KEMATH1 Calculus for Chemistry and Biochemistry Students Francis Joseph H Campeña, De La Salle University Manila February 9, 2015
Contents 1 Conic Sections 2 11 A review of the coordinate system 2 12 Conic Sections 4 121 Parabola 4 122 Ellipse 5 123 Circle 7 124 Hyperbola 8 13 * 9 2 Review of Functions and Their Graphs 10 21 Functions 10 22 Operations on Functions 11 23 Special Types of Functions 12 3 Limits 14 31 Limits 18 32 One Sided Limits 21 33 Limits at Infinity 23 1
Chapter 1 Conic Sections 11 A review of the coordinate system To identify points in the plane, we set up a coordinate system We start by setting up a pair of perpendicular lines The vertical line is called the y-axis, while the horizontal line is called the x-axis The point of intersection of these two axes is called the origin, and this point is assigned the pair of coordinates (0, 0) If P is any other point on the plane, then P is assigned a pair of coordinates (x, y) that will give its address or location in the plane The first coordinate x gives the directed distance of P from the y-axis, which means that x is positive when P is to the right of the y-axis, and x is negative if P lies to the left of the y-axis Similarly, the y coordinate of P gives its directed distance from the x-axis Figure 11: The quadrants of the cartesian coordinate system Second Quadrant First Quadrant Third Quadrant Fourth Quadrant 2
Given the two points P (x 1, y 1 ) and Q(x 2, y 2 ) then the distance of P to Q is given by d(p Q) = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 and the midpoint of the line segment with endpoints at P and Q is the point given by ( x1 + x 2 M, y ) 1 + y 2 2 2 Exercise 1 Do the following 1 Plot the points A and B and deterime the length of the segment AB and its midpoint a) A(1, 1), B(3, 7) b) A( 1, 0), B(3, 2) c) A(5, 5), B(0, 2) d) A(10, 10), B(10, 5) 2 Find the length of the medians of the triangle having vertices A( 4, 5), B(3, 7) and 1, 3 Hint: A median of a triangle is a line segment with endpoint at the midpoint of a side of a triangle and the other endpoint is at the vertex of the angle opposite that side 3 Prove that the triangle whose vertices are A( 6, 2), B(1, 5) and C(4, 5) is an isoscles triangle 4 Prove that the triangle whose vertices are A(1, 8), B( 3, 2) and C(4, 1) is a right triangle 3
12 Conic Sections The graphs of some special second degree equations in two variables are called conic sections and are illustrated below 121 Parabola The set of all points P (x, y) on the plane equidistant from a fixed point F and a fixed line l forms a continuous curve called a parabola The fixed point is called the focus while the line l is called the directrix of the parabola Example 1 The figure shows the graph of a parabola with vertex at (0, 0) and focus at (1, 1) focus (0,2) The following are the standard forms of a parabola (x h) 2 = 4p(y k) where the axis of symmetry is vertical and the parabola open upward if p > 0 or downward if p < 0 4
(y k) 2 = 4p(x h) where the axis of symmetry is vertical and the parabola open to the right if p > 0 or to the left if p < 0 In the above standard forms of a parabola, the vertex is at (h, k) and the distance between the vertex and the foci is given by p and the length of the focal chord is 4 p Exercise 2 Do the following 1 Find the focus, vertex, focal distance ( p ), the end points of the latus rectum and the equation of the directrix of the following parabolas a) x 2 = 24y b) (y 4) 2 = 16(x 3) c) x 2 + 20y + 4x 60 = 0 d) x 2 2y 12x + 25 = 0 2 Find the equation of the parabola that satisfies the given conditions a) vertex at (1, 1), opens upward with length of latus rectum equal to 8 b) focus at ( 2, 5) and vertex at (2, 5) c) end points of latus rectum at (4, 1) and 4, 5 d) axis of symmetry is horizontal and vertex at ( 1, 1) and passes through (2, 2) 122 Ellipse If F 1 and F 2 are two fixed points on the plane, the set of all points P (x, y) for which the sum P F 1 + P F 2 of the distances from P to each of the two fixed points is a constant forms a continuous closed curve called an ellipse Each fixed point is called a focus (plural: foci) of the ellipse The following are the standard forms of an ellipse (x h)2 (y k)2 + = 1, a > b where the center is at (h, k) and the orientation a 2 b 2 of the ellipse is horizontal, that is, the major axis is horizontal (y k)2 (x h)2 + = 1, a > b where the center is at (h, k) and the orientation a 2 b 2 of the ellipse is vertical, that is, the major axis is vertical The following are some properties of an ellipse The line passing through the two foci of an ellipse is called the principal axis of the ellipse The points of intersection of the ellipse and the principal axis are called the vertices (singular: vertex) of the ellipse 5
The line segment of the principal axis whose endpoints are the vertices is called the major axis of the ellipse The length of the major axis is denoted by 2a The midpoint of the major axis is called the center of the ellipse, usually denoted by C(h, k) The distance from the center to each of the two foci is denoted by c, while the distance from the center to each vertex is a The line segment passing through the center, perpendicular to the major axis and whose endpoints are on the ellipse is called the minor axis, and its length is denoted by 2b The relationship between the real numbers a, b, c is given by the equation a 2 = b 2 + c 2 Exercise 3 Do the following 1 For each of the following ellipses, find the coordinates of the vertices, foci, points of the minor axis, and center a) x2 9 + y2 4 = 1 (y 3)2 (x + 1)2 b) + 81 64 c) 16y 2 + 25x 2 = 400 = 1 d) 9x 2 + y 2 18x + 4y 23 = 0 e) 4x 2 + y 2 + 32x 6y + 69 = 0 f) 25x 2 + 16y 2 150x 128y + 381 = 0 2 Find the equation of the ellipse that satisfies the following conditions a) center at (0, 0), one focus at ( 12, 0) and one vertex at (13, 0) b) one focus at (3, 6) and vertices at (3, 8) and (3, 4) c) Foci at (1, 2) and ( 1, 2) and a = 2 d) one vertex at (5, 1), one focus at (4, 1) and center at (0, 1) e) one vertex at (10, 1) and endpoints of the minor axis at (3, 4) and (3, 6) f) center at (2, 3), a = 10, b = 8 and the major axis is parallel to the y-axis 6
123 Circle A circle is the set of all points P (x, y) in plane equidistant from a fixed point The fixed point is called the center of the circle, and the constant distance is called the radius The standard equation of a circle with center at (h, k) and radius r is given by: (x h) 2 + (y k) 2 = r 2 Example 2 The figure shows the graph of a circle centered at (1,1) and radius 2cm 2 cm Exercise 4 Do the following 1 Find the center and radius of each of the given equation of circles a) (x 3) 2 + (y + 8) 2 = 40 b) 5(x + 3) 2 + 5(y 7) 2 = 100 c) x 2 + y 2 = 6x 8y 15 = 0 d) 4x 2 + 4y 2 + 12x + 80y + 265 = 0 e) 2x 2 + 2y 2 10x + 8y = 3 2 Find the equation of the circle that satisfies the following conditions a) center at (5, 2) and radius 3 b) center at (7, 1) and passing through the point (3, 2) c) has diameter with endpoints at (4, 2) and ( 3, 5) d) center at ( 3, 4) and has a diameter with length 4 3 e) center at (8, 10) and tangent to the x-axis 7
124 Hyperbola If F 1 and F 2 are two fixed points on the plane, then the set of all points P (x, y) for which the absolute value of the difference, that is, P F 1 P F 2, of the distances from P to each of the two fixed points is a constant, forms a pair of continuous curves called the branches of a hyperbola Each fixed point is called a focus (plural: foci) of the hyperbola The following are some properties of a hyperbola The midpoint of the line segment joining the two foci is called the center of the hyperbola, usually denoted by C(h, k) The distance from the center to each of the two foci is denoted by c The line through the two foci intersects the hyperbola at two distinct points called the vertices of the hyperbola The distance from the center to each vertex is denoted by a The line segment joining the two vertices of the hyperbola is called the transverse axis Let b = c 2 a 2 The line segment of length 2b whose midpoint is the center of the hyperbola and perpendicular to the transverse axis is called the conjugate axis of the hyperbola The rectangle formed by drawing vertical/horizontal lines through the endpoints of the transverse axis and the conjugate axis is called the auxiliary rectangle of the hyperbola The two lines joining opposite vertices of the auxiliary rectangle are called the asymptotes of the hyperbola Example 3 Figure 12: Parts of a hyperbola In general, the standard equations of a hyperbola are given as follows: 8
(x h)2 (y k)2 = 1, where the center is at (h, k), c 2 = a 2 + b 2, and a a 2 b 2 horizontal transverse axis (y k)2 (x h)2 a 2 b 2 transverse axis = 1, where the center is at (h, k) c 2 = a 2 +b 2, and a vertical Exercise 5 Do the following 1 Find the coordinates of the vertices, foci, center and equations of the asymptotes a) x2 36 y2 25 = 1 (y 7)2 (x + 2)2 b) = 1 81 49 c) 16x 2 9y 2 + 18y + 64x = 91 d) 9x 4y 2 36x + 8y 4 = 0 e) 4y 2 x 2 = 4x f) 4x 2 9y 2 + 8x 36y 4 = 0 13 Chapter Exercises 9
Chapter 2 Review of Functions and Their Graphs 21 Functions We give two equivalent definitions of a function If there is an association or a correspondence between each element x of a set X to exactly one element y of a set Y then the association is a function from X to Y In symbols we have : f : X Y and is read as a function f from a set X to a set Y Remarks We use the following notation The first set is called the domain, while the second set is called the codomain The set of all values y Y that corresponds to a value in the domain is called the range If an element y of a set Y corresponds to the element x of a set X, we call y the value of the function at x or the image of x under the function, and we write is as follows: f(x) = y, read as f of x is equal to y The following statement is an alternative definition of a function A function, f, from X to Y is a set of ordered pairs of real numbers,(x, y), in which not two distinct ordered pairs have the same first number Remarks 1 An element of the range may correspond to more than one element of the domain 10
2 Some elements in the set Y (co-domain) may not be in the range 3 An element x of the domain NEVER corresponds to more than one element in the co-domain If f is a function, then the graph of f is the set of all points (x, y) in the plane R 2 for which (x, y) is an ordered pair in f 22 Operations on Functions Given two functions f and g, 1 the sum of f and g denoted by f + g is the function defined by (f + g)(x) = f(x) + g(x) 2 the difference of f and g denoted by f + g is the function defined by (f g)(x) = f(x) g(x) 3 the product of f and g denoted by f + g is the function defined by (fg)(x) = f(x)g(x) 4 the quotient of f and g denoted by f + g is the function defined by ( ) f (x) = f(x) g g(x) where g(x) 0 5 the composite function of f by g denoted by f g is defined by (f g) (x) = f (g(x)) Example 4 Given the functions f(x) = 2x + 1 and g(x) = x 2 1 then (f + g)(x) = f(x) + g(x) = (2x + 1) + (x 2 1) = x 2 + 2x (fg)(0) = f(x) g(x) = (2(0) + 1) ((0) 2 1) = (1) ( 1) = 1 (f g)(x) = f (g(x)) = f (x 2 1) = 2(x 2 1) + 1 = 2x 2 2 + 1 = 2x 2 1 11
( ) f (2y) = f(2y) g g(2y) = 2(2y) + 1 (2y) 2 1 = 4y + 1 4y 2 1 23 Special Types of Functions In this section we study some special types of functions, namely polynomial function, radical function, greatest integer function, signum function, absolute value function and piece-wise defined function A polynomial function is a function defined by an equation of the form f(x) = a n x n + a n 1 x n 1 + + a 1 x + a 0, where n Z + when n = 1 : f(x) is a linear function and Domain : R, Range : R n = 2 : f(x) is a quadratic function ( and Domain : R,Graph: Parabola opening upward or downward, Vertex at a 1, 4a ) 2a 0 a 2 1 2a 2 4a 2 Example 5 The following are examples of polynomial functions: 1 f(x) = 2 2 f(x) = 1 x x 2 + 2x 5 3 f(x) = 1 4 x + 5 4 f(x) = x3 3 2x 5 The greatest integer function x or the floor function is defined by f(x) = x where x = n if n x < n + 1 The domain of this function is R and its range is the set of integers Z Example 6 The following are examples of the greatest integer of a specified number 1 03 = 0 2 19 = 1 3 21 = 3 4 002 = 1 5 5 = 5 6 009 = 0 12
Figure 21: The graph of the function f(x) = [x] is shown below The signum function is a piecewise-defined function defined by 1 if x < 0 Sgn(x) = 0 if x = 0 1 if x > 0 We note that the domain of the signum function is the set of all real numbers, R and its range is the set { 1, 0, 1} A function f defined by f(x) = { x if x < 0 x if x 0 is called an absolute value function We note that the domain of the absolute value function is the set of all real numbers, R and its range is the set [0, + ) 13
Chapter 3 Limits In this chapter we will study the concept of the limit of a function Intuitively, we can think if the limit of a function as a value that the function approaches to as the values of the independent variable approach a certain real number Let us first consider the following example Example 7 Given the function f(x) = x 2 We obtain the values of the function f given some values arbitrarily close to 2 as seen in the table below x f(x) x f(x) 10000 10000 30000 90000 15000 22500 25000 62500 17000 28900 23000 52900 18000 32400 22000 48400 19000 36100 21000 44100 19900 39601 20100 40401 19990 39960 20010 40040 19999 39996 20001 40004 Notice that the value of f(x) becomes close to 4 as we let the value of x approach to 2, in symbols, lim x 2 x 2 = 4 read as, the limit of f(x) as x approaches to 2 is 4 Example 8 Consider the function defined by f(x) = x2 + x 6 which is defined everywhere except at 2 When x 2, we may write f(x) = x + 3, since x 2 x 2 + x 6 (x 2)(x + 3) = = x + 3 Consider the following table of function values x 2 (x 2) for f(x) at numbers close (but not equal) to a = 2 14
x f(x) x f(x) 2500000 5500000 1500000 4500000 2100000 5100000 1900000 4900000 2003000 5003000 1997000 4997000 2000040 5000040 1999900 4999900 2000030 5000030 1999997 4999997 Notice that the value of f(x) becomes close to 5 as we let the value of x be close to 2 This tells us that the limit of the function is 5 as the value of x approaches to x 2 + x 6 2, in symbols we have, lim = 5 x 2 This tells us that even if the function is undefined at a number a, the limit of the function at a may exist Example 9 Let f be defined by x 2 1 f(x) = x 1, if x 1 1, if x = 1 Example 10 Discuss the behavior of the following functions as x approaches a = 0 { 0, x < 0 1 U(x) = 1, x 0 15
{ 1/x, x 0 2 g(x) = 0, x = 0 Remarks A function may fail to have a limit at a point in its domain Example 11 For the given function f(x) with its graph shown, find the following limits 1 lim x 1 f(x) 2 lim x 2 f(x) 3 lim x 3 f(x) Figure 31: The graph of the function f(x) 16
Example 12 For the given function g(x) with its graph shown, find the following limits 1 lim g(x) 2 lim g(x) x 2 x 1 3 lim g(x) x 0 Figure 32: The graph of the function g(x) Example 13 For the given function h(x) with its graph shown, find the following limits 1 lim x 2 h(x) 2 lim x 1 h(x) 3 lim x 0 h(x) Figure 33: The graph of the function h(x) 17
31 Limits Let f be a function defined at every number in some open interval containing a, except possibly at the number a itself The limit of f(x) as x approaches a is L written as lim f(x) = L if ε > 0, δ > 0 such that f(x) L < ε whenever 0 < x a < δ Example 14 Prove that lim(2x + 1) = 7 x 3 Preliminary to the proof: We choose ε = 1 We need to find a δ > 0 such that f(x) L < ε whenever 2 x a < δ We take note of the following: f(x) = 2x + 1, L = 7,and a = 3 Thus, we are to find δ > 0 such that 2x + 1 7 < 1 whenever 0 < x 3 < δ Now 2 consider Proof: Let ε > 0 be given such that We take δ = ε 2 Hence: (2x + 1) 7 < 1 2 2x 6 < 1 2 2(x 3) < 1 2 2 x 3 < 1 2 x 3 < 1 4 (2x + 1) 7 < ε 2x 6 < ε 2(x 3) < ε 2 x 3 < ε x 3 < ε 2 0 < x 3 < δ x 3 < ε 2 2 x 3 < ε 2(x 3) < ε 2x 6 < ε (2x + 1) 7 < ε Some Limits Theorems We present here some of the limits theorems and the reader is referred to TC7 Chapter 1 section5 for their proofs Theorem 18
(1) The limit of a constant at any number a is also the constant c = c, c R lim (2) lim x = a (3) lim mx + b = ma + b For No s (4)-(8), assume that lim [ ] (4) lim cf(x) = c lim f(x), c R (5) lim [f(x) + g(x)] = lim f(x) + lim g(x) (6) lim [f(x)g(x)] = (7) lim [f(x)] n = (8) lim [ ] f(x) = g(x) ( ) ( ) lim f(x) lim g(x) ( ) n lim f(x), n Q lim f(x) lim 1 (9) lim x = 1 a, a 0 g(x), where lim f(x) and lim g(x) exists g(x) 0 (10) lim n x = n a if n is odd or if n is even and a > 0 (11) If P (x) is a polynomial function then lim P (x) = P (a) (12) If R(x) = N(x) is a rational function then lim R(x) = R(a) provided that a is D(x) in the domain of R(x) Example 15 Consider the following functions f(x) = 3x 2 and g(x) = 2x 7 lim x 1 f(x) = 3(1) 2 = 1 lim g(x) = 2( 3) 7 = 1 x 3 Example 16 Evaluate the following limits: 19
(a) lim x 1 (3x 2 2x + 1) x 2 (c) lim x 4 x 4 (b) lim x 0 7x 1 2x 1 (d) lim x 1 x + 5 2 x + 1 Items (a) and (b) can be evaluated using direct substitution but direct substitution fails in items (c) and (d) For (c) and (d), using direct substitution leads to a form 0 but by using some algebraic manipulation we can avoid this and rewrite the expression in a form where we 0 can use direct substitution For (c), consider the following: x 2 x 2 x + 2 x 4 = x 4 = ( x + 2)( x 2) x + 2 (x 4)( x 4 = x + 2) (x 4)( x + 2) = 1 x + 2 Thus, For (d), consider the following: x + 5 2 x + 1 Thus, x 2 lim x 4 x 4 x + 5 + 2 x + 5 + 2 = lim x 1 x + 5 2 x + 1 Exercise 6 Evaluate the following limits = lim 1 = 1 x 4 x + 2 4 (x + 5) 4 (x + 1)( x + 5 + 2) = 1 x + 5 + 2 1 = lim = 1 x 1 x + 5 + 2 4 3 1 lim x 1 x 2 lim x 1 2x2 3 3 lim x 2 x 2 + 2x 1 4 lim x 2 x3 + 8 5 lim x 0 2x 3 3x 2 + x + 1 6 lim x 2 (4x + 3) 7 lim x 0 5x + 7 3x 2 + 1 8 lim x 2 2x + 1 x + 2 (x 1) 2 9 lim x 3 (2x 5) 3 (2x + 1) 10 lim x 1 (3x 2) 11 lim x 3 3 5 + 2x 5 x 8x + 1 12 lim x 1 x + 3 x 13 lim x 1 x 14 lim x 1 3 x 1 x 1 20
1 1 + x 15 lim x 0 x Remarks The following should be clear given the functionf(x) = x If a is not an integer then lim If a is an integer then x = lim x = lim x = a + lim x = a 1 and lim x = a + 32 One Sided Limits Let f be a given function, and let a be a fixed number (a) The limit of f(x) as x approaches a from the left is equal to L 1 and we denote this by lim f(x) = L 1 if the values f(x) becomes closer and closer to L 1 as x becomes closer to a with x < a L 1 is called the left-hand limit of f(x) at a (b) The limit of f(x) as x approaches a from the right is equal to L 2 and we denote this by lim + f(x) = L 2 if the values f(x) becomes closer and closer to L 2 as x becomes closer to a with x > a L 2 is called the right-hand limit of f(x) at a Exercise 7 Evaluate the following limits 1 Let f(x) = x x (a) (b) lim f(x) x 0 lim f(x) x 0 + (c) lim x 0 f(x) 2 Let f(x) = (a) lim f(x) x 0 (b) lim f(x) x 0 + (c) lim f(x) x 0 (d) lim f(x) x 1 { x 2 1, x 1 1 x, x > 1 21
(e) lim f(x) x 1 + (f) lim x 1 f(x) x + 5, x < 3 3 Let f(x) = 9 x 2, 3 x 3 3 x, x > 3 (a) (b) (c) lim f(x) x 0 + lim f(x) x 3 lim f(x) x 3 + (d) lim f(x) x 3 (e) lim f(x) x 3 (f) lim f(x) x 3 + (g) lim f(x) x 3 4 Let f(x) = (a) lim f(x) x 3 (b) lim x 0 f(x) { x 2 9, x 3 4, x = 3 Remarks The following are some facts regarding the limit of a function The limit of a function f(x) as x approaches a may exists even when f(a) is not a real number Inversely, the limit as x approaches a may fail to exists even when the function is defined at the number a, that is, even if f(a) is a real number A function f can only have on limit at a number a The limit of a function f(x) as x approaches a fails to exists when f(x)is unbounded at a 22
I Evaluate the following limits 1 lim x 2 (x 2 4x) 2 lim x 1 x3 2x 2 + x 1 ( ) x + 1 3 lim x 0 x 2 + 6x + 9 ( ) 3 x 3 x 4 lim x 0 3 x + 3 x ( ) x 2 4 5 lim x 2 x 2 5x + 6 ( ) x 2 6 lim x 2 x2 4 ( 1 7 lim 1 ) x 2 x 3 x 3 x 1 8 lim x 1 x2 + 3 2 x + 4 2 9 lim x 0 x 3 10 lim (x + 1) 6 x 4 PRACTICE EXERCISES 7 x 11 lim x 7 x 7 12 lim x + 1 x 1 13 lim x + 3 + 1 x 2 14 lim x 3 1 x 15 lim 3x 2 x 5 16 lim x 1 1 x (1 x) 2 1 x 2 17 lim x 1 + 1 x 18 lim x 4 + 16 x 2 16 x 2 1 + x 1 x 19 lim x 0 x x 20 lim x 0 x x II Let f(x) = x be the greatest integer function For what values of a does the lim f(x) exists? III Find the limits lim f(x) and lim f(x) for each integer n given the following x n x n + functions: 1 f(x) = ( 1) x x 2 f(x) = 2 IV Determine the vertical asymptotes of the following functions Describe the behavior of the graph of f(x) as the values of x becomes close the asymptote/s: 1 f(x) = 3x2 x + 5 x 2 4 2 f(x) = x2 4 x + 1 + x x 2 3x + 2 33 Limits at Infinity 3 f(x) = 2 3x2 6x 2 2x 5 4 f(x) = x 2 4 x 2 In the preceding sections of this chapter, we focused on the behavior of a function as the values of the independent variable x approach a fixed number a We now look at 23
how the function behaves when we allow the values of x to either increase or decrease without bound Let f be a function If we allow the values of x to increase or decrease indefinitely, and the corresponding values of f(x) becomes very close to a unique number L, then we say that the limit of f(x) as x approaches ± is equal to L, and we denote this by lim x + f(x) = L or by Theorem Let r be any positive integer Then 1 1 lim x + x = 0 r 1 2 lim x x = 0 r lim f(x) = L x Example 17 Consider the function f(x) = 1 We see here that lim x 1 lim x + x = 0 1 x x = 0 and Figure 34: The graph of the function f(x) = 1 x 24