the exercises and also we did in our lecture. Q37. Please tell me Navigation using Spherical Co-ordinates? Ans: First of all you should know what do we mean by the Navigation? Well you know that our earth is assumed to be Spherical and we define the Longitude and Latitude axis on that sphere. And the process or method of finding the position of a place on the earth is known as Navigation. Spherical coordinate are also related by Longitude and Latitude axis and we consider the earth as a sphere of fixed radius and try to find out the position of any place using spherical coordinate which we suppose that are located in the center of the earth. Since for any point on the earth we have "r" being the radius of the earth as constant thus for every point on the earth we have one coordinate fixed and the remaining coordinates namely θ and ϕ can be obtained by the Longitude and Latitude. And this process is known as Navigation using Spherical coordinates that is the process of finding the position of an object on the earth or space. Q38. What's the proper definition of three dimensions Analytical Geometry? Why do we use polar coordinate and Parametric Equations as well as difference between them? What do we mean by Arc length? Ans: You define the real numbers and know that there is one to one correspondence between the number of points on any line segment and real numbers. If you are dealing only with real numbers it means that you are considering one dimension and you need only one real number for the location of any point on the line because there is one to one correspondence as mention above. Now if you have that is Cartesian product of two real lines, which we define as two dimensions and every point in two dimensions required two real numbers for its position that s why we have two perpendicular lines as reference axis in 2d. And there is one to one correspondence between the set of points of and the points in the plane. Now we have that is Cartesian product of the real line three times, every point in this set will have three real numbers and this set has elements as order triples, there is one to one correspondence between the space and the set which we define as 3d space. Prepared By: Irfan Khan Page 19

20 Since in Mathematics we draw the graph of the function y =f(x) and any vertical line crosses the graph of the function only at one point (This is also known as the vertical line test for a function). But there are many important graphs for which the vertical line test fails but they have much more importance in the real life. For example any equation of the circle say x 2 + y 2 = a 2 now any vertical line cross this circle at two points and we want to write it in the form of functions so we define another variable say "t" and define the both coordinate points "x" and "y" in terms of this new variable "t" which is known as parameter. And you can see that the relation between them is x=a cost and y=sint where as these are functions and are known as the parametric equations of the circle. You can locate the position of any particle in the plane by two perpendicular lines namely x and y axis. Besides this coordinate system we can also find out the position of any particle in the plane by the polar coordinate axis and there is no relation between parametric equations and Polar coordinates. These are two different terms. When you move along a circle then you are not describing the straight line and in any time interval you are describing a part of the circle then we say that the path you have describe is an arc or simply you can say that arcs are the curved path. Q39. What is the use of LIMITS and why we are interested in LIMITS of FUNCTIONS? Ans: The concept of limits is the basic concept in Calculus and Analysis. You already study the concept of limit, Continuity, Differentiability and integration in one variable and note that all these concepts based on the limits and all these use the limits in their definition. And now you have the application of Differentiability and continuity in real life and so the limits are in use. For example Suppose that a particle is moving with some velocity in space then you can defined the motion of the particle by a function of two variable and if you want to find out the instantaneous velocity of the particle then you need the concept of the limit. What is Parabola and Hyperbola? Parabola A parabola is the curve obtained when the plane cuts parallel to the cone side Hyperbola Hyperbola is the curve obtained when the plane cuts almost parallel to the axis What is Reference Axis System? Reference axis system is one that is used to locate the position of a point in a plane or space. Prepared By: Irfan Khan Page 20

21 Q40. What is the definition of limit of a function at a point? Ans: You have concept of Limit of a function of one variable at a given point. Suppose that f(x) is a function of one variable and you want to find out whether the limit at x=x 0 exists or not. Then you approach towards that point from both side and in the case of one variable you can approach a point from only two ways from its left and write because domain of your function is a subset of real numbers, hence can be located on a line. Then you check the functional values(i.e the values of f(x) if these values approaches towards a fixed number either you approach towards x=x 0 from either side then you say that the limit of the function exists at x=x 0. And in the language of mathematics we say that the function must have left and right limit and these limits must be same. Now come to the case of more then one variable, say you want to see the limit of the function of two variables namely f(x,y) at some point (x 0,y 0 )in its domain, now as your domain is a subset of plane and you can approach towards this point through any curve which is in that plane and passes through the point (x 0,y 0 ), so you will check that if we approach towards that point along any curve then if the value of the function approaches towards the same point in the range then we will say that the limit of the function exists. There is formal definition of the limit of a function of more then two variables which we skip in the lecture and we think that you don't need this concept. Q41. Why the function w=xylnz, has Domain Half space z > 0 and Range (, )? Ans: Since you know that the ln (that is the log with base e where e is irrational number) of negative real numbers and 0 doesn't exists and in the formula of function we have ln z. So we made the restriction over the domain of the function that the variable z>0. And the range of the function is whole of the real line because for every real number you can find out values of x y and z such that the given equation satisfied. What is real mean if i saw x is equal y scare or y= x scare in graph.? The graph of y=x2 or x=y2 will consist of all those points which satisfy these equations. Both equations will represent parabolas. What is the Hyperbola? Ellipse we have two fixed points, similarly we have two fixed points for the Hyperbola all the points in the plane are included in the Hyperbola if the difference of distance from the fixed points is a given positive number and that positive number is less then the distance between the fixed points. The plane curves line, circle, Ellipse,Parabola and Hyperbola are known as the conic Sections because these all curves are obtained by the intersection of a cone and a plane Prepared By: Irfan Khan Page 21

22 Q42. What is a function? Ans: Function is a rule which assigns every element of one set (which is known as the domain of the function) to a unique element of the other set (known as the co domain). The subset of the co domain which consists of only those elements of the co domain which are the image of some element of domain under f are known as the range of the function. You can also treat function as a machine which takes input from a set and gives us out put of the other set but this out put is unique. In calculus we will study the real valued functions only it means that we will consider those functions which have real numbers as their domain and range. Such functions are known as real valued functions. What is A function? WHEN ONE THING DEPENDS ON ANOTHER, as, for example, the area of a circle depends on the radius, or the temperature on the mountain depends on the height, then we say that the first is a "function" of the other. The area of a circle is a function of -- it depends on -- the radius. More precisely: A function is a relationship between two variables, typically called x and y. For example, y = 2x + 3. The values that x may assume are called the domain of the function, and the values of y (that correspond to each value of x) are called the range. Thus if 5 is a value in the domain of that function, then = 13 is the corresponding value in the range. y is called a function of x if and only if there is a rule that assigns to every value of x in the domain one -- and only one -- value of y. A "function" is therefore single-valued ("one and only one"). The rule is typically in the form of an equation, e.g. y = 2x + 3. It is customary to call x the independent variable, because we are given, or we must choose, the value of x first. y is then called the dependent variable, because its value will depend on the value of x. By the value of the function, we mean the value of y. The range, then, is composed of the values of the function. Q43. When the domain is Entire space, and if the domain is entire space then why the Range is (0, )? Ans: This is not necessary that if you have function of three variables and its domain is entire space then the range is (0, ), but in particular this is possible that you have a Prepared By: Irfan Khan Page 22

23 function which has domain as whole space can have the range which you have mentioned For example the function f( xyz,, ) = x + y + z have entire space as domain where as range as positive real numbers. And the example of the function f( xyz,, ) = x+ y+ z has whole of the space as domain and also whole of the space as range. Q44. What is composition of two functions (Both of one variable) and if the two functions are continuos then what s true about their composition? Ans: You have learned the difference between the product of two functions and Composition of two functions in your calculus I course. But i will illustrate it for you by some example. 2 Suppose that we have two functions one is f( x) = x+ 1 and the other is gx ( ) = x ( I took these functions but you can any two functions). Now the composition f g will also be a function defined by the formula f gx ( ) = f( gx ( )) 2.But we have gx ( ) = x so we get 2 f gx ( ) = f( x ) and by the definition of the function f(x) we get Also note that now the input for the function f is x 2 not x) f gx ( ) = x 2 +1 which is the composition of f and g. It means that when we are writing the composition of two functions then first the input is used in one function and then out put of that function is used as input of the 2nd function as clear from above example. Now you should compute g f your self using the definition g f( x) = g( f( x)). Also if the two functions are continuous then their composition will also be continuous. What are Conic sections? When a solid figure, which in this case is a cone, is cut by a plane, the section which is obtained is called a conic section. Conic sections could be circles, ellipses, hyperbolas, and parabolas depending upon the angle of intersection between axis of the cone and the plane. Q45. What is difference between the coutntinuty in two dimensions (Plane) and three dimensions (space)? Ans: The idea is the same but in 2d you approach towards the point in domain where you want to find out the limit from two ways and the calculate the limit. Because the function y =f(x) has some subset of real number as domain and on real line we can approach a point either from its left side or from its right side. After calculating the limit you see the function is defined at that point or not if the function is defined then we see Prepared By: Irfan Khan Page 23

24 the limiting value of the function and the functional values are same or not. If both values are same then we say that the function is continuous at that point. Similarly in 3d if we want check the continuity at a point we find the limit and do the same things but we perform all these steps in 3d.Now suppose that you have a function f(x, y) of two variable then as you know that the graph of the function will be a surface in 3d space and the domain of that function will be some subset of plane and if you want to find out the continuity of the that function at some point (x 0,y 0 ) then first you will calculate the Limit of the function f(x,y) at the point(x 0,y 0 ) now since our point is in plane and we can approach towards that point through the infinite curves which passes through that point and along all these curves if your functional values approaches towards the same point then we say that the Limit of the function exist. Then we will check that the function is defined at that point or not, if the function is not defined at that point then function can t be continuous and if the function is defined at that point and has the same value as we have the limit of the function at the point (x 0,y 0 ).Then we say that the function is continuous at the point (x 0,y 0 ). Now you can note that the basic steps are same in the investigation of the continuity of functions whether in 2d or 3d. Q46. If we want to take the double partial derivative of a function f(x, y) with respect to x then how we can get this? Ans: If you have a function of two variable say "f(x,y)" and then you find the partial derivative of that function with respect to "x" that you find the "f x " Now again you want to take the partial derivative of that function. Then you should simply partially differentiate the "f x "you will get the "f xx ". For example if you have the function say f(x, y) = sin (xy) then f x = ycos(xy) and for " f xx " you will partially differentiate " f x " with respect to x which is f xx = y 2 sin(xy) Similarly you can find out f xxx by again partially differentiating f xx with respect to x and so on. WHAT IS THE "LONGITUDE AND LATITUDE CO-ORDINATES"? Any location on Earth is described by two numbers--its latitude and its longitude. The ancient Greek geographer Ptolemy created a grid system and listed the coordinates for places throughout the known world in his book Geography. When looking at a map, latitude lines run horizontally. Latitude lines are also known as parallels since they are parallel and are an equal distant from each other. Each degree of latitude is approximately 69 miles (111 km) apart; there is a variation due to the fact that the earth is not a perfect sphere but an oblate ellipsoid (slightly egg-shaped). The vertical longitude lines are also known as meridians. They converge at the poles and are widest at the equator (about 69 miles or 111 km apart). Zero degrees longitude is located at Greenwich, England (0 ). The degrees continue 180 east and 180 west where they meet and form the International Date Line in the Pacific Ocean. Prepared By: Irfan Khan Page 24

25 Q47. Explain the Geometrical meaning of partial derivatives of multivariable function. Ans: You know that the derivative of one variable function say f(x) at some point x=x 0 represent the slope of the line at the point (x 0,f(x 0 )). Now in the case of functions of two variables, you have two independent variables and can differentiate with respect to both variables, keeping the one variable constant and the other variable varies in the formula of function. Suppose that you have a function f(x,y) of two variables then its graph will be a surface in 3d space. First of all you should note that if we have (x 0,y 0 ) be any point in the domain of the function f(x,y) then the point (x 0,y 0,f(x 0,y 0 )) will lie on the surface of the z =f(x,y). When you fix one variable then you got a plane whose intersection with your surface z=f(x,y) gives a curve in which one variable is constant and the other varies and finally you can find the slope of the tangent line at that curve and we call it the partial derivative of the function f(x,y) with respect to the variable which varies. As shown in the figure below. For example if you want to find out the slope of the tangent line which passes through the point (x 0,y 0,f(x 0,y 0 )) of your surface by keeping y=y 0 in the formula of function, it means that we are going to calculate the partial derivative of the function f(x,y) with respect to x. And the intersection of plane y=y 0 and the surface gives us a curve on the surface passing Prepared By: Irfan Khan Page 25

26 through (x 0,y 0,f(x 0,y 0 ) and partial derivative represent the slope of the tangent line along this curve and passing through this point. In the above figure we show both tangent lines. What are RULES FOR CONTINUOUS FUNCTIONS? (a) If g and h are continuous functions of one variable, then f(x, y) = g(x)h(y) is a continuous function of x and y (b) If g is a continuous function of one variable and h is a continuous function of two variables, then their composition f(x, y) = g(h(x,y)) is a continuous function of x and y. A composition of continuous functions is continuous. A sum, difference, or product of continuous functions is continuous. A quotient of continuous function is continuous, except where the denominator is zero. Q48. Can we take square root of 0? Ans: Since we can write 0 2 =0 hence there is no restriction of writing square root of 0. But as you know that when we take the square root of a positive number then we write ± with the answer. And as you know that 0 is neither positive nor negative number so we will write 0 = 0. Also remember that if you have a function in which you have square root with some variable as input in the denominator then you can't use here 0 as input because it will create 1/0 in the defining formula of the function. For example we have a function f( x) = 1 x then we will not take x=0 as input because it will create 1/0 form. Q49. What is the use of Limits in our daily life? Ans: You can find out numerous applications of derivatives and integrals in our real life and definition of derivative as well as integral of a function uses the limits. Prepared By: Irfan Khan Page 26

27 For example suppose that a particle is moving with some velocity in space then you can defined the motion of the particle by a function of two variable and if you want to find out the instantaneous velocity of the particle then you need the concept of the limit. In order to find out the rate of decay of a radioactive element we use derivatives and hence concept of limit is in action. As well as when we use integrals to find out the area of a certain region then again limits are in use. Q50. Why we always discuss domains of functions and give no importance to the range relatively? Ans: Range of the function is dependent on the domain of the function. You know the definition of the range. Since in Calculus we will consider only real valued functions. And the range of a real valued function is the subset of real numbers and the real numbers in this set are obtained by putting the values of the domain values into the defining formula of the function. It means that if you find out the domain of a function then you can easily find out its range.that`s why we did examples in which you are given the formula of the function and we find out the domain of a function and stress over it. Q51. Please tell me about different table calculation which are used in lecture #5 for the finding out the limit of function at a point. Ans: You have the concept of Limit of a function of one variable. Suppose that f(x) is a function of one variable and you want to find out whether the limit at x=x 0 exists or not. Then you approach towards that point from both side and in the case of one variable you can approach a point on the real line from only two ways that is from its left and right sides. Then you check the functional values (the values of f(x))if these values approach towards a fixed number either you approach towards x=x 0 from either side then you say that the limit of the function exists at x=x 0.And in the language of mathematics we say that the function must have left and right limit and these limits must be same. As shown in the figure below in the formal language using and δ here we are approaching towards the point a. Now come to the case of more then one variable, say you want to see the limit of the function of two variables namely f(x,y) at some point (x 0,y 0 ),now as your domain is some subset of plane and you can approach towards this point through any curve which is in Prepared By: Irfan Khan Page 27

28 the plane and passes through the point (x 0,y 0 ) so you will check that if we approach towards that point along any curve then if the values of the function approaches towards the same point in the range then we will say that the limit of the function exists. There is formal definition of the limit of a function of more then two variables which we skip in the lecture and we think that you don't need this concept. We use the table just to show you the approaching behavior of the limits. And you can note if the limit of the function exists then if we approach towards the point through any path then the functional value approaches towards the same number. And these are the just calculations in the example we have a function and move along different paths and reach towards the point at which we are trying to find out the limit. Each time we have a point and put this value of the point in the given function and get the value of the function at that point and we note that if by any path we approach towards a point in the domain of the function (the at which we want to to see whether the limit exist or not) and if the functional values approaches towards the same value then we will say that the function has a limit. If you remember the way how we find the limit of a function in one variable then there will be no problem to you in under standing the concept of limit in 3d. In function of one variable we have only two directions left or right to the real number on which you want to find out the limit but in the case of 3d we have infinite curves which passes through the point where we are trying to find out the limit so we can approach the point along different curves. Some curves are shown in the figure below. y ( x 0, y 0 ) ( xy, ) And check the functional value along different curves. x Q53. Which one is correct saying about the function sin 1 (x) sin inverse (x) or inverse sin(x)? Prepared By: Irfan Khan Page 28

29 Ans: I will explain this with an example i hope that you know the Hyperbolic Trigonometric functions. For example sinh(x), when we say it we simply said that sine hyperbolic of "x".but this is not right and the correct is Hyperbolic of sinx. But you can note in your real life that every one use the same wrong terminology. Similarly in the case of inverse functions of trigonometry, its just a trend of saying the inverse trigonometric functions. But definitely "inverse sin "is the correct saying of the inverse trigonometric function of sine. Q54. Why we say partial derivative when we take the derivative of a function of two or more variable with respect to one variable? Ans: As you know that if we have a function of more then one variable then the graph of that function will be a surface in 3d space. Now as you have two independent variables so you can get the change in the function by varying "x" as well as "y".(if your function has independent variables "x" and "y"). So if we fix a variable then the function of two variables will become your function of one variable and geometrically this represent a curve on the surface of the function of two variables. Now you have a curve and you can define the tangent line on this curve. Since we keep a variable constant so we say that this is the partial derivative of the function. What is relation of Limit and Value? A limit can be defined in simple words as it is a certain value to which a function approaches. Finding a limit means finding what value of the function y =f(x) as x approaches a certain number. You would typical say that the limit of a certain function is a number to which the function approaches as x approaches to a certain number. e.g y=x^2 is a certain function the limit as x approaches to 2, f(x) approaches to the value 4. What are Ordinary derivatives? We take partial derivative of function where more than one independent variables are involved. In other words, partial derivatives are taken for functions of several variables. Now, to answer that what is a partial derivative, I would say that partial derivative of function of several variables is just a derivative with respect to one of those variables involved in the function, while other variables are taken as constants. Lets consider an example for a better understanding of above description. Example Prepared By: Irfan Khan Page 29

30 Consider a function f(x,y) = 2x3y2+2y+4x. This is a function of two variables x and y. We can take partial derivative of this function with respect to both of the variables x and y in a way that while taking derivative with respect to x, we will consider y as a constant and vice versa. Lets first take partial derivative w. r. t x. fx(x,y) = 6x2y2+4. Here, you see that while taking derivative w. r.t x, we have treated the variable y as a constant. Similarly, we can have fy(x,y) = 4x3y+2. This is how we can find partial derivative of a function of several variables. Leture 7,8,9,10 Q55. What is the Implementation of Log in our real life? On which bases scientists have made these log tables? Ans: Suppose that you sit in your home, the building suddenly begins to shake. You are experiencing an earthquake. When it ends you notice that the ground has opened to expose a cave where the skeleton of an ancient individual is found. What mathematical tool can be used to describe the intensity of the earthquake as well as the age of the body? We can answer such type of questions using Logarithms and Exponential functions. A logarithm is an exponent used in mathematical calculations to depict the perceived levels of variable quantities such as visible light energy, electromagnetic field strength, and sound intensity. And we can define the Log as x = log b y <---> b x = y. The log b y and the exponent function b y are inverse of each other. There are certain real life problems which when converted into mathematical model they involve Exponential as well as Logarithm functions. For example if you want to find out the decay of a radioactive element then you have to find out the solution of an equation which involve exponential function, I mean equation involving "e"(to some power) and you know that the function s" and "ln" are inverse of each other. That s why you a Scientist need the tables of Logarithms. I give you one example but there are many real life problems in which Logarithms used. CONTINUOUS EVERYWHERE? A function f that is continuous at each point of a region R in 2-dimensional space or 3-dimensional space is said to be continuous on R. A function that is continuous at every point in 2-dimensional space or 3-dimensional space is called continuous everywhere or simply continuous. Prepared By: Irfan Khan Page 30

31 Q56. Define partial Derivative and also describe its applications. Ans: There is a formal definition of the partial derivatives which and we define in the lecture. The partial derivative of a function z = f(x, y) with respect to x exists and is z f( x+ xy, ) f( xy, ) = lim x 0 equal to x x if the limit exists. Similarly partial derivative of z f( xy, + y) f( xy, ) = lim x 0 function z = f(x, y) is defined as y y. How to find out the partial derivative of a function? Well, suppose you are given a function of two or more variable. Then you can find out the partial derivative of this function with respect to one of its variables and keeping the all other variable constant. For example consider the function f(x,y,z)=x + siny + cosz (note that your function has three variables) And if you want to find out the partial derivative with respect to "x" say then you will simply take the derivative of the function "f" and will treat "y" and "z" as constant. So the partial derivative of the function "f" with respect to "x" is "1"for this function. Now if you want to find out the partial derivative of the same function with respect to "y" then it will be "cosy" and partial derivative with respect to "z" will be "-sinz". And if we slightly change the function an consider the function g(x,y,z) =x y + x siny + x y cosz Then the partial derivative with respect to "x" becomes "sy+siny+ycosz".similarly you can find out any partial derivative with respect to a variable by keeping the other two variables constant. In real life you face many problems which involve the rate of change of quantity. For example when you are moving on the earth and you are not moving with constant speed and want to find out the rate of change of your speed at a particular time, then you need derivatives. In real life you have so many examples in which you talk about the rate of change with respect to time (say). These problems can be translated into Mathematical model and then using partial derivatives you can get the required rate of change. In simple words where you have to find out the "Change in some quantity with respect to the other" that is rate of change of some quantity, you can use there Partial derivatives. In your course "Differential equations you will see applications of Partial derivatives. Partial Derivative: In mathematics, a partial derivative of a function of several variables is its derivation with respect to one of those variables, with the others held constant (as opposed to the total derivation, in which all variables are allowed to vary). Partial derivatives are used in vector calculus and differential geometry. The partial derivative of a function f with respect to the variable x is variously denoted by The partial-derivative symbol is Prepared By: Irfan Khan Page 31

32 Continuous functions of two variables satisfy all of the usual properties familiar from single variable calculus: The sum of a finite number of continuous functions is a continuous function. The product of a finite number of continuous functions is a continuous function. The quotient of two continuous functions is a continuous function wherever the denominator is non-zero. Q57. We know that first derivative of a function gives slope of tangent line, what the second and third derivative gives? Ans: As you said that first derivative at some point say(x 0, y 0 ) gives the slope of the tangent line at that point. And remember that the y 0 =f(x 0 ) and the point will lie on the graph of f(x). Now you want to know about the second derivative, you can treat the first derivative of the function as another function say "g=d/dx(f(x))" then the first derivative of "g" will be the second derivative of the function "f". Now as you know that the first derivative of the function is the slope of the tangent line to the curve, so first derivative of "g" represent the slope of the tangent line at (x 0, y 0 ).But "g" is actually first derivative of "f" It means that second derivative of the function represent the slope of the tangent line to the first derivative at (x 0, y 0 ).Similarly the third derivative at some point (x 0, y 0 ) is the slope of the tangent line to the second derivative. As you have learnt in your first course of calculus that the first derivative also tells us about the graph of the function over a certain interval if your first derivative is always positive then your function is increasing over that interval and if the first derivative if negative over a certain interval then the graph of then function is decreasing over that interval. Also you get the critical points by solving f / (x) =0 Your second derivative geometrically tells us about the graph of the function that whether it is concave up or concave down. Where you have learnt these terms concave up and down in your first course of calculus. Concave up and down graphs are shown in figure below. Prepared By: Irfan Khan Page 32

33 Q58. How we can define Calculus? What is its purpose and in which field we can apply it. Ans: Calculus is the study of continuous rates of change. Or in simple wording we can say that in calculus we study how the quantities changes with respect to the other quantities. For example if you want to find out the rate of change of temperature with time. You may want to find out rate of increase in pollution. when you are moving, then you may be interested in to know how my speed is changing with respect to time. That is you want to know about the change in one quantity as the other changes. It has applications in many fields as described above. Cartesian coordinates (or rectangular coordinates) use an origin, O, a horizontal line (x axis) and a vertical line (y axis) to give a frame of reference. Any point can then be found by its (x, y) coordinate. Polar coordinates use an origin, O, and a horizontal line serving as the reference line from which to measure angles of rotation,, in an anti-clockwise manner. Any point can then be found by its (r, theta) coordinate. Suppose p(r, theta) is the point in polar coordinates, then OP=r and theta is the angle made by OP with +ve x-axis in anti-clockwise direction. Q59. Is the ideas and logic of both Calculus courses are same? Ans: We extend almost all the ideas of the one variable to the more then one variable and have different terms for these concepts also we will study some new concepts which are related to the functions of more then one variable. Q60. What is the difference between derivative and partial derivative? Prepared By: Irfan Khan Page 33

34 Ans: Actually there is no difference between these two terms; the concept behind them is same. We can say that these are the same terminologies used for different functions, depending on the number of independent variables of your function. We say the derivative of a function when our function is of single variable, and we use the term Partial derivative when our function is of more then one variable. And also remember that in both case we get the slope of the tangent line passes through the given point. Q61. Explain Euler Theorem and Laplace equation. Ans: Suppose that you have a function of two variable say f(x,y) then you can find its partial derivative with respect to any of its variable, suppose that you find the partial derivative w.r.t "x" then it will be again a function of two variables and you can again find out the partial derivative of this function (which is actually partial derivative of f(x,y) with respect to both variable. Now suppose that you take the partial derivative w.r.t "y" to that function which is the partial derivative w.r.t "x" of f(x,y). Then you find the mixed derivative in which "first we take partial derivative of f(x,y) w.r.t "x" the "y". You can also find out the mixed derivative by first taking the partial derivative of f(x,y) w.r.t "y" and then by taking the partial derivative of this function w.r.t "x". In general these two partial derivatives are not equal and Euler s Theorem tells us under what conditions these two different mixed derivatives are equal. In the end of our lecture we define Laplace equation, and which is very simple, if you have a function of three variable then its Laplace equation can be found by taking the double partial derivatives w.r.t to each of the variable and then by taking sum of these derivatives. If the sum is 0 then we will say that the function is a solution of the Laplace equation otherwise function is not a solution of the Laplace equation. Laplace equation physically formed in the potential Theory. Q62. What is the purpose of Laplace equation? Ans: Laplace equation is very much important and formed in potential theory and also interprets the heat flow from a rod. So its solution gives us the physical interpretation of the experiments. You can find its little application in Advanced Engineering Mathematics By Kreyszig. Q63. What is slope of the tangent? Ans: Since you know that slope of any line is the Tangent of the inclination of that line. Where as Inclination of a line is the angle which that line makes with the positive x-axis. In particular slope of the tangent line is the defined as the TAN of the angle which Tangent line makes with the x-axis.(which is also known as the inclination of the line).so slope of the line is simply Tan(inclination of tangent).and you also know that if we find out the derivative of a function and put particular values of a point which lie on the graph of that function whose derivative we have taken, then it gives us slope of the tangent line at that point. Prepared By: Irfan Khan Page 34

35 Q64. What are the Advantages of Euler s theorems? Ans: Advantages of Euler s Theorem are very evident from its statement, if you have a function then Euler s Theorem tells us under what conditions the mixed derivatives are equal for that functions and you can find the conditions under which the function has this property at LMS. Also note that in general this is not necessary that the mixed derivatives of a function are equal. Q65. How we can find out the distance from point (-5, 2, 3) to xy-plane and x-axis? Also How we can find out equation of sphere with center (-1, 3, 2) and passing through origin? Ans: First of all you should know that if we have a point P(x, y, z) Then x,y are the distances from origin to the foot of perpendicular on these axis of the projection of point P where the projection is in the xy-plane and z is the distance of the perpendicular from the point on the z-axis. As shown in the figure below. Thus if you are given a point P(-5,2,3) say and you are asked to find out the distance of that point from any planes formed by the coordinate axes, then the shortest distance between that plane and the point is the absolute value of coordinate of the point which is other then the axes by which our plane is formed. It means that if you want to find out the distance between xy-plane and the point P(-5,2,3) then it will be 3.Similarly the distance of the point P(-5,2,3)to yz-plane is 5. And if you are given a particular point which lie in the plane and asked to find out the distance between the given point and that point. In such a case you can find out the distance between two points by distance formula. Now any point on the x-axis will has coordinate (x, 0, 0) and the distance between (-5,2,3) and the point (x,0,0) can be obtained by using distance formula. In the end you know that equation of the sphere having center (a, b, c) and radius" is (x-a) 2 +(y-b) 2 +(z-c) 2 =r 2. Now you are given (-1, 3, 2) as center of the sphere, but you are not given radius but you are given another condition that the sphere is passing through (0,0,0). So origin will lie on the sphere and distance between center and origin will be equal to the radius of the sphere. Now you have center as well as radius so you can find out the equation of sphere. Prepared By: Irfan Khan Page 35

36 Q66. Can we find out the partial derivative of a function with more then three variable? Ans: Of course you can find out the partial derivatives to functions of more than three variables. It is not restricted to the functions of two or three variables. Example: Suppose that we have a function f(x 1, x 2, x 3, x 4, x n ) of n variables defined by f(x 1, x 2, x 3, x 4, x n ) = x x x x 2 n Then we can take partial derivative of that function with respect to any one of its variable. Suppose that we want to find out f f = 2x i xi where xi is a variable and 1 i n.then xi is the required partial derivative. Q67. What is meant by intermediate variables? Ans: Intermediate variables are such variables which are used to define a function and these variables themselves are the functions of some other variable, we can say that when the actual variable is hidden, then the apparent variable is called an "intermediate variable". As in our examples the actual variable is hidden and actually our variable is "t" in the first example of our lecture # 9. Similarly in all the other examples you can find out the intermediate variables easily. Q68. What is difference among x, dx, x, and δx? How we can find out the slope of the tangent line at a given point? Ans: The first symbol which we use in x which you use with "x" we use it to show the small change in the variable "x". Where as dx is used when we are talking about the differentials. Where as the next term is called "partial", used to represent partial derivative. And usually we use partial when your function is of more then one variable. Where as the last symbol is one of the difference operator used to approximate the partial derivative. When we are taking the derivative of a function at some particular point then it represents the slope of the tangent at that point. And it finds out the rate of change of a dependent variable with respect to the independent variable. For example if you have a function f(x)=sinx then its derivative is cosx represent the change in the function f(x) with respect to "x' and suppose that you have the point (0,0) on the curve, then the slope of the tangent line on the graph of the function f(x) at this point is "1" and obtained by simply putting the value in the derivative of the function. There is proper way to find out the differentials which you didn t study yet. Prepared By: Irfan Khan Page 36

37 Q69. I want to ask that let a function f(x, y)= xy+ x, can we plot its graph in a plane? Also how we can find out the rate of change of a function with respect to x? Ans: As I told you many times that the graph of the function of two variables will be a surface in the 3d and you can t draw a 3d graph on the plane. Because you need three perpendicular lines for this, two lines for the variables and the third line for the value of function. But some packages and software are available which draw the 3d graph and all the 3d graphs on your book are generated by computer. And the graph of the function f(x, y)= xy+ x where the 0 x 10 and 0 y 100 is shown above. Secondly it is very simple to find out the rate of change of a function in "x" only.in order to do this you will find out the Partial derivative of this function with respect to "x". Q70. Tell us online links where we get more and more material to study. Ans: We are following the Calculus by Haward Anton which we also follow during your first course of calculus. For more material please visit the links These links will be helpful to you in understanding the concepts for both calculus courses. Also you can find more Links and solution files of Home Assignments at Internet links on LMS. Q71. Tell me the definition of Mathematics? Which branch of Math deals with Calculus? Ans: Dear student there is no proper definition of the Mathematics, some people say that it is the subject of Nature, some say that it is the subject of reasoning and shapes of the objects, some says that it is the subject of definitions and Theorms.For more discussion on the definition of Mathematics please visit the link Now Calculus is a sub-branch of a branch of mathematics known as "Mathematical Analysis" in calculus we study the rate of change of one quantity with respect to the Prepared By: Irfan Khan Page 37

38 change in other quantity. Or we can say that it is the subject in which we study the continuous rates. For example some one may be interested in knowing how water flow changes w.r.t time etc. You want to know how this happens continuously etc. Q72. What is difference between partial derivatives and implicit differentiation? Ans: I think that you want to know the difference between the partial derivative and implicit differentiation. We take the partial derivative of a function of two or more dy variables and by implicit differentiation we find ( dx that is derivative) of a given equation if we can t express one variable "y" as a function of the other variable "x". Also note that in this case the equation does not represent a function. Where as we find out the partial derivative of a function of two or more variables. Q73. Why partial derivatives are used? Is their any method which can replace derivatives? Ans: The derivatives can t replace, but partial derivatives as well as derivatives of one variable can be approximated. And we use forward, backward and central differences to approximate these derivatives. These approximations are used in the numerical solutions of partial differential equations and in many other branches of mathematics. But you will perhaps go through these approximations of derivatives and partial derivatives in your course of Numerical Analysis. Q74. Give examples of vectors from real life. Ans: There are so many examples in real life for which we use vectors because by using vectors we can solve the real life problems in which direction matters. For example you can talk about the angular velocity, power, weight of an object, voltage etc these are all vector quantities and problems involving these terms are solved by considering vectors correspond to these quantities. Example from handouts: w = ( 4 x 3y + 2z) 5 First partially differentiate w.r.t x by using the power rule =5.( 4 x 3y + 2z) 4 (4-0+0) =20 ( 4 x 3y + 2z) 4 now again partially differentiate the function 20 ( 4 x 3y + 2z) 4 w.r.t "y" by using the power rule =20.4 ( 4 x 3y + 2z) 4 (0-3+0) Prepared By: Irfan Khan Page 38

39 = ( 4 x 3y + 2z) 3 now again partially differentiate the function ( 4 x 3y + 2z) 3 w.r.t "z" by using the power rule = ( 4 x 3y + 2z) 2.(0+0+2) = ( 4 x 3y + 2z) 2 Q75. What are the rectangular components of a vector? Ans: As you can find out the resultant of a number of vectors similarly you can find out a number of components of a given vector. Whose resultant is obviously is the given vector. Now among these components of a vector rectangular components are those components which are perpendicular to each other, and we have only two components of a vector which are perpendicular to each other. Q76. What are orthogonal coordinates? Ans: A system of curvilinear coordinates in which each family of surfaces intersects the others at right angles is known as the orthogonal coordinate system. A curvilinear coordinate system composed of intersecting surfaces. If the intersections are all at right angles, then the curvilinear coordinates are said to form an orthogonal coordinate system. If not, they form a skew coordinate system. Q77. What is difference in dot product and cross product of two vectors? Ans: If you have two vector quantities then you can define their product. Now what are the possibilities for the answer of that product, either the product will be a Scalar or a vector. Corresponding to these two possibilities we have two types of the product of two vectors. One is known as Dot Product when the answer of the product is a scalar. On the other hand we name the product as Cross product if the answer of product of two vectors is a vector. I hope that now the difference between these two products will be clear to you. And the difference from the above discussion is that" in Dot product your answer is scalar and in the cross product your answer is a vector. For example in real life when you take the dot product of "Force" with "Displacement" then you got the work done by a body which is scalar quantity. On the other hand if we take the cross product of "Current" and "voltage" then you will get a vector quantity as their cross product which is "Power" and you study this in your phy301 course. Q78. What are Null vectors? Prepared By: Irfan Khan Page 39

40 Ans: The vectors which have zero magnitude are known as Null Vectors or zero vectors. Q79. What is law of polygonal? Ans: If you have a number of vectors acting on a body and you want to find out the resultant effect of these vectors. Then you want to find out the resultant of all the vectors acting on it and for the resultant of these vectors you will use head to tail method which is also known as law of polygon. In which you draw a vector at origin and then draw the other vector such that its tail is at the head of the first, similarly you draw all the vectors. Finally in order to get the resultant of these vectors you join the tail of first vector to the head of the last vector. This vector represent the resultant of your all the vectors. And you should note that after we draw the resultant we get a closed figure which is called polygon that s why we call this method as polygon method for getting the resultant.(remember that a polygon is a closed figure with a number "n"of sides) Q80. What do we mean by the projection of a vector on the other? Ans: Suppose that you have two vectors namely "a" and "b" and you draw them so that they start from the same point and if you are not given the vectors in this form then note that you can redraw the given vectors such that they start from the same point say "O". Now you can draw two rectangular components of any vector with respect to the other, by dropping a perpendicular from the head of one vector ("a" say) to the other ("b"). One component of the vector "a" will be along the other vector which is "b" and one component will be perpendicular to that component. Now the component of the vector "a" along the vector "b" is known as the projection of the first vector "a" along the vector "b". And if we are given the angle between these two vectors say (θ )then you can calculate this projection by using right angle triangle and the projection of "a "on "b" will be " acos(θ ).As shown in figure below Explain the formula for finding projection of A on B and projection of B on A? Prepared By: Irfan Khan Page 40

41 Suppose that you have two vectors namely "a" and "b" and you draw them so that they start from the same point and if you are not given the vectors in this form then note that you can redraw the given vectors such that they start from the same point say "O". Now you can draw two rectangular components of any vector with respect to the other, by dropping a perpendicular from the head of one vector ("a" say) to the other ("b"). One component of the vector "a" will be along the other vector which is "b" and one component will be perpendicular to that component. Now the component of the vector "a" along the vector "b" is known as the projection of the first vector "a" along the vector "b". And if we are given the angle between these two vectors say ( )then you can calculate this projection by using right angle triangle and the projection of "a "on "b" will be " What is meant by the function is defined at a given point? If function has valid output for any input then we say that function is defined at that point. e.g. f(x) = 1/(1-x) is not defined at x =1 because the function value is inifinity ( or not defined at 1/0). differentiable function and an example of function no differentiable? The derivative of the function f at the point a in its domain is given by f `(a) = lim h --> 0 [f(a+h) f(a)]/ h We say that function f is differentiable at the point a in its domain if f '(a) exists. f(x) = x^2 is a differentiable function.. f(x) = 1/x is not differentiable at x=0 Prepared By: Irfan Khan Page 41

42 Lecture no 11 to 15 Q81. How to find out the gradient of a scalar function? Can we define Gradient of a vector function? What gradient of a function tells us? Also give examples of gradient of a function. ϕ = i+ j+ k ϕ Ans: Gradient of a scalar function is defined as x y z = i+ j+ k where x y z is known as Del operator and is a vector and ϕ( xyz,, ) is a scalar function. Remember that a function is said to be scalar function if its range consist of scalar quantities and as I told you many times that during the study of Calculus we will consider real valued function that is we will consider the functions whose domain and range both are subset of real numbers, unless we mention the type of function. Example: xy Suppose that we have a scalar function ϕ ( xyz,, ) = e sin z then you can see that this function is scalar. Because correspond to each input we have a real number as out put. Thus we can find out the gradient of that function. And which is xy xy xy ( e sin z) ( e sin z) ( e sin z) xy xy sin sin xy ϕ = i + j + k = ye zi + xe z j + e cos zk x y z Note that the gradient of a scalar function is a vector quantity and it has certain properties which are list below. The gradient of a function at a point that is ϕ is the direction of maximum increase of the value of function at that point. The negative of gradient of a function at a point that is ϕ is the direction of minimum increase of the value of function at that point. The gradient can be easily generalized to apply to functions of three or more variables. The gradient of a scalar function is perpendicular to the level curve Since as you can see that the Del operator is a vector operator so if we have vector valued function then we can take the dot or cross product of that vector valued function with the Del operator and the resulting answer is known as the Divergence or Curl of the vector respectively. Q82. What is directional derivative and when we use it? Also give examples of it. Prepared By: Irfan Khan Page 42

43 2 Ans: Suppose that you have a function f ( x, y) = x + xy of two variables (You can also consider the function of more then one variable) then as you know that you can find out its partial derivatives with respect to a particular variable. Let us calculate the partial derivative of the above function with respect to x which is fx = 2x+ 1. Now as we know that unit vector along x-axis is u = i and f = (2 x+ y) i+ xj.in the end f. u = ((2 x+ y) i+ xj).( i) = 2x+ y we find which is the same as the partial derivative of the function with respect to x. And we can also say that this is the derivative in the direction of x-axis. If you are given a point on the surface and you are also given a unit vector. You are asked to find out the derivative of the function in the direction of the given unit vector at the point which is given on the surface of the function, this one is known as the Directional Derivative of the function in the direction of the given unit vector. We can say that Directional derivative is the Instantaneous rate of change of in f in the direction of given unit vector. As shown in the figure below. Directional Derivative of a function f in the direction of unit vector u is denoted by D f D f= fu. u and is obtained by u. Example: 2 Suppose that we have f ( x, y) = 2xy 3y and want to find out the directional derivative of that function in the direction of u = 4 i+ 3 j at the point P(5,5). First of all we note that the given vector u = 4 i+ 3 j is not a unit vector so we will 1 u = ( 4 i+ 3 j) convert it into unit vector and the unit vector is 5.Now Prepared By: Irfan Khan Page 43

44 f = 10 i 20 j (5,5) and hence the Directional Derivative in the direction of the given 1 D f= fu. = (10 i 20 j). ( 4 i + 3 j) = 4 u vector is 5. In general we can write the expression for the Directional Derivative of a function f (x, y) in the direction of the unit vector u = ui 1 + u2 j D f = fx u1+ fy u u P 2 0 P0 at the point P 0 is Q83. What is a Tangent plane at a given point of a surface? Ans: You have the concept of a tangent line at a point on the curve of a function of one variable; similarly you can talk about the tangent line at a point in the function of more then one variable. But note that in the case of two variables we have any point on the surface of the function has three coordinates like that P(x 0, y 0, z 0 ) and now there are infinite many curves passing through this point and note that these curves can be treated as the curves corresponding to the function of one variable, so we can draw a tangent line passing through that point P along that curve. Since we have infinite many curves passing through that point it means that we can find out the infinite tangent lines passing through the point P on the surface. Some of these tangent lines are shown in the figure below. Now any plane which contains all these tangent lines at a point is known as Tangent Plane to the surface at that point. Now how to find out the equation of tangent Plane? Well this I will explain with an example. Example: Suppose that we have to find out the equation of tangent plane to the surface x 2 + y 2 + z 2 = 25 at P( 3,0,4). We also know that the general equation of tangent x ( 0) ( 0) ( 0) 0 P y 0 P z P 0 0 Plane is f x x + f y y + f z z = f where z P 0 means the value of Prepared By: Irfan Khan Page 44

45 0 0 0 partial derivative at point P 0.So f = 6, f = 0, f = 8 x P y P z P hence required equation of tangent plane is 6( x+ 3) + 0( y 0) + 8( z 4) = 0 6 x+ 8 z = 50 3 x 8 z+ 25 = 0. Also note that the formula of the equation of tangent plane given in your book is a special case of the above formula which I write and in book your formula is for the function z=f (x, y) which can be written as f (x, y)-z = 0 and hence f z = -1 and by putting this value of partial derivative with respect to z you can see that above formula becomes f ( x x 0) + f ( y y x y 0) ( z z 0) = 0 P0 P0 which is same as in your book. So you should not be confused by the formula. Q84. What is difference between Tangent Planes and Normal Lines to Surfaces? Ans: We define normal line with respect to tangent line as in the case of functions of one variable. You have a tangent line passing through a point and any line perpendicular to that tangent line is known as normal line at that point. Now in the case of more then one variable as you know that we have more then one tangent lines passing through a point and in fact we have infinite tangent lines through a point in the case of more then one variable. And any plane which contains all these tangent lines is known as Tangent Plane at that point. Now you know that when we define a plane then we talk about a normal vector to that plane, since the tangent plane at a point is also plane and we will have a line which is perpendicular to that plane and that line is known as the normal line to that plane at the given point. As shown in the figure below. Q85. What are the conditions for two Surfaces to be orthogonal? What is the role of orthogonal Surface? Prepared By: Irfan Khan Page 45

46 Ans: Two surfaces are said to be orthogonal at the point of their intersection, if their normal vectors at that point are perpendicular to each other. And two surfaces are said to be orthogonal if they are orthogonal at each point where they intersect. What is the condition for two Surfaces to be orthogonal at a point? Well we say that if the normal lines are perpendicular then we will say that surfaces are orthogonal at the point of intersection, also we know that if the two lines are perpendicular to each other then the sum of product of the direction cosines or direction ratios of these lines will be zero. So if we have (g x, g y, g z ) be the direction ratios of normal line to one surface and (f x, f y, f z ) are the direction ratios of normal line to the other surface then by the condition of perpendicular lines we must have the sum of the products of these directions ratios fg x x + fg y y + fg z z = 0 equal to zero which is which is the required condition for the two surfaces to be orthogonal at the point of intersection. Q86. What is difference between dx and x? Why dx = x and this not true in the case of variable y? Ans: The term x represent the actual change or increment in variable x where as dx is the differential that is change in the variable x calculated along the tangent line or we can say that approximated change. There is no difference between these two terms if we are talking about the variable "x" which is independent variable in the function y = f(x) or in general if you are talking about independent variable, then there is no difference between these two terms. And differential and increment are not same if we are talking about the dependent variable. As you can see from the figure below From the above figure it is clear that dx = x in the case of "x". But you can also note from the above figure that this is note the case when we are talking about "y". That is increment of "y" is not the same as the differential of "y". Where you know that increment in "y" is denoted by " y " and differential is denoted by "dy". Prepared By: Irfan Khan Page 46

47 Q87. Explain the concept of orthogonal vectors. Ans: Two vectors are said to be orthogonal if the angle between them is 90. And you know that Dot or scalar product of two vectors a and b is given by ab. = a bcosθ where θ is the angle between the vectors. Now if the vectors are orthogonal then θ = 90 and as we know that cos(90 ) = 0 So finally we have if the two vectors are orthogonal then ab=. 0 (That is their dot product is zero). So if the dot product of two non zero vectors is zero then we will say that the vectors are orthogonal. Q88. What is difference between normal of a curve and Gradient of that function? When we write x = x 0 + at and what are x 0, x, a and t? Ans: We define the normal line of a surface at a point and that normal line is perpendicular to the tangent plane at that point. That is that line is perpendicular to all the tangent lines at that point and also you know that all these tangent lines are in the tangent plane. And this normal line varies from point to point. It means that if you take a point other then the first point then the normal line of first point may not be a normal line at the second point. Where as Gradient of scalar function is a vector which is normal to that surface at all the points. Secondly I think you are talking about the parameter "t" which we involve in the parametric representation of a line. Note that x = x 0 + at represent the x coordinate of any point on the line. Also for a particular point we will have a particular value of the parameter "t" finally a is the x-coordinate of direction ratio of the line. Q89. What is difference between Absolute maxima, minima and relative maxima, minima? Ans: When you have a function which is not a constant function then it will has different values at different points in its domain and we can compare these values of the function. Now the point in the domain of function for which the function has maximum value is known as Absolute maxima. Similarly the point in the domain of the function for which the function has minimum value is known as Absolute minima of the function. Relative maxima or minima are the maximum or minimum values corresponding to a small neighborhood around that point in the domain which has maximum or minimum value in that neighborhood. (Here neighborhood means small disk about a point). Where as Absolute maxima or minima is the maximum or minimum value in the whole of the domain of the function. Now from this you can easily conclude that every Absolute maxima or minima are also relative maxima or minima, but the converse may not true. That is it is not necessary that a relative maxima or minima is also an absolute maxima or minima. Prepared By: Irfan Khan Page 47

48 In the above figure both relative Extrema and absolute Extrema are explained. Note that these are the points in the domain of the function where function can have maximum or minimum values and these points of the domain are known as Extrema of the function and you can find out the maximum or minimum value ate these points simply by putting these points of the domain to the function. Q90. What do we mean by open disk? Explain the concept of relative Extrema using open disks. Ans: Well open disc is the set of all points in the plane whose distance from a fixed point is less then a particular positive real number (That particular real number is known as radius of the disc) and this is mathematically defined in the lecture. Q91. What is difference between Extrema of single variable and multivariable Extrema? Ans: There is no major difference between the extrema of functions of one variable and function of multi variable except that in the function of one variable we say a point as inflection point where the function change its shape or change its concavity (That is function changes from concave up to concave down).as sown in the figure below. Prepared By: Irfan Khan Page 48

49 In the above figure we analyze the graph of a function of one variable and the first line with + and sign represents the sign of first derivative of the function and the arrows below that line show where function is increasing and where it is decreasing. Also not that where the arrow is horizontal it represents that the function has first derivative zero at that point and these points are the critical points. And you can note that the point x=0 is the inflection point. In the case of multivariable functions we have saddle points correspond to inflection point in the case of one variable. And these are those critical points for which in the open disk around that point in the domain of the function there are points such that f( x0, y0) f( x, y) and f( x0, y0) f( x, y) satisfy. The figure shown below will explain you what is a saddle point. Prepared By: Irfan Khan Page 49

50 In the above figure point (0, 0) is the saddle point because if we take any open disk around that point we will have points in that disk whose out put lie on the blue as well black parabolas. Now if you it is quite clear that if we move along the black parabola then the point (0,0) is the minimum point and is we move along the blues parabola then (0,0) is the maximum point. Hence (0, 0) is the Saddle point for that surface. Where as you can note that in order to get the maximum or minimum values we put first derivative equal to zero and get the critical points in the case of one variable similarly in the case of multivariable function we put all the partial derivatives equal to zero and get the critical points. And then apply certain tests to conclude that the point is maximum or minimum. Q92. Is there any role of tangent or normal lines to finding the Maxima and minima? Ans: Off course yes tangent lines play an important role in finding out the maxima and minima of a function. You should note that at the Extreme points of the surfaces the tangent lines at that point must be parallel to their respective coordinate axis. It means that the tangent line must be parallel to the x and y axis if your function is of two variables. And mathematically you know that the partial derivative with respect to an independent variable represent the slope of the tangent line and the above condition shows that slope of these tangent lines must be zero. (Which will make sure to us that the tangent line are parallel to the axis). For example at the maximum point the partial derivative of the function with respect to "x" must be equal to the zero. (Which we get from the information that the tangent line at the maximum point must be parallel to the x-axis). Similarly the partial derivative with respect to "y" must also be zero. As shown in the figure below. In the above figure you can note that both tangent lines correspond to partial derivative with respect to x and y are parallel to the x and y axis respectively. And you can note that in order to find out the maxima or minima we put partial derivative equal to zero and then get the critical points after that we check further conditions. Prepared By: Irfan Khan Page 50

51 Q93. What are saddle points? How critical points can exist when a function is zero? Ans: If you have a function of two variables the as you know that its graph is a surface in 3d and its domain is a subset of 2d (As you have two independent variables). Suppose that we have a point on the surface it will be of the form (x 0,y 0,f(x 0,y 0 )) and this point will be correspond to the point (x 0,y 0 ) in the domain of function. Now you also know that we have more then one curve passing through the same point in a 3d surface. And if through the point (x 0,y 0,f(x 0,y 0 )) we have two curves such that in one curve the point (x 0,y 0,f(x 0,y 0 )) has maximum value and if we go along the other curve the (x 0,y 0,f(x 0,y 0 )) has minimum value. Then we will say that point (x 0,y 0 ) in the domain of the function is a saddle point. As shown in the figure below and I also explain Saddle point in the above question. Secondly you are saying that if the function is zero then how to find out the saddle point of that function. It means that your function is a constant function in its domain so we can t talk about the maximum or minimum value of that function because on whole domain function has the same value. Q94. What is the difference between critical points and points on which Extreme values occur? Ans: Critical points are those points of the domain at these may be extrema of the function. But it is not necessary that all critical points are either relative extrema or absolute extrema. That is we can say that critical points are the candidates for the relative extrema or Absolute extrema of a function. But the converse is not true that is a critical point may not be an extrema of the function. When Ii am saying that relative extrema then it means both relative maxima and relative minima. Also note that every absolute extrema is relative extrema but the converse is not true. Lecture no 16 to 22 Q95. How we can find out the Critical points of a function? Give examples. Prepared By: Irfan Khan Page 51

52 Ans: Well if you have a function of two variables say f(x, y) then to get the critical points for this function first we will find out the partial derivatives of that function. After finding these partial derivatives we will equate to zero both expressions of partial derivatives and so we will have two equations one correspond to the partial derivative with respect to "x" and the second equation correspond to the partial derivative with respect to "y". After that we will solve both of these equations and will get the values of x and y which are the coordinates of the points in the critical points. So you have the critical points now you have to check whether these are maxima or minima. Example: Consider the function f(x, y) = x 3-3x + xy 2 and we have to find out the critical points of this function. Well first of all we will find out the partial derivatives of that function and equate them to zero. So partial derivatives are 2 2 fx = 0 2x 3+ y = 0 f and y = 0 2xy = 0 x = 0or y = If x = 0 then putting this value in the equation 2x 3+ y = 0 we get y = ± 3.Hence critical points are (0, 3) (0, 3). 2 2 If y = 0 then putting this value in the equation 2x 3+ y = 0 we get y = ± 1.Hence critical points are (1, 0) ( 1, 0).Thus the total critical points are (0, 3),(0, 3), (1, 0) and ( 1, 0). Q96. What do you mean by boundary points? What is the purpose of boundary points? Ans: When you have a function of two variable it must have a domain which is a subset of plane being the subset of plane domain of the function must has some boundary lines and points on that lines are known boundary points. Now your Extrema of the functions may occur on the critical points or on the boundary points, that s why in order to find out the extreme values we check both critical points as well as boundary points. Example: 2 2 Consider the function f( xy, ) = 4 x y then as you know that its Natural domain is the disk x 2 + y 2 4. Now the boundary of the Domain in this case is the circle x 2 + y 2 = 4. As shown in the figure below boundary with black line. Prepared By: Irfan Khan Page 52

53 Q97. In example on page # 21 of Lecture # 16 we find critical point (1, 2). But this point not lies on the surface. How are we sure that it is critical point? Why we check the critical points on the line segment (0, 0) and (3, 0)? Ans: First of all you should know if we want to find out the Extreme values of a function then what steps we should follow. (i) Find all the critical points with in the domain of the function. (ii) Find all the points on the boundary of the domain of your function where Extreme values may occur. (iii) Test critical points for relative Extrema. (iv) For Absolute Extrema find the value of the function at critical points and then decide which point is Absolute maxima and minima. So our extreme values may occur at the critical points and also at the boundary points of domain of your function. Now you know that for the critical points we find the partial derivatives and equate them to zero and then we get the two equations from which we get the values of "x" and "y". These are the x and y coordinates of your critical point. Now you can see that we find the point (1, 2) by equating partial derivative equal to zero. And this critical point also lies in the domain of the function so we will check the value of the function at that point. Also we have the boundary points and will check the value of the function at these points. Q98. Is there any use of differential calculus in real life? Give some examples. Ans: Differential calculus has many Applications in real life. For example you are listening the Weapons of mass destruction now a day which consists of chemical and biological weapons and if some one use these weapons against other then we want to Prepared By: Irfan Khan Page 53

54 know at what rate it will destroy the human population or what is rate of affection of this on the Environment. Then you need differential calculus to solve this and get the approximation. Another very interesting example is to find out the": decay of radio active element" which also can be determined by using differential calculus. Also the applications of differential calculus can be found in any book of differential equations. Many fundamental problems in science, engineering, and other areas such as economics are described by differential equations. Other fields where differential calculus is applicable are population growth, mixing problems, mechanics and electrical circuits. Q99. What do you mean by the critical points in the interior of R where R is domain of the function? Ans: For the determination of critical points you solve the equations f x = 0and f y = 0 simultaneously while solving these equations this is possible for a function in which you restrict the domain of the function that the critical points which you find by equating the partial derivatives equal to zero may not be in the restricted region. Then such critical points will not be considered because these are out of our region and the critical points which lie inside the region are to be considered. As we did in our examples of Lecture # 17 first of all we solve the two equations and get the points correspond to the solution of these two equations then we check whether these points are with in the domain of the function or not. Q100. Why we use integrals? What is purpose of it? Ans: So far you define the derivative of a function in case of one variable and also define the partial derivative of a function of more then one variable, now it is natural to ask a question, Can we find out the original function from its derivative? Well in other words is there some reverse process of Derivatives which gives us the original function back. This reverse process is known as Integration. That s why we use integrals of certain functions. Geometrically integration is actually area under the curve in the case of one variable as shown in the figure below. The above shaded region is obtained by the integral b a f ( x) dx. Prepared By: Irfan Khan Page 54

55 Now suppose that you want to find out the area of a certain region then you can apply the technique of integration there. Also the integration for the functions of more then one variable gives us volume of the solid bounded by the surface and the domain of the integrand. And we use it when we have to find out the volume of the 3d object and we know the domain of the function of two variables. In the above figure if we want to find out the volume of the 3d region bounded by the light blue surface which is correspond to the function z = 4 x - y and the rectangle which is in the xy-plane, then we will use the double integral and this volume is equal to the value of double integral (4 x y) dxdy. Q101. What is the derivative of e x? What is its derivative when it is used with sinx or cosx? x Ans: e is very interesting function in Mathematics and is the only function whose x derivative is equal to the function. That is the derivative of e is e x. You should also know that the "e" is a number lies between 2 and 3 and there is a proper definition of this x 1 e= lim x+ x number and we define it as x in order to get the information about history of "e" x x Secondly you want to know about the derivative of e sin x or e cos x. You will use here product rule of differentiation which you know as u into v formula. And the x x x x derivative of e sin xis e sin x+ e cos xand the derivative of e cosx is x x e cos x e sin x Q) Prepared By: Irfan Khan Page 55