MATH 1902: Mathematics for the Physical Sciences I

Transcription

1 MATH 1902: Mathematics for the Physical Sciences I Dr Dana Mackey School of Mathematical Sciences Room A305 A Dana.Mackey@dit.ie Dana Mackey (DIT) MATH / 46

2 Module content/assessment Functions and limits (2 weeks) Differential and Integral Calculus (4 weeks) Sequences and Series (1 week) Linear Algebra (3 weeks) Introduction to Probability (2 weeks) Lecture notes: dmackey/lectures.html Assessment: End of module exam : 70% Test : 30% Date of test : Tuesday, 24th March (Week 8 = Week after Review Week) Dana Mackey (DIT) MATH / 46

3 Numbers The natural numbers (or the counting numbers) are 1,2,3,4,5,... The integers are... 3, 2, 1,0,1,2,3,... The rational numbers are those numbers that can be written as the ratio a/b of two integers (b 0). An easy way to visualize all these numbers and understand their properties is to represent them on the number line (an unbroken and endless straight line, with an origin and a positive unit of length) / The line consists of all real numbers, that is the rational numbers as well as other numbers, such as 2, π, etc which are called irrational. Dana Mackey (DIT) MATH / 46

4 Intervals If a and b are two points on the real number line, we call the set of all points that lie between a and b an interval. We have the following types of intervals: (a,b) = the open interval from a to b (this interval consists of all the points between a and b, without a and b); [a,b] = the closed interval from a to b (all points between a and b, including a and b); (a,b] = the half open interval from a to b (b is included while a is not included). [a,b) = the half open interval from a to b (a is included while b is not included). Dana Mackey (DIT) MATH / 46

5 Hence, An open interval is an interval which does not contain its endpoints A closed interval does contain both its endpoints. The differences between open and closed intervals can be quite significant, as the following example shows. Example Consider the closed interval [0, 1]. The smallest number in this interval is 0. What is the smallest number in (0,1)? Dana Mackey (DIT) MATH / 46

6 Infinite intervals The set of numbers which are greater than a given number a also forms a segment of the number line which, in interval notation, is written as (a, + ). + (infinity) is an abstract point which is supposed to be at the right end of the number line (and so has the property of being larger than any real number). Similarly, we invent a point,, which is at the left end of the number line and the interval consisting of points which are less than a is written (,a). Sometimes we represent the whole number line as the interval (,+ ). Dana Mackey (DIT) MATH / 46

7 Sets A set is a collection of objects. The objects are referred to as elements of the set. For example, the real numbers form a set which is usually denoted by R. There are two ways of specifying a set. The simplest way is to list all the elements of the set, in any order, between two braces { and }. For example A = {2,4,6,8,10} Z = {... 3, 2, 1,0,1,2,3,4,...} = the integers The second way to define a set is by specifying a property that all its elements must satisfy, for example A = {n : 2 n 10, n is even} = the even numbers between 2 and 10 Q = { a : a, b Z, b 0} = the rational numbers b Dana Mackey (DIT) MATH / 46

8 If a is an element of a set A we write a A (a belongs to A). Two sets A and B are equal if they contain the same elements. We have {1,5,4} = {4,5,1,1} since the order in which the elements are listed has no importance and it does not matter if an element appears more than once. A set M is called a subset of another set A if any element of M is also an element of A (so M is contained" in A). We write M A. Example {1,2,3} [1,6] [1,+ ) R The empty set is the set which contains no elements and is denoted by /0. Dana Mackey (DIT) MATH / 46

9 Operations with sets If A and B are two sets then we can form new sets as follows A B (A union B) = the set consisting of all the elements which belong to either A or B; A B (A intersection B) = the set consisting of the elements which belong to both A and B; A \ B (A minus B or A less B) = the set consisting of the elements which belong to A but do not belong to B. Example Let A = {1,2,3,4,5}, B = {3,6}. Find A B, A B, A \ B and B \ A. Dana Mackey (DIT) MATH / 46

10 Review of Functions A FUNCTION takes input and produces output A function assigns to each element of an input set (usually a number) a unique element of an output set (usually another number) Calculus is mostly concerned with how the output of a function changes when we vary the input Dana Mackey (DIT) MATH / 46

11 Examples of functions The temperature at a given location is a function of time The gravitational pull of a planet is a function of its mass The squaring function = the function which accepts any number as input and returns its square as output The reciprocal function = the function which accepts any number other than 0 as input and returns 1 divided by that number as output. The f(x) notation: x is the input, f is the name of the function, f(x) is the output. The squaring function: f(x) = x 2 The reciprocal function: f(x) = 1 x Dana Mackey (DIT) MATH / 46

12 Domain of a function Often, the formula for a function doesn t make sense when some numbers are input. We say that the function is not defined for those numbers. Example: The reciprocal function, f(x) = 1 x, is not defined for x = 0 (because we cannot divide by zero). Example: The square root function, g(x) = x, is not defined for x 0 ( we cannot take square roots of negative numbers) The domain of a function is the set of all values for which the function is defined (makes sense). Therefore we say that the domain of f is R \ {0} and the domain of g is [0,+ ) = R +. Dana Mackey (DIT) MATH / 46

13 Example Find the domain of f(x) = 1 3 x We cannot have square root of a negative number so we must have 3 x 0 so x 3 We cannot divide by zero so x 3 So, the domain consists of all numbers x such that x < 3, or (, 3) Dana Mackey (DIT) MATH / 46

14 Example A cat falls from a building 40m high. The height of the cat at time t (height is a function of time) is given by h(t) = t 2 How long does it take the cat to hit the ground? We want to know t for which h(t) = t 2 = 0 t 2 = = t = = seconds Dana Mackey (DIT) MATH / 46

15 Exercise Think about: How far does the cat fall in the first second? How far does the cat fall during the last second before it hits the ground? Does the cat fall half the height in half the time? Can you work out the average speed of the cat? Can you work out the average speed of the cat for the last second of the fall? Is it possible to work out the speed of the cat when it hits the ground? Dana Mackey (DIT) MATH / 46

16 Let s write d(t) = 4.9t 2. This represents the distance fallen at time t. The distance fallen in the first second is then equal to d(1) = 4.9 t 2 = 4.9m The cat hits the ground at t = 2.857s. The distance travelled during the last second is d(2.857) d(1.857) = = 23.10m To find the time required to fall to 20m (half the height), solve the equation 20 d(t) = 20 or 4.9t 2 = 20 so t 2 = 4.9 = 2.02s So the cat does not fall half the height in half the time! Dana Mackey (DIT) MATH / 46

17 The average speed of the cat over this journey is the distance travelled divided by time = 14.0m/s The average speed of the cat for the last second of its journey is d(2.857) d(1.857) = = 23.10m/s How about the average speed during the last tenth of a second? d(2.857) d(2.757) 0.1 = = 27.54m/s Dana Mackey (DIT) MATH / 46

18 Limits of functions Example Consider the function f(x) = x 1 3 x 1. The domain of this function is R \ {1}. The function is not defined at 1. What happens when x is close to 1? x f(x) x f(x) approaching 1 approaching 1.5 approaching 1 approaching 1.5 Dana Mackey (DIT) MATH / 46

19 As x gets closer to 1, either from the left or the right hand side, f(x) gets closer to 1.5. We say that THE LIMIT AS x APPROACHES 1 OF f(x) IS 1.5 and write either f(x) 2 as x 1 or lim f(x) = 1.5. x 1 Example Find lim x 3 (x 2 + 1) We can calculate: f(2.9) = 9.41, f(2.99) = , f(2.9999) = ,... So we guess that lim x 3 f(x) = 10. Note that this is exactly what we get if we substitute x = 3, i.e. f(3) = = 10. Dana Mackey (DIT) MATH / 46

20 However, we can t always do this! The function ( x 1)/( 3 x 1) is not defined at x = 1 so we cannot set x = 1. Example: Find lim x 2 x 2 4 x 2 If we substitute x = 2 we get = 0, which doesn t make sense as 0 the function is not defined for x = 2. However, we notice that we can simplify the function as follows f(x) = x 2 4 x 2 = (x 2)(x + 2) x 2 = x + 2 so if we want to find out what happens when x gets close to 2 we substitute x = 2 to get lim(x + 2) = 2+2 = 4. x 2 Dana Mackey (DIT) MATH / 46

21 Back to the cat example To find the speed of the cat as it hits the ground, we have to calculate the limit Factorize the numerator d(2.857) d(t) t 2 lim = lim t t t t (2.857 t)(2.857+t) lim 4.9 = lim 4.9(2.857+t) = 28 t t t The speed is 28 m/s. Dana Mackey (DIT) MATH / 46

22 A general rule that holds for finding limits of functions made up of polynomials or quotients of polynomials is the following: Z Given the question of finding lim x a f(x) check first if f(a) exists. (1) If it does then f(a) will be the desired limit. (2) If f(a) does not exist because f(a) = c 0 the limit does not exist. for some number c 0 then (3) If f(a) does not exist because f(a) = 0 0 then the limit may or may not exist, we have to examine f more closely. Dana Mackey (DIT) MATH / 46

23 Example State whether each of the following limits exist and evaluate those that do. x 3 8 lim x 2 x + 1, lim x 3 lim x 1 x 2 2 x 2 9, x 2 + x x + 1, x 2 5x + 6 lim x 2 x 2 3x + 2. Note The factorisation trick works fine for finding limits of quotients of polynomials. For other functions, say functions which involve square roots or trigonometric functions, other tricks are needed. Dana Mackey (DIT) MATH / 46

24 Limits involving square roots Given the expression a+b, we call the expression a b the CONJUGATE of a+b. For example, the conjugate of x is x The useful property is that ( a+b)( a b) = a b 2 contains no square root sign. Example x 5 2 Find lim. x 9 x 9 If we substitute 9 into the function we get = 0 0. So we don t know if the limit exists or not, and there is no quadratic to factorise. What do we do? The trick needed is to MULTIPLY ABOVE AND BELOW BY THE CONJUGATE OF THE LINE INVOLVING THE SYMBOL. In this case we multiply above and below by the conjugate of x 5 2, i.e. x Dana Mackey (DIT) MATH / 46

25 Example Say whether each of the following limits exist and evaluate those that do. x 1 1 lim, x 2 x 2 x + 4 lim, x x 4 lim x 0 2x 1+x 1 x. Dana Mackey (DIT) MATH / 46

26 Trigonometric limits Most trigonometric limits will use the following property sinx lim x 0 x x = 1. or lim x 0 sin x = 1. Justification: Consider the unit circle and a small angle x. (Recall x radians is the angle we get on the unit circle by walking a distance x around the circumference.) For x very close to zero, x and sinx will be quite similar, and hence sinx x will be close to one. (Draw a picture!) The smaller x gets, the closer sinx x gets to 1. If you re still not convinced, choose a number close to zero,.001 for example. Calculate sin(.001) = , and thus sin = which is very close to 1. Dana Mackey (DIT) MATH / 46

27 This limit will also appear disguised in other forms: sinh lim h 0 h sin = 1, lim 0 sin(kx) = 1, lim = 1. kx 0 kx Of course, for any non-zero number k, x 0 is the same thing as sin(kx) kx 0 and so the above limit also appears as lim = 1. x 0 kx Example Calculate each of the following trigonometric limits. sin(4x) lim, x 0 x tan(3x) lim, x 0 x sin(5x) sin(3x) lim, x 0 x.sin 2x sin 2 (3x) lim x 0 x sin(5x). Dana Mackey (DIT) MATH / 46

28 More limits Recall the function We found that f(x) = x 1 3 x 1 lim f(x) = 1.5 x 1 To show that this is true, use the identity a 3 b 3 = (a b)(a 2 + ab+b 2 ) If we let a = 3 x and b = 1 we get x 1 = ( 3 x 1)( 3 x x + 1) Dana Mackey (DIT) MATH / 46

29 Now multiply both below and above the line by the conjugate 3 x x + 1. f(x) = x 1 3 = ( x 1)( 3 x x + 1) x 1 ( 3 x 1)( 3 x x + 1) = ( x 1)( 3 x x + 1) x 1 Also multiply above and below by the conjugate of x 1 f(x) = ( x + 1)( x 1)( 3 x x + 1) ( x + 1)(x 1) The factor (x 1) cancels out and we see that = (x 1)( 3 x x + 1) ( x + 1)(x 1) lim x 1 f(x) = 3 2 Dana Mackey (DIT) MATH / 46

30 Limits at infinity We have lim x + 1 x = lim x 1 x = 0 x f(x) x f(x) approaching + approaching 0 approaching approaching 0 Dana Mackey (DIT) MATH / 46

31 Limits of rational functions A rational function is a quotient of two polynomials. Example Find the limit lim x + x + 1 2x 1 If we divide above and below by x we have lim x + x + 1 2x 1 = lim x x 2 1 x = 1 2 Dana Mackey (DIT) MATH / 46

32 Example Find the limit lim x + 2x 1 2x 2 + x 3 If we divide above and below by x 2 we have lim x + 2x 1 2x 2 + x 3 = lim x + 2 x 1 x x 3 x 2 = 0 2 = 0 Dana Mackey (DIT) MATH / 46

33 Example Find the limit 4x 3 + x 2 lim x + x 2 1 If we divide above and below by x 3 we have 4x 3 + x 2 lim x + x 2 1 = lim x x 1 x 1 x 3 = 4 0 = Dana Mackey (DIT) MATH / 46

34 The limit at infinity of the rational function P(x) Q(x) where P(x) and Q(x) are polynomials, is equal to If the degree of P is higher than the degree of Q, the limit is infinite (does not exist) If the degree of P is lower than the degree of Q, the limit is zero If the degree of P is equal to the degree of Q, the limit is equal to the coefficient of the highest power term in P divided by the coefficient of the highest power term in Q. Dana Mackey (DIT) MATH / 46

35 Differentiation Rates of Change Given a function we are often interested in how the values of the function f(x) are changing as we change x For example, does f(x) increase or decrease when we increase x? What is the rate of increase/decrease? This arises is real-world applications such as A car travels in a straight line. Let f(t) be its distance from the starting point after time t. Then the speed of the car is the rate of change of this position. Let p(t) be the price of a basket of goods at time t. The rate of change of this price is the inflation rate Dana Mackey (DIT) MATH / 46

36 Differentiation from First Principles The average rate of change of f between x 1 = a and x 2 = a+h is f(x 2 ) f(x 1 ) x 2 x 1 = f(a+h) f(a) (a+h) a = f(a+h) f(a) h We move to the instantaneous rate of change or derivative of f at x by taking the limit of this as x 2 approaches x 1, that is as h 0. Z The derivative of the function f at the point a is f (a) = lim h 0 f(a+h) f(a) h if the limit exists. Dana Mackey (DIT) MATH / 46

37 Geometric Description Graphically, the average rate of change of f is the slope of the secant line joining two points on the graph of f As we move one of these closer to the other, the secant line becomes a tangent line and the derivative is the slope of the tangent line Dana Mackey (DIT) MATH / 46

38 Motivating Example Recall the example of the cat falling from a building. The height of the cat at time t was h(t) = t 2 The average speed of the cat for the last second of its journey is h(2.857) h(1.857) = = To get the instantaneous speed of the cat when it hits the ground we need the limit of the average speed as t approaches the time of impact: h(2.857) h(t) lim = h (2.857) = 28 t t The negative sign indicates that as time increases, the height decreases. Dana Mackey (DIT) MATH / 46

39 Definition Let f be a function and let a a number in its domain. If the limit, f(a+h) f(a) lim h 0 h exists then we say f is differentiable at a. The value of the limit is then called the rate of change or derivative of f at a and is denoted by f (a), or by df dx (a). If f is differentiable at each point of its domain then we say it is a differentiable function. Example Let f(x) = x 2. Let a R. Is f differentiable at a? If so, what is the derivative? Dana Mackey (DIT) MATH / 46

40 Example Let f be the function f(x) = 2x + 5. Determine whether f is differentiable at a and calculate its derivative at a. Note The method of calculating a derivative by using this limit formula is called DIFFERENTIATION FROM FIRST PRINCIPLES. Later, we will introduce faster techniques for differentiating. Example Find f (a) from first principles where f(x) = 1 x and a R \{0}. Example Let f(x) = x 3. Find f (x) from first principles. Dana Mackey (DIT) MATH / 46

41 Rules for Differentiation Differentiating from first principles, while effective, is rather slow and tedious. However, there are rules which can be proved (from first principles) which allow us to differentiate more quickly. Example Let f(x) = x 3. Find f (x) from first principles. f(a+h) f(a) (a+h) 3 a 3 a 3 + 3a 2 h+3ah 2 + h 3 a 3 lim = lim = lim h 0 h h 0 h h 0 h h(3a 2 + 3ah+h 2 ) = lim h 0 h = lim 3a 2 + 3ah+h 2 = 3a 2. h 0 Thus the derivative function is f (x) = 3x 2. Dana Mackey (DIT) MATH / 46

42 Z Power of x rule d dx xa = ax a 1 So, if f(x) = x 93 then f (x) = 93x 92. Notice this rule works even when the power is negative or even when the power is not an integer. For example the derivative of f(x) = x can be found by writing f(x) = x = x 1 2 which gives f (x) = 1 2 x = 2 x 1 2 = 1 2 x. Using the f notation, this rule is (x a ) = ax a 1. Dana Mackey (DIT) MATH / 46

43 Z The Constant Multiple Rule d dx (mf(x)) = m d dx f(x) If f and g are differentiable functions, then Z Addition rule d dx (f(x)+g(x)) = d dx f(x)+ d dx g(x) or (f + g) = f + g Dana Mackey (DIT) MATH / 46

44 The rule for differentiating the product of two differentiable functions, f and g is: Z Product Rule: or ( ) ( ) d ( ) d d f(x)g(x) = dx dx f(x) g(x)+f(x) dx g(x) (fg) = f g + fg Example Differentiate the polynomial x x 4 + x 3 3x + 2. Example Differentiate the function f(x) = (x 2 + 1) x. Dana Mackey (DIT) MATH / 46

45 How do we differentiate the function f(x) g(x)? Z Quotient rule or d f(x) dx g(x) = g(x) dx d f(x) f(x) dx d g(x) g(x) 2 ( f g ) = gf fg g 2 Note that f g is differentiable only at the points x such that g(x) 0. Example Differentiate h(x) = x x Dana Mackey (DIT) MATH / 46

46 Derivatives of trigonometric functions The functions sin and cos can be differentiated from first principles: Z d sinx = cosx, dx d cosx = sinx. dx All other trigonometric functions can now be differentiated by using the basic rules of differentiation. Example Find the derivative of the function tanx. Dana Mackey (DIT) MATH / 46