Section 1.1: A Preview of Calculus When you finish your homework, you should be able to

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1 Section 1.1: A Preview of Calculus When you finish your homework, you should be able to π Understand what calculus is and how it compares with precalculus π Understand that the tangent line problem is basic to calculus π Understand that the area problem is also basic to calculus Warm-up: Find the area of the triangle which has sides that measure 8 m, 12 m, and 18 m, respectively. If needed, you may round to the nearest tenth. Feel free to use your technology. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 1

2 Calculus is the mathematics of. What does this mean?? Well, think about traveling on highway 5 when there s no traffic (if that s even possible). Suppose you travel from MiraCosta College to go to the races in Del Mar. It takes you 40 minutes to travel and also park at the racetrack. The total distance you traveled is 19 miles. From your precalculus knowledge, are you able to find this rate with the given information?. If yes, find the rate. What type of rate of change did you find? The rate of change, assuming that you traveled at a speed. Is this realistic?. Suppose you passed by the highway patrol while you were on the freeway, and he used his radar gun to find out your rate? This would measure your at the exact moment he got the reading. This type of rate of change is. This leads us to the line problem. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 2

3 f x x 2 at Example 1: Find the equation of the line secant to graph of 2 x 0 and x 3. The main difference between precalculus and calculus is the addition of the process to your calculations, and thinking of change versus change. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 3

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6 The problem combines and calculus. Example 2: Consider the region bounded by the graphs of 2 x 1. f x x, y 0, and CREATED BY SHANNON MYERS (FORMERLY GRACEY) 6

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8 Section 1.2: Finding Limits Graphically and Numerically When you finish your homework you should be able to π Estimate a limit using a numerical or graphical approach. π Learn different ways that a limit can fail to exist. π Study and use a formal definition of limit. Warm-up: Consider the function f x 3 x 1 x 1 1. Graph the function by hand. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 8

9 2. f Graph the function on your graphing calculator and sketch the result. 4. Now, please complete the chart below. x f x 3 x 1 x 1 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 9

10 Guess what? We just did some calculus!!! The first three entries on the chart above represent the as -1 from the of. In mathese, we write this as or for clarity, we write.the next three entries on the chart above represent the as -1 from the of. In mathese, we write this as or for clarity, we write. Now did we get the same approximate result for each one-sided limit?. Cool! This means that we have discovered that the as approaches of is. This implies that if there is no designation of approaching form the or approaching from the, the - limits approach the same -. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 10

11 Example 2: Complete the table and use the result to estimate the limit. lim a. x3 1 x 2 x 3 x f x b. x 0 1 lim cos x x f x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 11

12 c. Consider the function Let s graph the function: f x 2 1 x 9 x f x What observations would you make about the behavior of this function at the vertical asymptote x 3? As approaches from the of the function without and as approaches from the of the function without. Therefore, and do not exist, and thus also. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 12

13 COMMON TYPES OF BEHAVIOR ASSOCIATED WITH NONEXISTENCE OF A LIMIT 1. f x approaches a different number from the right side of c than it approaches from the left side. f x increases or decreases without bound as x approaches c f x oscillates between two fixed values as x approaches c. Example 3: Consider the function sin x, x 0 g x 1 cos x, 0 x cos x, x a. Please graph the function: b. Now identify the values of c for which the lim g x notation. xc exists in interval CREATED BY SHANNON MYERS (FORMERLY GRACEY) 13

14 DEFINITION OF LIMIT Let f be a function defined on an open interval containing c (except possibly at c) and let L be a real number. The statement lim f x L x c means that for each small positive number epsilon, denoted, there exists a small positive number delta, denoted, such that if 0 x c, then f x L. Let s check it out! CREATED BY SHANNON MYERS (FORMERLY GRACEY) 14

15 Example 4: Find the limit L. Then find 0 x c. 2 lim x 4 x5 such that f x L 0.01 whenever CREATED BY SHANNON MYERS (FORMERLY GRACEY) 15

16 1.3: Evaluating Limits Analytically When you are done with your homework you should be able to Evaluate a limit using properties of limits Develop and use a strategy for finding limits Evaluate a limit using dividing out and rationalizing techniques Evaluate a limit using the Squeeze Theorem Warm-up: 1. Factor and simplify. a. 3 8x 27 2x 3 b. 3 2 x x x c. 2 3x 11x 4 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 16

17 2. Rationalize the numerator. Simplify if possible. x 1 2 x 3 Example 1: Find the following function values. a. f x 2 f 4 b. f x x f 4 3 c. f x x f 4 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 17

18 Example 2: Evaluate the following limits using your graphing calculator. a. lim 2 x4 lim x b. x 4 c. lim x x 4 3 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 18

19 DIRECT SUBSTITUTION If the limit of f x as x approaches c is f c, then the limit may be evaluated using direct substitution. That is, f x f c lim. These types of xc functions are continuous at c. THEOREM: SOME BASIC LIMITS Let b and c be real numbers and let n be a positive integer. 1. lim b xc 2. lim x xc 3. n lim x xc THEOREM: PROPERTIES OF LIMITS Let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits. LIMIT lim f x L and lim g x K xc xc 1. Scalar multiple: lim xc bf x bl 2. Sum or difference: lim xc f x g x L K 3. Product: lim 4. Quotient: xc f x g x LK 5. Power: lim f x L lim, K 0 xc g x K xc n f x L n CREATED BY SHANNON MYERS (FORMERLY GRACEY) 19

20 Example 3: Find the limit. Identify the individual functions and the properties you used to evaluate the limit. a. lim5 x x2 f x, g x Properties used: 6x 5 b. lim x0 3 x 2 f x, g x Properties used: CREATED BY SHANNON MYERS (FORMERLY GRACEY) 20

21 THEOREM: LIMITS OF POLYNOMIAL AND RATIONAL FUNCTIONS If p is a polynomial and c is a real number, then lim p x p c. xc If r is a rational function given by r x p x q x and c is a real number such that qc 0, then lim r x r c. xc Example 4: Find the following limits. a. lim x 4 6x 2 2 x3 b. x 3 1 lim x4 2 x 7 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 21

22 THEOREM: THE LIMIT OF A RADICAL FUNCTION Let n be a positive integer. The following limit is valid for all c if n is odd, and is valid for c 0 if n is even. lim n xc x n c Example 5: Find the following limits. lim a. x 225 x b. lim x x THEOREM: THE LIMIT OF A COMPOSITE FUNCTION Let f and g be functions with the following limits. lim g x L and lim f x f L xc xl lim lim f g x f g x f L xc xc CREATED BY SHANNON MYERS (FORMERLY GRACEY) 22

23 Example 6: Consider the composite function f x and g x Evaluate the following. a. g 4 2 sin h x f g x x. So, b. x lim c. lim sin x 4 2 x x lim CREATED BY SHANNON MYERS (FORMERLY GRACEY) 23

24 THEOREM: LIMITS OF TRIGONOMETRIC FUNCTIONS Let c be a real number in the domain of the given trigonometric function. 1. limsin x sin c 2. limcos x xc xc 3. lim tan x tan c 4. lim cot x xc xc 5. lim sec x sec c 6. lim csc x xc xc cosc cot c csc c Example 7: Evaluate the following limits. lim tan x a. 2 x 3 b. limcos x x c. lim csc x x 6 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 24

25 THEOREM: FUNCTIONS THAT AGREE AT ALL BUT ONE POINT Let c be a real number and let f x g x for all x c in an open interval containing c. If the limit of of f x also exists and g x as x approaches c exists, then the limit lim f x lim g x. xc xc Example 8: Find the following limits, first using direct substitution. If you get an indeterminate result, try using the theorem above and identify g x. a. Direct substitution: 3 x 1 lim x 1 x1 lim x1 3 x x 1 1 b. Direct substitution: lim x3 x 2 5x 6 x 3 lim x3 x 2 5x 6 x 3 THE SQUEEZE THEOREM If h x f x g x for all x in an open interval containing c, except possibly at c itself, and if lim h x L lim g x then lim f x xc xc xc exists and is equal to L. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 25

26 THEOREM: TWO SPECIAL TRIGONOMETRIC LIMITS sin x lim 1 x 1. x0 2. x0 1 cos x lim 0 x sin tan tan cos CREATED BY SHANNON MYERS (FORMERLY GRACEY) 26

27 Example 9: Find the following limits. a. Direct Substitution: lim x 0 tan x 2 x lim x 0 tan x 2 x b. Direct Substitution: 1 tan x xlim sin x cos x 4 1 tan x xlim sin x cos x 4 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 27

28 STRATEGIES FOR FINDING LIMITS 1. Try using first. If this works, you are done! If not, go to step If you obtain an result when using direct substitution ( ), try a. the numerator and denominator, out common factors, and then use direct on the new expression. b. the numerator, and then use direct substitution on the new expression. 3. If you obtain an indeterminate result when using direct substitution (0/0), on a expression try a. Rewriting the expression using trigonometric, and then use direct substitution on the new expression. b. Rewriting the expression using trigonometric identities, and then use the special trigonometric limits and. 4. your result by graphing the function on your graphing calculator. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 28

29 Example 10: Find the following limits. lim a. Direct substitution: x4 x 2 2 x 4 lim x4 x 2 2 x 4 b. Direct Substitution: lim x0 2 2 x x x 2 2 x lim x0 2 2 x x x 2 2 x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 29

30 lim c. Direct Substitution: x0 1 1 x 3 3 x lim x0 1 1 x 3 3 x d. Direct Substitution: 0 cos lim tan 0 cos tan lim CREATED BY SHANNON MYERS (FORMERLY GRACEY) 30

31 1.4: Continuity and One-Sided Limits When you are done with your homework you should be able to Determine continuity at a point and continuity on an open interval Determine one-sided limits and continuity on a closed interval Use properties of continuity Understand and use the Intermediate Value Theorem Warm-up: Consider the function f x x x a. Sketch the graph. b. f 2 c. lim x2 2 x x 4 2 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 31

32 DEFINITION OF CONTINUITY CONTINUITY AT A POINT: A function f is continuous at c if the following three conditions are met. 1. f c 2. lim f x xc is defined. exists. 3. lim f x f c xc. Consider the warm-up problem. Use the definition of continuity at a point to determine if f is continuous at x 2? CREATED BY SHANNON MYERS (FORMERLY GRACEY) 32

33 CONTINUITY ON AN OPEN INTERVAL: A function is continuous on an open interval (a,b) if it is continuous at each point in the interval. A function that is continuous on the entire real line is everywhere continuous. Example 1: Draw the graph of the following functions with the given characteristics on the open interval from a to b: a. The function has a removable discontinuity at x 0 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 33

34 b. The function has a nonremovable discontinuity at x 0 THEOREM: THE EXISTENCE OF A LIMIT Let f be a function and let c and L be real numbers. The limit of approaches c is L if and only if lim f x L and lim f x L. xc xc f x as x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 34

35 CONTINUITY ON A CLOSED INTERVAL: A function f is continuous on the closed interval [a,b] if it is continuous on the open interval (a, b) and lim f x f a and lim f x f b. xa xb The function f is continuous from the right at a and continuous from the left at b. Example 2: Graph each function and use the definition of continuity to discuss the continuity of each function. a. g x tan x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 35

36 b. x, x 2 y x, 2 x 4 2 x, x 4 4 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 36

37 THEOREM: PROPERTIES OF CONTINUITY If b is a real number and f and g are continuous at x functions are also continuous at c. 1. Scalar multiple: bf 2. Sum or difference: f g 3. Product: fg f g g 4. Quotient: c, 0 c then the following The following types of functions are continuous at every point in their domains. p x a x a x a x x n n 1 1. Polynomial functions: r x 2. Rational functions: n p x q x n 3. Radical functions: f x x n1 1 0, q x 0 4. Trigonometric functions: sin x, cos x, tan x, cot x, sec x, csc x THEOREM: CONTINUITY OF A COMPOSITE FUNCTION If g is continuous at c and f is continuous at g c then the composite function f g x f g x is continuous at c. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 37

38 THEOREM: THE INTERMEDIATE VALUE THEOREM If f is continuous on the closed interval a, b and k is any number between f a and f b, a, b such that f c k. then there is at least one number c in Example 3: Consider the function f x cos 2x a. Sketch the graph of f by hand. on the closed interval 0,. 2 b. State the reason why we can apply the intermediate value theorem (IVT). f c c. Use the IVT to find c such that 1. 2 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 38

39 Example 4: Evaluate the one-sided limits. a. lim x2 2 x 2 b. lim x4 x 2 x 4 c. lim x10 x 10 x 10 d. lim x0 1 x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 39

40 1.5: Infinite Limits When you finish your homework you should be able to π Determine infinite limits from the left and from the right π Find and sketch the vertical asymptotes of the graph of a function Warm-up: Evaluate the following limits analytically. a lim x x 0 x b. lim 0 2 tan CREATED BY SHANNON MYERS (FORMERLY GRACEY) 40

41 c. sin t 2 lim t 0 t DEFINITION OF INFINITE LIMITS Let f be a function that is defined at every real number in some open interval containing c (except possibly at c). The statement lim xc f x means that for each M 0 there exists a 0, such that f x M whenever 0 x c. Similarly, the statement lim xc f x means that for each N 0 there exists a 0, such that f x N whenever 0 x c. To define the infinite limit from the left, replace 0 x c by c x c. To define the infinite limit from the right, replace 0 x c by c x c. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 41

42 Example 1: Determine the infinite limit. lim a. 2 x x 3 x 9 b. x 3 x lim sec 6 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 42

43 DEFINITION OF VERTICAL ASYMPTOTE If f x approaches infinity or negative infinity as x approaches c from the right or the left, then the line x c is a vertical asymptote of the graph of f. THEOREM: VERTICAL ASYMPTOTES Let f and g be continuous on an open interval containing c. If f c 0, 0 g c, and there exists an open interval containing c such that g x 0 for all x c in the interval, then the graph of the function given by has a vertical asymptote at x c. h x f x g x Example 2: Find the vertical asymptotes (if any) of the graph of the function. a. g cos CREATED BY SHANNON MYERS (FORMERLY GRACEY) 43

44 b. h x 2 x x x x THEOREM: PROPERTIES OF INFINITE LIMITS Let b and L be real numbers and let f and g be functions such that lim f x and lim g x L xc xc 1. Sum or difference: lim f x g x xc 2. Product: 3. Quotient: lim f x g x, L 0 xc x g x lim 0 xc f Similar properties hold for one-sided limits and for functions for which the limit of f x as x approaches c is. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 44

45 2 Example 3: Let lim f x, lim g x and lim h x 5. Determine the xc 3 xc xc following limits: a. lim x c f x g x b. lim f x g x 2 xc c. lim xc h x f 3 x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 45

46 2.1: The Derivative and the Tangent Line Problem When you are done with your homework you should be able to Find the slope of the tangent line to a curve at a point Use the limit definition to find the derivative of a function Understand the relationship between differentiability and continuity Warm-up: Find the following limits. 3x a. lim x0 2 x 2x b. lim x x 4 4 x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 46

47 c. lim x0 3 3 x x x x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 47

48 DEFINITION OF TANGENT LINE WITH SLOPE m If f is defined on an open interval containing c, and if the limit lim x0 x0 y f c x f c lim m x x exists, then the line passing through line to the graph of f at the point c, f c with slope m is the tangent c, f c. **The slope of the tangent line to the graph of f at the point called the slope of the graph of f at x = c. c, f c is also 2 Example 1: Find the slope of the graph of f x 1 x at the point the limit definition. 1,2 using CREATED BY SHANNON MYERS (FORMERLY GRACEY) 48

49 DEFINITION FOR VERTICAL TANGENT LINES If f is continuous at c and f c x f c f c x f c lim or lim x x x0 x0 the vertical line x c passing through the point c, f c is a vertical tangent line to the graph of f. DEFINITION OF THE DERIVATIVE OF A FUNCTION The derivative of f at x is given by f x lim x0 f x x f x x provided the limit exists. For all x for which this limit exists, f is a function of x. The process of finding the derivative of a function is called. A function is at x if its derivative exists at x and is differentiable on an open interval (a, b) if it is differentiable at in the interval. NOTATION FOR THE DERIVATIVE OF y f x : CREATED BY SHANNON MYERS (FORMERLY GRACEY) 49

50 Example 2: Find the derivative of 3 f x 4 x using the limit process. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 50

51 ALTERNATIVE LIMIT FORM OF THE DERIVATIVE f c lim xc f x f c x c This form of the derivative requires that the one-sided limits f x f c f x f c lim and lim xc x c xc x c exist and are equal. 2 3 Example 3: Is the function f x x differentiable at x = 0? THEOREM: DIFFERENTIABILITY IMPLIES CONTINUITY If f is differentiable at x c, then f is continuous at x c. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 51

52 2.2: Basic Differentiation Rules and Rates of Change When you are done with your homework you should be able to Find the derivative of a function using the constant rule Find the derivative of a function using the power rule Find the derivative of a function using the constant multiple rule Find the derivative of a function using the sum and difference rules Find the derivative of the sine function and of the cosine function Use derivatives to find rates of change Warm-up: Find the following derivatives using the limit definition of the derivative. a. f x 2 2 b. f x x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 52

53 c. f x cos x THEOREM: THE CONSTANT RULE The derivative of a constant function is zero. That is, if c is a real number, d dx then c 0 Hmmm isn t this theorem the equivalent of saying that the of a line is zero? Example 1: Find the derivative of the function g x 6. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 53

54 THEOREM: THE POWER RULE If n is a rational number, then the function n f x x is differentiable and d x n nx n1 dx For f to be differentiable at x 0 defined on an interval containing zero. Example 2: Find the following derivatives. 5 a. f x x, n must be a number such that n 1 x is 1 2 b. f x x 5 3 c. f x x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 54

55 THEOREM: THE CONSTANT MULTIPLE RULE If f is a differentiable function and c is a real number, then cf is also differentiable and Proof: d cf x cf x dx f x 2x Example 3: Find the slope of the graph of 3 at a. x 2 b. x 6 c. x 0 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 55

56 THEOREM: THE SUM AND DIFFERENCE RULES The sum (or difference) of two differentiable functions f and g is itself differentiable. Moreover, the derivative of f g (or f g ) is the sum (or difference) of the derivatives of f and g. Proof: d f x g x f x g x dx d f x g x f x g x dx CREATED BY SHANNON MYERS (FORMERLY GRACEY) 56

57 Example 4: Find the equation of the line tangent to the graph of f x x x at x = Find the at. 2. Find. 3. Write equation of the line to at. THEOREM: DERIVATIVES OF THE SINE AND COSINE FUNCTIONS d dx d sin x cos x cos x sin x dx Example 5: Find the derivative of the following functions with respect to the independent variable: f x a. sin x 6 b. r 5 3cos CREATED BY SHANNON MYERS (FORMERLY GRACEY) 57

58 RATES OF CHANGE We have seen how the derivative is used to determine. The derivative may also be used to determine the of of one with respect to another. A common use for rate of change is to describe the motion of an object moving in a straight line. In such problems, it is customary to use either a horizontal or a vertical line with a designated origin to represent the line of motion. On such lines, movement to the or is considered to be in the positive direction, and movement to the left or downwards is considered to be in the direction. THE POSITION FUNCTION is denoted by s and gives the position (relative to the origin) of an object as a function of time. If, over a period of time t, the object changes its position by s s t t s t, then, by the familiar formula rate distance time the average velocity is change in distance change in time s t CREATED BY SHANNON MYERS (FORMERLY GRACEY) 58

59 s t 16t v t s. Example 6: A ball is thrown straight down from the top of a 220-foot building with an initial velocity of -22 feet per second. The position function for freefalling objects measured in feet is 2 What is its velocity after 3 seconds? 0 0 What is its velocity after falling 108 feet? CREATED BY SHANNON MYERS (FORMERLY GRACEY) 59

60 2.3: The Product and Quotient Rules and Higher-Order Derivatives When you are done with your homework you should be able to Find the derivative of a function using the product rule Find the derivative of a function using the quotient rule Find the derivative of a trigonometric function Find a higher-order derivative of a function Warm-up: Find the derivative of the following functions. Simplify your result to a single rational expression with positive exponents. 2 3x x 2 a. f x x b. g x 5x 3 2 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 60

61 f x x 4 c. cos CREATED BY SHANNON MYERS (FORMERLY GRACEY) 61

62 THEOREM: THE PRODUCT RULE The product of two differentiable functions f and g is itself differentiable. Moreover, the derivative of fg is the derivative of the first function times the second function, plus the first function times the derivative of the second function. d dx f x g x f x g x f x g x This rule extends to cover products of more than two factors. For example the derivative of the product of functions fghk is d fghk f x g x h x k x f x g x h x k x f x g x h x k x f x g x h x k x dx Proof: CREATED BY SHANNON MYERS (FORMERLY GRACEY) 62

63 Example 1: Find the derivative of the following functions with respect to the independent variable. Simplify your result to a single rational expression with positive exponents. g x x x a. cos b. ht 3 t 2 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 63

64 c. f x x 3 x x 2 2 x 2 x 1 Do not simplify this guy THEOREM: THE QUOTIENT RULE The quotient of two differentiable functions f and g is itself differentiable at all values of x for which g x 0. Moreover, the derivative of f g is the derivative of the numerator times the denominator, minus the numerator times the derivative of the denominator, all divided by the square of the denominator. 2 d f x f x g x f x g x dx g x g x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 64

65 Example 2: Find the derivative of the following functions. Simplify your result to a single rational expression with positive exponents. 4 x a. g x x 1 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 65

66 b. ht t t 1 c. f x tan x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 66

67 THEOREM: DERIVATIVES OF TRIGONOMETRIC FUNCTIONS d dx d dx d dx 2 2 tan x sec x cot x csc sec x sec x tan x d dx x csc x csc x cot x Example 3: Find the derivative of the trigonometric functions with respect to the independent variable. a. g x 2csc x h t cot b. 2 t CREATED BY SHANNON MYERS (FORMERLY GRACEY) 67

68 c. r s sec s s HIGHER ORDER DERIVATIVES Recall that you can obtain by differentiating a position function. You can obtain an function by differentiating a velocity function. Or, you could also think about the acceleration function as the derivative of the function. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 68

69 Example 4: An automobile s velocity starting from rest is vt measured in feet per second. Find the acceleration at 100t 2t 15 where v is a. 5 seconds b. 10 seconds c. 20 seconds CREATED BY SHANNON MYERS (FORMERLY GRACEY) 69

70 NOTATION FOR HIGHER-ORDER DERIVATIVES First derivative Second derivative Third derivative Fourth derivative nth derivative Example 5: Find the given higher-order derivative. 2 x a. f x 2, f x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 70

71 b. 4 6 f x tan x, f x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 71

72 2.4: The Chain Rule When you are done with your homework you should be able to Find the derivative of a composite function using the Chain Rule. Find the derivative of a function using the General Power Rule. Simplify the derivative of a function using algebra. Find the derivative of a trigonometric function using the Chain Rule. Theorem: The Chain Rule If y f u is a differentiable function of u and u g x y f g x function of x, then is a differentiable function of x and dy dy du or d f g x f g x g x. dx du dx dx is a differentiable CREATED BY SHANNON MYERS (FORMERLY GRACEY) 72

73 Example 1: Use the methods learned in 2.2 and 2.3 to evaluate the derivative of the following functions. Then find the derivative using the Chain Rule. a. y 2 x 3 old way Chain Rule b. f x sin 2x old way Chain Rule CREATED BY SHANNON MYERS (FORMERLY GRACEY) 73

74 c. ht old way t t 1 Chain Rule CREATED BY SHANNON MYERS (FORMERLY GRACEY) 74

75 Theorem: The General Power Rule If y u x, number, then n where u is a differentiable function of x and n is a rational dy n1 du d n n1 n u x or u nu u. dx dx dx Example 2: Complete the table. u g x y f u y f g x y y y y 2 8x 3 25 x tan 3 2 csc x 5 2 x 6 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 75

76 Example 3: Find the derivative of the following functions. a. y sec x b. y sec 2x c. y 2 sec x d. y sec x 2 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 76

77 e. y x 5 3 f. y 2x 5 5 g. y x h. y cos x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 77

78 i. f x x 2 2 x 2 3 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 78

79 f x d x 15 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 79

80 h x xsin 4x e. 2 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 80

81 f. Find the equation of the tangent line at t 1 for the function 2 9 t 2 3 s t. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 81

82 2.5: Implicit Differentiation When you are done with your homework you should be able to Distinguish between functions written in implicit form and explicit form Use implicit differentiation to find the derivative of a function Warm-up: Find the derivative of the following relation. 2 2 x y 25 IMPLICIT AND EXPLICIT RELATIONS Up to this point, we have typically seen functions expressed in form. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 82

83 Some relations or functions are only by an equation. When you were differentiating the warm-up problem, you were able to explicitly write y as a function of x, using equations. Oftentimes, it is difficult to write y as a function of x. In order to differentiate we must use differentiation. To understand how to find dy dx implicitly, you must know which variable you are differentiating with to. You must use the rule on any variable which is different than the one you are differentiating with respect to. Example 1: Find the derivative of the following functions with respect to x. Simplify your result to a single rational expression with positive exponents. 5 a. y x b. x y 5 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 83

84 c. x y GUIDELINES FOR IMPLICIT DIFFERENTIATION (WITH RESPECT TO x) 1. Differentiate both sides of the equation with respect to x. 2. Collect all terms involving dy dx other terms to the right side of the equation. 3. Factor dy dx 4. Isolate dy dx. out of the left side of the equation. on the left side of the equation and move all CREATED BY SHANNON MYERS (FORMERLY GRACEY) 84

85 Example 2: Find the derivative of the following functions with respect to x. Simplify your result to a single rational expression with positive exponents. a. 4cos xsin y 1 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 85

86 b. 2 2 x y y x 2 c. 1 x csc y CREATED BY SHANNON MYERS (FORMERLY GRACEY) 86

87 Example 3: Use implicit differentiation to find an equation of the tangent line to 2 2 x y the hyperbola 1 at x 4. Sketch the graphs of the hyperbola and the 4 9 tangent line at x 4. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 87

88 Example 4: Find the points at which the graph of the equation 2 2 4x y 8x 4y 4 0 has a horizontal tangent line. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 88

89 Example 5: Find 1 xy x y 2 d y dx 2 in terms of x and y. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 89

90 2.6: Related Rates When you are done with your homework you should be able to Find a related rate Use related rates to solve real-life problems Warm-up 1: Find the derivative of V with respect to t. 2 V r h Warm-up 2: Find the volume of a cone with a radius of 24 inches and a height of 10 inches. Round to the nearest hundredth. FINDING RELATED RATES We use the rule to find the rates of change of two or more related variables that are changing with respect to. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 90

91 Some common formulas used in this section: Volume of a Sphere: Right Circular Cylinder: Right Circular Cone: Right Rectangular Prism: Right Rectangular Pyramid: Pythagorean Theorem: GUIDELINES FOR SOLVING RELATED-RATE PROBLEMS 1. Analyze: Identify all given quantities and quantities to be determined. Make a sketch and label the quantities. 2. Related Variables Equation: Write an equation involving the variables whose rates of change either are given or are to be determined. 3. Find the Related Rate: Using the Chain Rule, implicitly differentiate both sides of the equation with respect to time t. 4. Find Desired Rate: After completing step 3, substitute into the resulting equation all known values for the variables and their rates of change. Then solve for the required rate of change. 5. Conclusion: Write your conclusion in words. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 91

92 Example 1: Find the rate of change of the distance between the origin and a dx moving point on the graph of y sin x if 2cm/sec dt. 1. Analyze: 2. Related Variables Equation: 3. Find Related Rate: 4. Find Desired Rate: 5. Conclusion: CREATED BY SHANNON MYERS (FORMERLY GRACEY) 92

93 Example 2: Find the rate of change of the volume of a cone if dr/dt is 2 inches per minute and h = 3r when r = 6 inches. Round to the nearest hundredth. How is this problem different than the warm-up problem? 1. Analyze: 2. Related Variables Equation: 3. Find Related Rate: 4. Find Desired Rate: 5. Conclusion: CREATED BY SHANNON MYERS (FORMERLY GRACEY) 93

94 Example 3: Angle of Elevation. A fish is reeled in at a rate of 1 foot per second from a point 10 feet above the water. At what rate is the angle between the line and the water changing when there is a total of 25 feet of line out? 1. Analyze: 2. Related Variables Equation: 3. Find Related Rate: 4. Find Desired Rate: 5. Conclusion: CREATED BY SHANNON MYERS (FORMERLY GRACEY) 94

95 Example 4: Consider the following situation: A container, in the shape of an inverted right circular cone, has a radius of 5 inches at the top and a height of 7 inches. At the instant when the water in the container is 6 inches deep, the surface level is falling at the rate of -1.3 in/s. Find the rate at which the water is being drained. 1. Analyze: 2. Related Variables Equation: 3. Find Related Rate: 4. Find Desired Rate: 5. Conclusion: CREATED BY SHANNON MYERS (FORMERLY GRACEY) 95

96 Example 5: Shannon is trying to hang her Christmas lights. She has a ladder that is 22 feet long. It is leaning against a wall of her house. Since Shannon did not secure the ladder, it is moving away from the wall at a rate of 1.5 feet per second. How fast is the top of the ladder moving down the wall when its base is 10 feet from the wall? 1. Analyze: 2. Related Variables Equation: 3. Find Related Rate: 4. Find Desired Rate: 5. Conclusion: CREATED BY SHANNON MYERS (FORMERLY GRACEY) 96

97 3.1: Extrema on an Interval When you are done with your homework you should be able to Understand the definition of a function on an interval Understand the definition of relative extrema of a function on an open interval Find extrema on a closed interval f x 2x 8x 5 Warm-up: Determine the point(s) at which the graph of 2 has a horizontal tangent. EXTREMA OF A FUNCTION In calculus, much effort is devoted to the behavior of a function f on an I. Does f have a value or value on I? Where is f? Where is f? In this chapter, you will learn how can be used to answer these questions. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 97

98 DEFINITION OF EXTREMA Let f be defined on an open interval I containing c. 1. f c is the minimum of f on I if f c f x 2. f c is the maximum of f on I if f c f x for all x in I. for all x in I. The minimum and maximum of a function on an interval are the extreme values, or extrema (the singular form of extrema is extremum), of the function on the interval. The minimum and maximum of a function on an interval are also called the absolute minimum and absolute maximum, or the global minimum and global maximum, on the interval. Let s check out the functions below. Identify the maximum and minimum of each function. a. b. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 98

99 THEOREM: THE EXTREME VALUE THEOREM If f is continuous on a closed interval a, b, then f has both a minimum and a maximum on the closed interval. DEFINITION OF RELATIVE EXTREMA 1. If there is an open interval containing c on which f c is a maximum, then f c is called a relative maximum of f, or you can say that f has a relative maximum at c, f c. 2. If there is an open interval containing c on which f c is a minimum, then f c is called a relative minimum of f, or you can say that f has a relative minimum at c, f c. Example 1: Find the value of the derivative at the extremum 0,1, for the x function f x cos. 2 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 99

100 DEFINITION OF A CRITICAL NUMBER Let f be defined at c. If f c 0 OR if f is not differentiable at c, then c is a critical number. Example 2: Find any critical numbers of the following functions. f x 2x 8x 5 a. 2 b. g x sin 2 x sin x, 0, 2 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 100

101 s t c. 2 3t t 4 THEOREM: RELATIVE EXTREMA OCCUR ONLY AT CRITICAL NUMBERS If f has a relative minimum or relative maximum at x number of f. c, then c is a critical CREATED BY SHANNON MYERS (FORMERLY GRACEY) 101

102 GUIDELINES FOR FINDING EXTREMA (AKA ABSOLUTE EXTREMA OR GLOBAL EXTREMA) ON A CLOSED INTERVAL To find the extrema of a continuous function f on a closed interval a, b, use the following steps. 1. Find the critical numbers of f in a, b. 2. Evaluate f at each critical number in a, b. 3. Evaluate f at each endpoint of a, b. 4. The least of these numbers is the minimum. The greatest is the maximum. Example 3: Locate the absolute extrema of the following functions function on the closed interval. a. f x 3cos x,, 1. Find critical numbers on. 2. Evaluate =. 3. Evaluate the function at the. 4. Conclusion. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 102

103 t b. g t, 3,5 t 2 1. Find critical numbers on. 2. Evaluate =. 3. Evaluate the function at the. 4. Conclusion. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 103

104 Example 4: A retailer has determined that the cost C of ordering and storing x units of a product is 300, 000 C 2 x, 1 x 300. The delivery truck can x bring at most 300 units per order. a. Find the order size that will minimize cost. b. Could the cost be decreased if the truck were replaced with one that could bring at most 400 units? Explain. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 104

105 3.2: Rolle s Theorem and the Mean Value Theorem When you are done with your homework you should be able to Understand and use Rolle s Theorem Understand and use the Mean Value Theorem Warm-up: Locate the global extrema of the function 3 closed interval 0, 4. f x x 12x on the ROLLE S THEOREM The Value Theorem states that a continuous function on a closed interval must have both a and a. Both of these values, however, can occur at the. Rolle s Theorem gives conditions that guarantee the existence of an extreme value in the of a closed interval. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 105

106 THEOREM: ROLLE S THEOREM Let f be continuous on the closed interval a, b and differentiable on the open interval, f a f b a, b such that f c 0 a b. If then there is at least one number c in. Example 1: Determine whether Rolle s Theorem can be applied to f on the closed interval a, b. If Rolle s Theorem can be applied, find all values of c in the open interval a, b such that f c 0. a. f x cos 2 x,, CREATED BY SHANNON MYERS (FORMERLY GRACEY) 106

107 b. f x x, 4, 4 THEOREM: THE MEAN VALUE THEOREM (MVT) If f is continuous on the closed interval a, b, and differentiable on the open interval a, b, then there exists a number c in a, b such that f c f b f a b a CREATED BY SHANNON MYERS (FORMERLY GRACEY) 107

108 Example 2: Consider the function f x x on the closed interval from 1,9. a. Graph the function on the given interval. b. Find and graph the secant line through the endpoints on the same coordinate plane. c. Find and graph any tangent lines to the graph of f that are parallel to the secant line. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 108

109 3.3: Increasing and Decreasing Function Intervals and the First Derivative Test When you are done with your homework you should be able to Determine intervals on which a function is increasing or decreasing Apply the First Derivative Test to find relative extrema of a function Warm-up: Find the equation of the line tangent to the function f x tan x at 3 x. 4 INCREASING AND DECREASING FUNCTION INTERVALS A function is if, as x moves to the right, its graph moves up, and is decreasing if its graph moves. A positive derivative implies that the function is and a derivative implies that the function is decreasing. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 109

110 DEFINITION OF INCREASING AND DECREASING FUNCTIONS A function f is increasing on an interval if for any two numbers x 1 and x 2 in the in the interval, x1 x2 implies f x f x. 1 2 A function f is decreasing on an interval if for any two numbers x 1 and x 2 in the in the interval, x1 x2 implies f x f x. 1 2 THEOREM: TEST FOR INCREASING AND DECREASING FUNCTION INTERVALS Let f be a function that is continuous on the closed interval a, b, and differentiable on the open interval a, b. 1. If f x 0 for all x in, 2. If f x 0 for all x in, 3. If f x 0 for all x in, a b, then f is increasing on a, b. a b, then f is decreasing on a, b. a b, then f is constant on a, b CREATED BY SHANNON MYERS (FORMERLY GRACEY) 110

111 GUIDELINES FOR FINDING INTERVALS ON WHICH A FUNCTION IS INCREASING OR DECREASING Let f be continuous on the a, b. To find the open intervals on which f is increasing or decreasing, use the following steps. 1. Locate the numbers of f in a, b, and use these numbers to determine test intervals. 2. Determine the sign of at one test value in each of the intervals. 3. Use the test for increasing and decreasing functions to determine whether f is increasing or decreasing on each. These guidelines are also valid if the interval a, b is replaced by an interval of the form,b, a,, or,. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 111

112 Example 1: Identify the open intervals on which the function is increasing or decreasing f x x x a. Find the critical numbers of f. b. Run the test for increasing and decreasing function intervals. i. Find the open interval(s) on which the function is decreasing. ii. Find the open interval(s) on which the function is increasing. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 112

113 Example 2: Identify the open intervals on which the function is increasing or decreasing. 2 f x cos x cos x, 0 x 2 a. Find the critical numbers of f. b. Run the test for increasing and decreasing intervals. i. Find the open interval(s) on which the function is decreasing. ii. Find the open interval(s) on which the function is increasing. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 113

114 THEOREM: THE FIRST DERIVATIVE TEST Let c be a critical number of a function f that is continuous on an open interval I containing c. If f is differentiable on the interval, except possibly at c, then f 1. If f x c can be classified as follows: changes from negative to positive at c, then f has a relative minimum at 2. If f x c, f c. changes from positive to negative at c, then f has a relative maximum at 3. If f x f c, f c. is positive on both sides of c or negative on both sides of c, then c is neither a relative minimum or relative maximum. 1. f x 0 f x 0 f x 0 f x 0 a c b a c b f x 0 f x 0 f x 0 f x 0 a c b a c b CREATED BY SHANNON MYERS (FORMERLY GRACEY) 114

115 g x x 4. Example 3: Consider the function 2 3 a. Find the critical numbers of g. b. Run the test for increasing and decreasing intervals. i. Find the open interval(s) on which the function is decreasing. ii. Find the open interval(s) on which the function is increasing. c. Apply the First Derivative Test. i. Identify all relative minima. ii. Identify all relative maxima. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 115

116 Example 4: The graph of a function f is given. The scale of each axis is from -10 to 10. Sketch a graph of the derivative of f. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 116

117 3.4: Concavity and the Second Derivative Test When you are done with your homework you should be able to Determine intervals on which a function is concave upward or concave downward Find any points of inflection of the graph of a function Apply the Second Derivative Test to find relative extrema of a function Warm-up: Consider the graph of f shown below. The vertical and horizontal axes have a scale of -10 to 10. Round to the nearest tenth as need when answering the questions below. a. Identify the interval(s) on which f is i. increasing ii. decreasing b. Estimate the value(s) of x at which f has a relative i. minimum ii. maximum CREATED BY SHANNON MYERS (FORMERLY GRACEY) 117

118 DEFINITION OF CONCAVITY Let f be differentiable on an open interval I. The graph of f is concave upward on I if f is increasing on the interval and concave downward on I if f is decreasing on the interval. THEOREM: TEST FOR CONCAVITY Let f be a function whose second derivative exists on an open interval I. for all x in I, then f is concave upward on I. 1. If f x 0 for all x in I, then f is concave downward on I. 2. If f x 0 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 118

119 Example 1: Identify the open intervals on which the function is concave upward or concave downward. a. 3 2 y x 3x 2 1. Find the of the derivative. 2. Run the. 3. Conclusion. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 119

120 2,, sin x b. f x x 1. Find the of the derivative. 2. Run the. 3. Conclusion. DEFINITION OF POINT OF INFLECTION Let f be a function that is continuous on an open interval and let c be an element in the interval. If the graph of f has a tangent line at the point c, f c, then this point is a point of inflection of the graph of f if the concavity of f changes from upward to downward or from downward to upward at the point. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 120

121 THEOREM: POINTS OF INFLECTION If c, f c is a point of inflection of the graph of f, then either f c 0 f does not exist at x c. g x 2x 8x 3. Example 2: Consider the function 4 a. Discuss the concavity of the graph of g. or 1. Find the of the derivative. 2. Run the. 3. Conclusion. b. Find all points of inflection. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 121

122 THEOREM: SECOND DERIVATIVE TEST Let f be a function such that f c 0 an open interval containing c. and the second derivative of f exists on 1. If f c 0, then f has a relative minimum at, c f c. 2. If f c 0, then f has a relative maximum at, 3. If f c 0 TEST. c f c., the test FAILS and you need to run the FIRST DERIVATIVE Example 3: Find all relative extrema. Use the Second Derivative Test where applicable. f x x 5x 7x a. 3 2 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 122

123 b. f x x x 1 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 123

124 Example 4: Sketch the graph of a function f having the given characteristics. f 0 f 2 0 f x 0 if x 1 f 1 0 f x 0 if x 1 f x 0 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 124

125 3.5: Limits at Infinity When you finish your homework you should be able to π Determine finite limits at infinity π Determine the horizontal asymptotes, if any, of the graph of a function π Determine infinite limits at infinity Warm-up: Evaluate the following limits analytically a. lim x1 1 3 x sin 3t lim t b. t 0 LIMITS AT INFINITY This section discusses the behavior of a function on an interval. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 125

126 DEFINITION OF LIMITS AT INFINITY Let L be a real number. 1. The statement lim x f x L means that for each 0 M 0, such that f x L whenever x M. 2. The statement lim x f x L means that for each 0 N 0, such that f x L whenever x N. there exists an there exists an DEFINITION OF A HORIZONTAL ASYMPTOTE The line y L is a horizontal asymptote of the graph of f if lim f x L or lim f x L. x x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 126

127 THEOREM: LIMITS AT INFINITY If r is a positive rational number, then c lim 0 x r. x r Furthermore, if x is defined when x 0, then c lim 0 x r. x DEFINITION OF INFINITE LIMITS AT INFINITY Let f be a function defined on the interval a,. 1. The statement lim f x x means that for each M 0 corresponding number N 0, such that 2. The statement lim f x x corresponding number 0 there is a f x M whenever x N. means that for each M 0 N, such that there is a f x M whenever x N. GUIDELINES FOR FINDING LIMITS AT +/- INFINITY 1. If the degree of the numerator is less than the degree of the denominator, then the limit of the rational function is. 2. If the degree of the numerator is equal to the degree of the denominator, then the limit of the rational function is the of the leading. 3. If the degree of the numerator is greater than the degree of the denominator, then the limit of the rational function is plus or minus infinity, hence it does not. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 127

128 Example 1: Find the limit. a. 5 x lim x x 3 b. x 3 2 lim x 2 2 x 1 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 128

129 c. lim x x x d. 1 lim cos x x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 129

130 3x 1 e. lim x 2 x x x f. lim x 2 x 1 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 130

131 Example 2: Sketch the graph of the equation using extrema, intercepts, symmetry, and asymptotes. Then use a graphing utility to verify your result. f x a. 2 x 1 x 2 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 131

132 h x 2x b. 2 3x 1 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 132

133 3.6: A Summary of Curve Sketching When you finish your homework you should be able to π Curve sketch using methods from Calculus Example 1: Sketch the graph of the equation by hand. If a particular characteristic of the graph does not occur, write none. f x 2 x x 1 a. Intercepts (write as ordered pairs) i. x-intercept: ii. y-intercept: b. Vertical Asymptote(s) c. Behavior at vertical asymptote(s) d. Horizontal Asymptote(s) CREATED BY SHANNON MYERS (FORMERLY GRACEY) 133

134 e. Run the test for increasing/decreasing intervals i. f is increasing on ii. f is decreasing on f. Find the ordered pairs where relative extrema occur. i. Relative minima: ii. Relative maxima: CREATED BY SHANNON MYERS (FORMERLY GRACEY) 134

135 g. Test for concavity. i. f is concave upward on ii. f is concave downward on h. Find the points of inflection. i. Sketch the graph by hand. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 135

136 Example 2: Sketch the graph of the equation by hand. If a particular characteristic of the graph does not occur, write none. f x 2 2x 5x 5 x 2 a. Intercepts (write as ordered pairs) i. x-intercept: ii. y-intercept: b. Vertical Asymptote(s) c. Behavior at vertical asymptote(s) d. Horizontal Asymptote(s) CREATED BY SHANNON MYERS (FORMERLY GRACEY) 136

137 e. Run the test for increasing/decreasing intervals i. f is increasing on ii. f is decreasing on f. Find the ordered pairs where relative extrema occur. i. Relative minima: ii. Relative maxima: CREATED BY SHANNON MYERS (FORMERLY GRACEY) 137

138 g. Test for concavity. i. f is concave upward on ii. f is concave downward on h. Find the points of inflection. i. Sketch the graph by hand. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 138

139 Example 3: Sketch the graph of the equation by hand. If a particular characteristic of the graph does not occur, write none /3 f x x a. Intercepts (write as ordered pairs) i. x-intercept: ii. y-intercept: b. Vertical Asymptote(s) c. Behavior at vertical asymptote(s) d. Horizontal Asymptote(s) CREATED BY SHANNON MYERS (FORMERLY GRACEY) 139

140 e. Run the test for increasing/decreasing intervals i. f is increasing on ii. f is decreasing on f. Find the ordered pairs where relative extrema occur. i. Relative minima: ii. Relative maxima: CREATED BY SHANNON MYERS (FORMERLY GRACEY) 140

141 g. Test for concavity. i. f is concave upward on ii. f is concave downward on h. Find the points of inflection. i. Sketch the graph by hand. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 141

142 Example 4: Sketch the graph of the equation by hand. If a particular characteristic of the graph does not occur, write none. 2cos, 0, 2 f x x x a. Intercepts (write as ordered pairs) i. x-intercept: ii. y-intercept: b. Vertical Asymptote(s) c. Behavior at vertical asymptote(s) d. Horizontal Asymptote(s) CREATED BY SHANNON MYERS (FORMERLY GRACEY) 142

143 e. Run the test for increasing/decreasing intervals i. f is increasing on ii. f is decreasing on f. Find the ordered pairs where relative extrema occur. i. Relative minima: ii. Relative maxima: CREATED BY SHANNON MYERS (FORMERLY GRACEY) 143

144 g. Test for concavity. i. f is concave upward on ii. f is concave downward on h. Find the points of inflection. i. Sketch the graph by hand. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 144

145 3.7: Optimization Examples When you finish your homework you should be able to Solve applied minimum and maximum problems Guidelines for Solving Applied Minimum and Maximum Problems 1. Analyze: Identify all given quantities and all quantities to be determined. MAKE A SKETCH!!! 2. Write a primary equation for the quantity that is to be maximized or minimized. 3. Reduce the primary equation to one having a single independent variable. You may need to use secondary equations relating the independent variables of the primary equation. 4. Determine the feasible domain of the primary equation. 5. Optimize (Find zeros of the critical numbers) 6. Verify (Use first or second derivative test) 7. Find all maximum or minimum values by back substitution. 8. State your conclusion in words (Does your conclusion make sense). CREATED BY SHANNON MYERS (FORMERLY GRACEY) 145

146 1. Find two positive numbers that satisfy the following requirements: The sum of the first number squared and the second is 27 and the product is a maximum. Analyze: Primary Equation: Feasible Domain: Reduce to 1 variable: CREATED BY SHANNON MYERS (FORMERLY GRACEY) 146

147 Optimize and Verify: Conclusion: CREATED BY SHANNON MYERS (FORMERLY GRACEY) 147

148 2. A woman has two dogs that do not get along. She has a theory that if they are housed in kennels right next to each other, they ll get used to each other. She has 200 feet of fencing with which to enclose two adjacent rectangular kennels. What dimensions should be used so that the enclosed area will be a maximum? Analyze: Primary Equation: Feasible Domain: Reduce to 1 variable: CREATED BY SHANNON MYERS (FORMERLY GRACEY) 148

149 Optimize and Verify: Conclusion: CREATED BY SHANNON MYERS (FORMERLY GRACEY) 149

150 2 3. A rectangle is bounded by the x-axis and the semi-circle y 25 x. What length and width should the rectangle have so that its area is a maximum? Analyze: y 5 Primary Equation: Reduce to 1 variable: Feasible Domain: -5 5 x -5 Optimize and Verify: Conclusion: CREATED BY SHANNON MYERS (FORMERLY GRACEY) 150

151 4. The US Post Office will accept rectangular boxes only if the sum of the length and girth (twice the width plus twice the height) is at most 72 inches. What are the dimensions of the box of maximum volume the Post Office will accept? (You may assume that the width and height are equal.) Analyze: Primary Equation: Feasible Domain: Reduce to 1 variable: CREATED BY SHANNON MYERS (FORMERLY GRACEY) 151

152 Optimize and Verify: Conclusion: CREATED BY SHANNON MYERS (FORMERLY GRACEY) 152

153 3.9: Differentials When you finish your homework you should be able to Understand the concept of a tangent line approximation Compare the value of the differential, dy, with the actual change in y, y. Estimate a propagated error using a differential. Find the differential of a function using differentiation formulas. Let f be a function that is at c. The equation for the tangent line at point (c, f(c)) is: or We call this the tangent line approximation (or linear approximation) of f at c. Example 1. Find a tangent line approximation T(x) to the graph () =, at (2,4). Use a graphing calculator to complete the chart below: x f x x T x So at values to, provides a good of. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 153

154 When the tangent line to the graph of f at the point (c,f(c)) = () + ()( ) is used to approximate the graph of f, (x c) is called the change in x, and is denoted by. When is small, the change in y (or ) can be approximated by: = ( + ) () We denote, as dx and call dx the differential of x. The expression () is denoted dy and is called the differential of y. Example 2. Let f(x) = 6 2x 2 Compare and dy at x = -2 and = = 0.1. y x -5 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 154

155 Error Propagation Calculating Differentials: Example 3: Find the differentials Derivative Differential 1. = 2. = 3. = 2 4. = 1 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 155

156 Example 4. The measurements of the base and altitude of a triangle are found to be 36 and 50 centimeters, respectively. The possible error in each measurement is 0.25 centimeter. Use differentials to approximate the possible propagated error in computing the area of the triangle. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 156

157 Example 5. A surveyor standing 50 feet from the base of a large tree measures the angle of elevation to the top of the tree as How accurately must the angle be measured if the percent error in estimating the height of the tree is to be less than 6%? CREATED BY SHANNON MYERS (FORMERLY GRACEY) 157

158 Differentials can be used to approximate function values. To do this, we use the formula: ( + ) () + = () + () Example 6: Use differentials to approximate the following: a. 28 b. sin (0.1) CREATED BY SHANNON MYERS (FORMERLY GRACEY) 158

159 4.1: Antiderivatives When you finish your homework you should be able to π Write the general solution of a differential equation π Use indefinite integral notation for antiderivatives π Use basic integration rules to find antiderivatives π Find a particular solution of a differential equation Warm-up: For each derivative, describe an original function F. a. F x 2x F x sec x d. 2 3 b. F x x e. F x sin x F x 1 x c. 2 f. F x 6 DEFINITION OF ANTIDERIVATIVE A function F is an antiderivative of f on an interval I if F x f x x in I. for all Why is F called an antiderivative of f, rather than the antiderivative of f? CREATED BY SHANNON MYERS (FORMERLY GRACEY) 159

160 THEOREM: REPRESENTATION OF ANTIDERIVATIVES If F is an antiderivative of f on an interval I, then G is an antiderivative of f on the interval I if and only if G is of the form G x F x C, for all x in I where C is a constant. NOTATION: Example 1: Verify the statement by showing that the derivative of the right side equals the integrand of the left side x 2 2 2x dx x 2x C Example 2: Find the general solution of the differential equation. a. dy dx 3 2x b. dr d CREATED BY SHANNON MYERS (FORMERLY GRACEY) 160

161 BASIC INTEGRATION RULES Differentiation Formula d C dx d kx dx d kf x dx d f x g x dx Integration Formula 0dx kdx kf x dx f x g x dx d x n dx d sin x dx d cos x dx d tan x dx d sec x dx d cot x dx d csc x dx n x dx cos xdx sin xdx 2 sec xdx sec x tan xdx 2 csc xdx csc xcot xdx CREATED BY SHANNON MYERS (FORMERLY GRACEY) 161

162 Example 3: Find the indefinite integral and check the result by differentiation. a. 16 x dx b. x 5 3 2x dx x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 162

163 c. 3x 4 3 dx d. 1 u udu CREATED BY SHANNON MYERS (FORMERLY GRACEY) 163

164 e. sec tan sec t t t dt 2 f. 4 csc d CREATED BY SHANNON MYERS (FORMERLY GRACEY) 164

165 g. 1 1 sin x dx INITIAL CONDITIONS AND PARTICULAR SOLUTIONS You have already seen that the equation y f xdx has solutions, each differing from each other by a. This means that the graphs of any two of f are translations of each other. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 165

166 In many applications of integration, you are given enough information to determine a solution. To do this, you need only know the value of y F x for one value of x. This information is called an condition. How do the following differ? 2 x versus y x 2 How about: y 2 x versus dy dx x 2 Example 4: Solve the differential equation. 6, 0 1 a. g x x 2 g CREATED BY SHANNON MYERS (FORMERLY GRACEY) 166

167 b. f x x f f sin, 0 1, 0 6 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 167

168 4.2: Area When you finish your homework you should be able to π Use Sigma Notation to Write and evaluate a Sum π Understand the Concept of Area π Approximate the Area of a Plane Region π Find the Area of a Plane Region Using Limits Warm-up: Evaluate the following limits. a. 2 25x 5x 3 lim x 2 x b x 6x lim 10 x 3 3x SIGMA NOTATION The sum of n terms a, 1 a is written as i index of summation,, n bounds of summation are n and 1. Example 1: Find the sum. n a a a a an where i is the i1 a i is the ith term of the sum, and the upper and lower a. 5 i b. i1 4 2 i c. i1 3 i1 i CREATED BY SHANNON MYERS (FORMERLY GRACEY) 168

169 Summation Properties 1. n n n n n kai k ai 2. ai bi ai bi i1 i1 i1 i1 i1 Example 2: Find the sum. a b. 1 i i1 i1 Theorem: Summation Formulas (These formulas will be provided for the exam) n n nn 1 1. c cn 2. i 2 i1 i n 1 n n n 2 i 4. i1 6 n i1 i 3 n 2 n Example 3: Evaluate the following sums. n i i i1 2. n i1 6i 4i 3 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 169

170 Area In Euclidean geometry, the simplest type of plane region is a. The definition for the area of a rectangle is.from this definition, you can develop formulas for the areas of many other plane regions such as triangles. To determine the area of a triangle, you can form a whose area is that of the. Once you know how to find the area of a triangle, you can determine the area of any by subdividing the polygon into regions. Finding the areas of regions other than polygons is more difficult. The ancient Greeks were able to determine formulas for the areas of some general regions by the method. Essentially, the method is a process in which the area is between two polygons one in the region and one about the region. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 170

171 Theorem: Limits of the Lower and Upper Sums Let f be continuous and nonnegative on the interval a, b. The limits as n of both the lower and upper sums exist and are equal to each other. That is, where lim lim s n f m x n n i 1 lim n n n i 1 lim S n n b a x and i n i i f M x f m and maximum values of f on the subinterval. f M are the minimum and i CREATED BY SHANNON MYERS (FORMERLY GRACEY) 171

172 Example 4: a. Find the upper sum from x 2 to x 6. b. Find the lower sum from x 2 to x 6. fx = 5-x-4 2 fx = 5-x A: (2.00, 1.00) B: (2.50, 2.75) C: (3.00, 4.00) D: (3.50, 4.75) D E F 5 A: (2.00, 1.00) B: (2.50, 2.75) C: (3.00, 4.00) D: (3.50, 4.75) D E F 4 E: (4.00, 5.00) F: (4.49, 4.75) C G 4 E: (4.00, 5.00) F: (4.49, 4.75) C G G: (5.00, 4.00) G: (5.00, 4.00) 3 H: (5.50, 2.75) I: (6.00, 0.99) 3 H: (5.50, 2.75) I: (6.00, 0.99) B H B H A I 1 A I CREATED BY SHANNON MYERS (FORMERLY GRACEY) 172

173 Definition of an Area in the Plane Let f be continuous and nonnegative on the interval a, b. The area of the region bounded by the graph of f, the x-axis, and the vertical lines x a and x b is n lim f c x, x c x i i i i n where i 1 Area 1 b a x, n right endpoint: c i a i x, left endpoint: 1 c a i x 3 Example 5: Find the area of the region bounded by the graph f x x and the vertical lines x 0 and x 1. i, the x-axis, CREATED BY SHANNON MYERS (FORMERLY GRACEY) 173

174 4.3: Riemann Sums and Definite Integrals When you finish your homework you should be able to Understand the definition of a Riemann sum. Evaluate a definite integral using limits. Evaluate a definite integral using properties of definite integrals. n i1 c nc n i1 i n n 1 2 n i1 i 2 12n 1 n n 6 n i1 i 3 n 2 n Evaluate lim f ci n xi n over the region bounded by the graphs of i 1 3 f x x, y 0, x 0, x 1. Hint: Let width of each interval is i n i xi 3 3. n c i 3 i 3 and recall that the n CREATED BY SHANNON MYERS (FORMERLY GRACEY) 174

175 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 175

176 is called the of, the width of the largest subinterval. As,. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 176

177 2. Evaluate the definite integral by the limit definition x 1 dx CREATED BY SHANNON MYERS (FORMERLY GRACEY) 177

178 Theorem 4.4 Continuity Implies Integrability If a function is on the closed interval, then is on [a,b]. That is exists. 3. Write the limit as a definite integral on the interval a, b where c i is any point on the ith interval. n a. lim 8ci 15 xi, 2, 6 0 i 1 n b. lim 5c 2 i ci 2 xi, 0,12 0 i 1 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 178

179 4. Sketch the region whose area is given by the definite integral. Then use a geometric formula to evaluate the integral. a xdx b x dx CREATED BY SHANNON MYERS (FORMERLY GRACEY) 179

180 Given f xdx 4 and 6 0 f a. 0 x dx f x dx 1, evaluate 3 3 f b. 6 x dx 3 f c. 3 x dx CREATED BY SHANNON MYERS (FORMERLY GRACEY) 180

181 4.4: The Fundamental Theorem of Calculus When you finish your homework you should be able to Evaluate a definite integral using the Fundamental Theorem of Calculus. Understand and use the Mean Value Theorem for Integrals. Find the average value of a function over a closed interval. Understand and use the Second Fundamental Theorem of Calculus. Understand and use the Net Change Theorem. Theorem: The Fundamental Theorem of Calculus If a function f is continuous on the closed interval a, b and F is an antiderivative of f on the interval a, b, then b a f x dx F b F a CREATED BY SHANNON MYERS (FORMERLY GRACEY) 181

182 Guidelines for Using the Fundamental Theorem of Calculus Helpful hints: 1. Provided you can find an antiderivative of f, you now have a way to evaluate a definite integral without having to use the limit of a sum. 2. When applying the Fundamental Theorem of Calculus, the following notation is convenient: b a f x dx F x F b xb xa F a 3. It is not necessary to include a constant of integration C in the antiderivative because b a f x dx F x C xb xa F b C F a C F b F a Rational Functions must be written as the sum/difference of functions (neg. exponents are ok) 2 b 5 x x 2 dx a 2 = x b (5 1 2 ) a x x dx Products of functions (not a constant times a function) must be multiplied out. b ( x 3)(2x 1) dx a CREATED BY SHANNON MYERS (FORMERLY GRACEY) 182

183 Trig functions: Use identities try to change everything to sines and cosines Pythagorean Conjugates - generate a difference of squares situation so that you can change the denominator from 2 terms to 1. b a 1 dx 1 sin x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 183

184 Example 1. Evaluate the definite integral. a dx f x F x a b F b F a 6 2 b. 2x 1 1 dx f x F x a b F b F a CREATED BY SHANNON MYERS (FORMERLY GRACEY) 184

185 f t F t 2 c. 0 2 t tdt a b F b F a d. v 1 3 dv f v F v a b F b F a CREATED BY SHANNON MYERS (FORMERLY GRACEY) 185

186 Example 2: Evaluate the definite integral. a dx 1 cos x f x F x a b F b F a CREATED BY SHANNON MYERS (FORMERLY GRACEY) 186

187 b. 6 ( x 8) dx f x F x a b F b F a Example 3. Find the area of the region bounded by the graphs of the equations. f x F x y x x x y 3 1, 0, 8, 0. a b F b F a CREATED BY SHANNON MYERS (FORMERLY GRACEY) 187

188 THE MEAN VALUE THEOREM FOR INTEGRALS If f is continuous on the closed interval a, b, then there exists a number c in the closed interval a, b such that b a f x dx f c b a Definition of the Average Value of a Function on an Interval If f is integrable on the closed interval a, b, then the average value of f on the interval is b 1 f b a a x dx If you isolate in the for, the other side of the is the of the function. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 188

189 Example 3. Find the value(s) of c guaranteed by the Mean Value Theorem for f Integrals for the function x cos x,, 3 3 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 189

190 Example 4. Find the average value of the function f x 2 4 x 1 2, 1,3 and all the values of x in the interval for which the function equals its average value. x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 190

191 The second fundamental theorem of calculus: Consider x 1 F x 3 t dt CREATED BY SHANNON MYERS (FORMERLY GRACEY) 191

192 d dx a x f t dt f x Example 5. a. Find d dx x ( 2 t 7) 0 dt b. Find d dx x t t 6 dt CREATED BY SHANNON MYERS (FORMERLY GRACEY) 192

193 d dx ug x a f t dt c. Find d dx 1 x 2 ln tdt CREATED BY SHANNON MYERS (FORMERLY GRACEY) 193

194 d. Find d dx cos x t dt The Net Change Theorem: Example 6: At 1:00PM, oil begins leaking from a tank at a rate of gallons per hour. a. How much oil is lost from 1:00PM-4:00PM? t b. How much oil is lost from 4:00PM-7:00PM? c. What did you notice? CREATED BY SHANNON MYERS (FORMERLY GRACEY) 194

195 Example 7. Consider the integral cos xdx 0 What if we wanted to find the total area? CREATED BY SHANNON MYERS (FORMERLY GRACEY) 195

196 4.5: Integration by Substitution When you finish your homework you should be able to Use pattern recognition to find an indefinite integral. Use a change of variables to find an indefinite integral. Use the General Power Rule for Integration to find an indefinite integral. Use a change of variables to evaluate a definite integral. Evaluate a definite integral involving an even or odd function. Warm-up: Find the indefinite integral or evaluate the definite integral. a. sec x tan xdx b. sin 2 x cos 2 x dx c x 7x 6 3 x 2 dx CREATED BY SHANNON MYERS (FORMERLY GRACEY) 196

197 When trying to integrate functions, we can use a method called of, or, as we gain ore experience, we can use recognition to integrate the composite function without as much extra work. Just like with the Chain Rule in differentiation, the more you, the easier it is to the derivative without taking the time to write out all the steps. Antidifferentiation of a Composite Function Let g be a function whose range is an interval I, and let f be a function that is continuous on I. If g is differentiable on its domain and F is an antiderivative of f on I, then f ( g( x)) g '( x) dx F( g( x)) C By substitution, if we let u g( x) then du g '( x) and f ( u) du F( u) C In order to use u-substitution, you must recognize that you are trying to integrate a composite function. It is important for you to be able to decompose the composite function. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 197

198 f g x g x dx F x C f u du F u C Example 1. Find the indefinite integral. a. 2 2 ( x 3) (2 ) x dx b x x dx CREATED BY SHANNON MYERS (FORMERLY GRACEY) 198

199 f g x g x dx F x C f u du F u C Many integrands contain the essential part of g '( x ) but are missing a constant multiple. How will we deal with that when it happens? Example 2. Find the indefinite integral. 2 3 x sin( x ) dx CREATED BY SHANNON MYERS (FORMERLY GRACEY) 199

200 Change of Variables We can be more formal and rewrite the integral in terms of another variable (usually we use the variable u and therefore we refer to this as u-substitution). Guidelines for Making a Change of Variables 1. Choose a substitution u. Usually, it is best to choose the inner part of a composite function, such as a quantity raised to a power. d dx 2. Compute u du dx. 3. Rewrite the integral in terms of the variable u. 4. Find the resulting integral in terms of u. 5. Back substitute u by the original expression to obtain an antiderivative in terms of x (or whatever your original variable was). 6. Check your answers by differentiating. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 200

201 f g x g x dx F g x C f u du F u C Example 3. Evaluate the following: a. t 1dt b. t t 1 dt CREATED BY SHANNON MYERS (FORMERLY GRACEY) 201

202 f g x g x dx F g x C f u du F u C c. x 5 x x 5 dx CREATED BY SHANNON MYERS (FORMERLY GRACEY) 202

203 f g x g x dx F g x C f u du F u C d. 2 sin 2x cos 2 xdx e. 2 sin 2xdx CREATED BY SHANNON MYERS (FORMERLY GRACEY) 203

204 Shifting Rule for Integration: To evaluate f ( x a) dx first evaluate f ( u) du foru :, then substitute x a There are f ( x a) dx F( x a) C, where F( u) f ( u) du. substitutions that are so useful, that they are worth noting explicitly. We have already used them in the previous examples: Scaling Rule for Integration: To evaluate f ( bx) dx evaluate f ( u) du, divide by b and substitute for bx foru : 1 f ( bx) dx F( bx) C, where F( u) f ( u) du. b CREATED BY SHANNON MYERS (FORMERLY GRACEY) 204

205 Theorem: General Power Rule for Integration: If g is a differentiable function of x, then n1 n [ g( x)] [ g( x)] g '( x) dx C, n 1. n 1 Equivalently, if u g( x) then n1 n u ( u) du C, n 1. n 1 Theorem: Change of Variable for Definite Integrals If the function u g( x) has a continuous derivative on the closed interval [a,b] and f is continuous on the range of g, then xb ug ( b) f ( g( x)) g '( x) dx f ( u) du xa ug ( a) Example 5: Evaluate the definite integrals: a x ( x 5) 2 2 dx CREATED BY SHANNON MYERS (FORMERLY GRACEY) 205

206 b. 6 0 x tan x dx Theorem: Integration of Even and Odd Functions Let f be integrable on the closed interval [a,-a]. 1. If f is an even function, then a a f ( x) dx 2 f ( x) dx a 0 2. If f is an odd function, then f ( x) dx 0 a a CREATED BY SHANNON MYERS (FORMERLY GRACEY) 206

207 Example 5: Evaluate the definite integral. a. 2 2 sin x cos xdx b. 3 2 x x dx CREATED BY SHANNON MYERS (FORMERLY GRACEY) 207

208 5.1: The Natural Logarithmic Function: Differentiation When you are done with your homework you should be able to Develop and use properties of the natural logarithmic function Understand the definition of the number e Find derivatives involving the natural logarithmic function Warm-up: 1. Use the limit definition of the derivative to find the derivative of f x with respect to x. 3 x 2. Graph y ln x. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 208

209 DEFINITION OF THE LOGARITHMIC FUNCTION BASE e If a is a positive real number ( a 1) and x is any positive real number, then the logarithmic function to the base e is defined as log x a. The number e : The number e can be defined as a ; specifically, lim1 v 1 The natural logarithmic function and the natural exponential function are of each other. v0 v. PROPERTIES OF INVERSE FUNCTIONS x 1. y e iff x 3. ln e, for all x 2. e ln x, for x 0 PROPERTIES OF NATURAL LOGS 1. ln1 2. ln e 3. ln xy 4. x ln y n 5. ln x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 209

210 The inverse relationship between the natural logarithmic function and the natural exponential function can summarized as follows: Example 1: Condense the following logarithmic expressions. a. ln x 8 ln x 2 5ln x 1 b. ln x 8ln z ln y 2 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 210

211 Example 2: Expand the following logarithmic expressions. a. ln 4 2 x 1 2x 5 3 b. 1 cos x cos 2x 5 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 211

212 Example 3: Solve the following equations. Give the exact result and then round to 3 decimal places. a. x x ln 2 ln 2 16 b. 1 30e 2 3t CREATED BY SHANNON MYERS (FORMERLY GRACEY) 212

213 DERIVATIVE OF THE NATURAL LOGARITHMIC FUNCTION CREATED BY SHANNON MYERS (FORMERLY GRACEY) 213

214 THEOREM: DERIVATIVES OF THE NATURAL LOGARITHMIC FUNCTION Let u be a differentiable function of x such that u 0 d dx d dx 1. ln x 2. ln u, then Example 4: Find the derivative with respect to x. a. y ln 5x b. f x ln 1 2x c. f x ln x x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 214

215 d. y ln 2 x 1 1 x e. 2 y ln cos x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 215

216 f. 2 ln xy x y 10 THEOREM: DERIVATIVE INVOLVING ABSOLUTE VALUE Let u be a differentiable function of x such that u 0 Proof: d ln u dx, then CREATED BY SHANNON MYERS (FORMERLY GRACEY) 216

217 Example 5: Find the derivative of the function with respect to x. f x ln sec x tan x Example 6: Differentiate the following functions with respect to x. a. f x x 2 x 1 x 2 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 217

218 b. y x x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 218

219 Example 7: Find an equation of the tangent line of the function 2 f x sin 2x ln x 1,0. at the point CREATED BY SHANNON MYERS (FORMERLY GRACEY) 219

220 5.2: The Natural Logarithmic Function: Integration When you are done with your homework you should be able to Use the Log Rule for Integration to integrate a rational function Integrate trigonometric functions Warm-up: 1. Differentiate the following functions with respect to x. a. y x ln 5x b. ln xy ln x y. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 220

221 THEOREM: LOG RULE FOR INTEGRATION Let u be a differentiable function of x dx ln x C 2. x 1 dx ln u C u Example 1: Find or evaluate the integral. a. 10 dx x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 221

222 b. x 2 5 x 3 dx x c. dx 2 1 x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 222

223 d. e e 2 dx x ln x e. 1 1 ln 2 e x dx x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 223

224 f. x 1 1 x dx g. x 3 6x 20 dx x 5 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 224

225 h. tand i. cot d CREATED BY SHANNON MYERS (FORMERLY GRACEY) 225

226 j. secd k. cscd CREATED BY SHANNON MYERS (FORMERLY GRACEY) 226

227 INTEGRALS OF THE SIX BASIC TRIGONOMETRIC FUNCTIONS sin udu cos udu tan udu cot udu sec udu csc udu Example 2: Solve the differential equation. a. y x 1 x 1 b. 2 r tan CREATED BY SHANNON MYERS (FORMERLY GRACEY) 227

228 Example 3: The demand equation for a product is average price on the interval 40 x 50. p 90, x Find the CREATED BY SHANNON MYERS (FORMERLY GRACEY) 228

229 5.3: Inverse Functions When you are done with your homework you should be able to Verify that one function is the inverse of another function Determine whether a function has an inverse function Find the derivative of an inverse function Warm-Up: 5 1. Use the Horizontal Line Test to show that f ( x) x 3 is one-to-one. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 229

230 2. Let 1 2x g( x) and h( x) x x x 2 find g( h( x )) Recall from Precalculus: CREATED BY SHANNON MYERS (FORMERLY GRACEY) 230

231 Important Observations about inverse functions: 1. If g is the function of f, then is the inverse function of. 2. The of 1 f 1 f is equal to the of. is equal to the of, and the of 3. If a function has an inverse, the inverse is. 4. f 1 ( f ( x)) and f f x 1 ( ( )). 5. A function has an inverse if and only if it is - -. Proof: CREATED BY SHANNON MYERS (FORMERLY GRACEY) 231

232 Example 1: Graph each function and verify that the following functions are inverses of each other. 2 f ( x) 16 x, x 0 g( x) 16 x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 232

233 Example 2: Find 1 f of 3x 2 f ( x). 1 x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 233

234 Example 3: Prove that 3 f ( x) x 2x 3 has an inverse function (you do not need to find it). Derivative of an Inverse Function: Proof: CREATED BY SHANNON MYERS (FORMERLY GRACEY) 234

235 Proof: CREATED BY SHANNON MYERS (FORMERLY GRACEY) 235

236 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 236

237 Example 4. Let f x x x 3 ( ) 2 3 a. What is the value of f 1 ( x) when x = 0? b. What is the value of f ( x) when x = 0? c. What is the value of 1 ( f ) ( x) when x = 0? d. What do you notice about the slopes of f and f 1 Graphs of inverse functions have slopes. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 237

238 5.4: Exponential Functions: Differentiation and Integration When you are done with your homework you should be able to Develop properties of the natural exponential function Differentiate natural exponential functions Integrate natural exponential functions Warm-up: 1. Differentiate the following functions with respect to x. a. y x 5x x f x ln e. b. cos2 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 238

239 DEFINITION: THE NATURAL EXPONENTIAL FUNCTION The inverse function of the natural logarithmic function ln the natural exponential function and is denoted by f x x is called That is, The inverse relationship between the natural logarithmic function and the natural exponential function can summarized as follows: Example 1: Solve the following equations. Give the exact result and then round to 3 decimal places. a. ln 6x e 20 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 239

240 5000 b. 2 x 1 e 2 c. ln 8x 3 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 240

241 d. e 2 x x 2e 8 0 f x 2e x Example 2: Sketch the graph of 1 calculator. without using your graphing CREATED BY SHANNON MYERS (FORMERLY GRACEY) 241

242 DERIVATIVE OF THE NATURAL EXPONENTIAL FUNCTION THEOREM: DERIVATIVES OF THE NATURAL EXPONENTIAL FUNCTION Let u be a differentiable function of x. 1. d e x e x dx d 2. e u dx Example 3: Find the derivative with respect to x. a. y 5 x e 3 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 242

243 3x b. f x xe c. y 1 e ln 1 e x x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 243

244 d. e y x e 2 x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 244

245 e. e x y xy x e f. ln 1 F x t dt 0 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 245

246 Example 4: Find an equation of the tangent line of the function 1 at the point 1,1. ln xy x y e CREATED BY SHANNON MYERS (FORMERLY GRACEY) 246

247 Example 5: Find the extrema and points of inflection of the function g x 1 x3 2 2 e. 2 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 247

248 THEOREM: INTEGRATION RULES FOR NATURAL EXPONENTIAL FUNCTIONS Let u be a differentiable function of x. x u 1. e dx 2. e du Example 6: Find the indefinite integrals and evaluate the definite integrals. a. e 12x dx b. x e x dx CREATED BY SHANNON MYERS (FORMERLY GRACEY) 248

249 c. e 1 e 2x 2x dx d. e 2x x 2e 1 dx x e CREATED BY SHANNON MYERS (FORMERLY GRACEY) 249

250 e. e sec 2xdx tan 2x 2 f x e e x dx CREATED BY SHANNON MYERS (FORMERLY GRACEY) 250

251 Example 7: Solve the differential equation. dy dx x x 2 e e dx Example 8: The median waiting time (in minutes) for people waiting for service in a convenience store is given by the solution of the equation the equation. 0 x 0.3e 0.3t 1 dt 2. Solve CREATED BY SHANNON MYERS (FORMERLY GRACEY) 251

252 5.5: Bases other than e and Applications When you are done with your homework you should be able to Define exponential functions that have bases other than e Differentiate and integrate exponential functions that have bases other than e Use exponential functions to model compound interest and exponential growth Warm-up: 1. Evaluate the expression without using a calculator. a. log b. log c. log Sketch the graph of f x 2 x without using your graphing calculator. DEFINITION: EXPONENTIAL FUNCTION TO BASE a If a is a positive real number ( a 1) and x is any real number, then the exponential function to the base a is denoted by x a lnax e. x a, then y 1 1 If 1 is a constant function. x a and is defined by CREATED BY SHANNON MYERS (FORMERLY GRACEY) 252

253 DEFINITION OF LOGARITHMIC FUNCTION TO BASE a If a is a positive real number ( a 1) and x is any positive real number, then the logarithmic function to the base a is denoted by log a x and is defined as log a x 1 ln ln a x. PROPERTIES OF LOGS 1. log a1 2. log a a 3. log xy a 4. log x n a PROPERTIES OF INVERSE FUNCTIONS x 1. y a iff 2. a log a a, for x 0 x 3. log, for all x a a THEOREM: DERIVATIVES FOR BASES OTHER THAN e Let a be a positive real number ( a 1) and let u be a differentiable function of x. d 1. a x dx d 2. a u dx d dx d dx 3. log a x 4. log a u CREATED BY SHANNON MYERS (FORMERLY GRACEY) 253

254 Example 1: Differentiate the following functions with respect to x. y a. 2 x 2 8 b. f x x3 x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 254

255 c. f t 2 4 t t d. g t log t CREATED BY SHANNON MYERS (FORMERLY GRACEY) 255

256 e. 2 1 y log x x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 256

257 ANTIDERIVATIVES OF EXPONENTIAL FUNCTIONS, BASE a 1. x ln a x a dx e dx 2. x 1 x a dx a C ln a Example 2: Find the indefinite integrals and evaluate the definite integrals. a. 3 x dx b. x x dx CREATED BY SHANNON MYERS (FORMERLY GRACEY) 257

258 c. 5 x dx e d x x dx CREATED BY SHANNON MYERS (FORMERLY GRACEY) 258

259 Example 3: Find the area of the region bounded by the graph of y 0, x 0, and x. y cos 3 x sin x, CREATED BY SHANNON MYERS (FORMERLY GRACEY) 259

260 Example 4: After t years, the value of a car purchased for $25,000 is V t t 3 25, a. Use your graphing calculator to graph the function and determine the value of the car 2 years after it was purchased. b. Find the rates of change of V with respect to t when t 1and t 4. c. Use your graphing calculator to graph V t asymptote of V t and determine the horizontal. Interpret its meaning in the context of the problem. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 260

261 5.6: Inverse Trigonometric Functions: Differentiation When you are done with your homework you should be able to Develop properties of the six inverse trigonometric functions Differentiate an inverse trigonometric function Review the basic differentiation rules for elementary functions Warm-up: Draw the following graphs by hand from,. List the domain and range in interval notation. f x x a. Graph sin Restricted Domain: Range: b. Graph g x arcsin x. Domain: Range: CREATED BY SHANNON MYERS (FORMERLY GRACEY) 261

262 c. Graph f x csc x. Restricted Domain: Range: d. Graph g x arc csc x. Domain: Range: CREATED BY SHANNON MYERS (FORMERLY GRACEY) 262

263 e. Graph f x cos x. Restricted Domain: Range: f. Graph g x arccos x Domain: Range: CREATED BY SHANNON MYERS (FORMERLY GRACEY) 263

264 g. Graph f x sec x. Restricted Domain: Range: h. Graph g x arcsec x. Domain: Range: CREATED BY SHANNON MYERS (FORMERLY GRACEY) 264

265 i. Graph f x tan x. Restricted Domain: Range: j. Graph g x arctan x. Domain: Range: CREATED BY SHANNON MYERS (FORMERLY GRACEY) 265

266 k. Graph f x cot x. Restricted Domain: Range: l. Graph g x arc cot x. Domain: Range: CREATED BY SHANNON MYERS (FORMERLY GRACEY) 266

267 Example 1: Evaluate each function. a. arccot 1 d. arctan 3 b. arcsin 2 2 e. 1 arc cos 2 c. 2 3 arc sec 3 f. arc csc 2 Example 2: Solve the equation for x. arctan 2x 5 1 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 267

268 Example 3: Write the expression in algebraic form. (HINT: Sketch a right triangle) a. secarctan 4x b. cos arcsin x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 268

269 Example 4: Differentiate with respect to x. a. y arcsin x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 269

270 b. y arccos x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 270

271 c. y arctan x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 271

272 d. y arc csc x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 272

273 e. y arcsec x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 273

274 f. y arc cot x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 274

275 What have we found out?! DERIVATIVES OF INVERSE TRIGONOMETRIC FUNCTIONS Let u be a differentiable function of x. d dx arcsin u u 1 u u 1 u u 1 u 1. 2 d dx arccos u 2. 2 d dx arctan u 3. 2 d u arc cot u dx 1 u d u arc secu dx u u d u dx u u 1 arc cscu 2 1 Example 5: Find the derivative of the function. Simplify your result to a single rational expression with positive exponents. f t arcsin t a. 3 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 275

276 b. g x arcsin x arccos x c. y x arctan 2x ln 1 4x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 276

277 d. x y 25arcsin x 25 x 5 2 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 277

278 Example 6: Find an equation of the tangent line to the graph of the function y arccos x at the point, CREATED BY SHANNON MYERS (FORMERLY GRACEY) 278

279 5.7: Inverse Trigonometric Functions: Integration When you are done with your homework you should be able to Integrate functions whose antiderivatives involve inverse trigonometric functions Use the method of completing the square to integrate a function Review the basic integration rules involving elementary functions Warm-up: 1. Differentiate the following functions with respect to x. a. y x arctan x 4 1 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 279

280 b. arctan xy arcsin x y. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 280

281 2. Complete the square. 2 a. 3 4x x b. 2 2x 6x 9 What did you notice about the derivatives of the inverse trigonometric functions? THEOREM: INTEGRALS INVOLVING INVERSE TRIGONOMETRIC FUNCTIONS Let u be a differentiable function of x, and let a 0. du u arcsin C a u a du 1 u arcsec C u u a a a du 1 u arctan C a u a a CREATED BY SHANNON MYERS (FORMERLY GRACEY) 281

282 Example 1: Find the indefinite integral, or evaluate the definite integral. dx a. 2 x x 1 xdx b. 2 x 1 dx c. 2 1 x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 282

283 d. dx xln x e. ln x 2 x dx f. ln xdx CREATED BY SHANNON MYERS (FORMERLY GRACEY) 283

284 Example 2: Find the integral by completing the square. dx a. 2 x 4x 13 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 284

285 dx b. 4 x x 4 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 285

286 2dx c. 2 x 4x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 286

287 2x 5 d. dx 2 x 2x 2 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 287

288 x e x x dx CREATED BY SHANNON MYERS (FORMERLY GRACEY) 288

289 f dx x 1 x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 289

290 g. 2 0 cos x 2 1 sin dx x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 290

291 Example 3: Find the area of the region bound by the graphs of x 4e y, x 0, y 0 and x ln 3. 2x 1 e CREATED BY SHANNON MYERS (FORMERLY GRACEY) 291

292 7.1: Area of a Region Between Two Curves When you finish your homework you should be able to Find the area of a region between two curves using integration. Find the area of a region between intersecting curves using integration. Describe integration as an accumulation process. Consider two functions that are continuous on an interval [a,b]. Also, the graph of g(x) lies below the graph of f(x). We can see that the area of the region between the graphs can be thought of as the area of the region under g subtracted from the area of the region under f. To verify this in general, partition the interval into n subintervals, each of width x. Notice the area of the rectangle formed is A ( height)( width) [ f ( x ) g( x )] x. Because f and g i i i are both continuous on [a,b], so is f g, and the limit exists. So CREATED BY SHANNON MYERS (FORMERLY GRACEY) 292

293 n b Area = lim [ f ( xi ) g( xi )] x = [ f ( x) g( x)] dx n i 1 a Example 1. Find the area bounded by the graphs of x 0, and x 2. y 3 x, 2 y 3x 2x, y x -5 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 293

294 Example 2. Sketch and find the area of the bounded region between the two curves 2 x y x 2 and y 1. 2 y x -5 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 294

295 Example 3. Find the area between the graphs 2 x y y x 2 and. y x -5 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 295

296 Functions of y Now, consider the trick of regarding x as a function of y to make the calculations much simpler. 2 x y y x 2 and. y x -5 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 296

297 f y y 1 and f y y 1. Example 4. Find the area between the graphs y x -5 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 297

298 7.2: Volume: The Disk Method When you are done with your homework, you should be able to π Find the volume of a solid of revolution using the disk method π Find the volume of a solid of revolution using the washer method π Find the volume of a solid with known cross sections The Disk Method When a rejoin in the plane is revolved about a line, the resulting solid is a, and the line is called the. Examples of solids of revolution commonly seen are axles, bottles, funnels, etc. but the simplest solid of revolution is a disk which is formed by revolving a rectangle about an axis adjacent to one side of the rectangle. Volume of a disk = (area of disk)(width of disk) 2 = R w Consider revolving the rectangle about the axis Of revolution. When we do this, a disk is generated whose volume is: CREATED BY SHANNON MYERS (FORMERLY GRACEY) 298

299 2 V= R x Approximating the volume of a solid by N disks of witdh us x and radius R( x ) i 2 gives n i1 [ R( x )] i 2 x And since the approximation gets better as 0 we can define: Volume of a Solid = lim [ R( x )] x = [ R( x)] dx 0 n b 2 2 i i1 a CREATED BY SHANNON MYERS (FORMERLY GRACEY) 299

300 Example 1 The region under the graph of Sketch the resulting solid and find its volume. 2 x on [0,1] is revolved about the x axis. y x -5 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 300

301 We can also use the vertical axis as the axis of revolution. The Washer Method The washer method is a way of finding the volume of a solid of revolution when it has a hole in the middle. Thus named because a disk with a hole in the middle is a washer. To see how we find the volume, we must consider the inner and outer radii. Let s call the outer radius R( x ) and the inner radius r( x ). If we revolve the region about the axis of revolution, the volume of the solid is: CREATED BY SHANNON MYERS (FORMERLY GRACEY) 301

302 b 2 2 V = ([ ( )] [ ( )] ) a R x r x dx CREATED BY SHANNON MYERS (FORMERLY GRACEY) 302

303 Example 2 Sketch and then find the volume of the solid formed by revolving the region bounded by the graphs of y sin x and y x on [0, /2] about the x-axis. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 303

304 Example 3. Find the volume of the solid generated by revolving the region bounded by the graphs of y x y x 2 2, 0, 2 about the a) y-axis, and then b) the line y = 8. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 304

305 Solids with Known Cross Sections We have looked at finding the volume of a solid having a circular cross section where A 2 R But we can generalize this method to solids of any shape as long as we know a formula for the area of an arbitrary cross section. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 305