SECTION 3.2 THE PRODUCT AND QUOTIENT RULES 1 8 3

Transcription

1 SECTION 3.2 THE PRODUCT AND QUOTIENT RULES 8 3 L P f Q L segments L an L 2 to be tangent to the parabola at the transition points P an Q. (See the figure.) To simplify the equations you ecie to place the origin at P.. (a) Suppose the horizontal istance between P an Q is 00 ft. Write equations in a, b, an c that will ensure that the track is smooth at the transition points. (b) Solve the equations in part (a) for a, b, an c to fin a formula for f. ; (c) Plot L, f, an L 2 to verify graphically that the transitions are smooth. () Fin the ifference in elevation between P an Q. 2. The solution in Problem might look smooth, but it might not feel smooth because the piecewise efine function [consisting of L for 0, f for 0 00, an L 2 for 00] oesn t have a continuous secon erivative. So you ecie to improve the esign by using a quaratic function q a 2 b c only on the interval 0 90 an connecting it to the linear functions by means of two cubic functions: t k 3 l 2 m n 0 0 h p 3 q 2 r s CAS (a) Write a system of equations in unknowns that ensure that the functions an their first two erivatives agree at the transition points. (b) Solve the equations in part (a) with a computer algebra system to fin formulas for q, t, an h. (c) Plot, t, q, h, an, an compare with the plot in Problem (c). L L THE PRODUCT AND QUOTIENT RULES The formulas of this section enable us to ifferentiate new functions forme from ol functions by multiplication or ivision. THE PRODUCT RULE Î u Î u Îu Î Îu By analogy with the Sum an Difference Rules, one might be tempte to guess, as Leibniz i three centuries ago, that the erivative of a prouct is the prouct of the erivatives. We can see, however, that this guess is wrong by looking at a particular eample. Let f an t 2. Then the Power Rule gives f an t 2. But ft 3, so ft 3 2. Thus ft f t. The correct formula was iscovere by Leibniz (soon after his false start) an is calle the Prouct Rule. Before stating the Prouct Rule, let s see how we might iscover it. We start by assuming that u f an v t are both positive ifferentiable functions. Then we can interpret the prouct uv as an area of a rectangle (see Figure ). If changes by an amount, then the corresponing changes in u an v are u Îu FIGURE The geometry of the Prouct Rule u f f v t t an the new value of the prouct, u uv v, can be interprete as the area of the large rectangle in Figure (provie that u an v happen to be positive). The change in the area of the rectangle is uv u uv v uv u v v u u v the sum of the three shae areas

2 8 4 CHAPTER 3 DIFFERENTIATION RULES If we ivie by, we get uv u v u v v u N Recall that in Leibniz notation the efinition of a erivative can be written as y y lim l 0 If we now let l 0, we get the erivative of uv: uv uv lim lim l 0 l 0u v u v v u v u u lim v lim l 0 l 0 v lim ulim l 0 l 0 u v u v v 0 2 v u uv u v (Notice that u l 0 as l 0 since f is ifferentiable an therefore continuous.) Although we starte by assuming (for the geometric interpretation) that all the quantities are positive, we notice that Equation is always true. (The algebra is vali whether u, v, u, an v are positive or negative.) So we have prove Equation 2, known as the Prouct Rule, for all ifferentiable functions u an v. N In prime notation: ft ft tf THE PRODUCT RULE If f an t are both ifferentiable, then f t f t t f In wors, the Prouct Rule says that the erivative of a prouct of two functions is the first function times the erivative of the secon function plus the secon function times the erivative of the first function. N Figure 2 shows the graphs of the function f of Eample an its erivative f. Notice that f is positive when f is increasing an negative when f is ecreasing. 3 EXAPLE (a) If f e, fin f. (b) Fin the nth erivative, f n. SOLUTION (a) By the Prouct Rule, we have f e e e e e e fª _3.5 f _ FIGURE 2 (b) Using the Prouct Rule a secon time, we get f e e e e e 2e

3 SECTION 3.2 THE PRODUCT AND QUOTIENT RULES 8 5 Further applications of the Prouct Rule give f 3e f 4 4e In fact, each successive ifferentiation as another term e, so f n ne N In Eample 2, a an b are constants. It is customary in mathematics to use letters near the beginning of the alphabet to represent constants an letters near the en of the alphabet to represent variables. EXAPLE 2 Differentiate the function f t st a bt. SOLUTION Using the Prouct Rule, we have f t st bst t a bt a bt t (st ) st b a bt 2 t 2 a bt 2st a 3bt 2st SOLUTION 2 If we first use the laws of eponents to rewrite irectly without using the Prouct Rule. f t, then we can procee f t ast btst at 2 bt 32 f t 2at bt 2 which is equivalent to the answer given in Solution. Eample 2 shows that it is sometimes easier to simplify a prouct of functions than to use the Prouct Rule. In Eample, however, the Prouct Rule is the only possible metho. EXAPLE 3 If f s t, where t4 2 an t4 3, fin SOLUTION Applying the Prouct Rule, we get f 4. So f [s t] s s t t 2 2 t t [s ] s t t 2s f 4 s4 t4 t s4 2 2 THE QUOTIENT RULE We fin a rule for ifferentiating the quotient of two ifferentiable functions u f an v t in much the same way that we foun the Prouct Rule. If, u, an v change by amounts, u, an v, then the corresponing change in the quotient uv is u v u u v v u v u uv uv v vv v vu uv vv v

4 8 6 CHAPTER 3 DIFFERENTIATION RULES so v u v u uv lim lim l 0 l 0 u v vv v As l 0, v l 0 also, because v t is ifferentiable an therefore continuous. Thus, using the Limit Laws, we get v u v lim l 0 u v u lim l 0 v lim v v l 0 v u u v v 2 N In prime notation: t f tf ft t 2 THE QUOTIENT RULE If f an t are ifferentiable, then f t f f t t t 2 In wors, the Quotient Rule says that the erivative of a quotient is the enominator times the erivative of the numerator minus the numerator times the erivative of the enominator, all ivie by the square of the enominator. The Quotient Rule an the other ifferentiation formulas enable us to compute the erivative of any rational function, as the net eample illustrates. N We can use a graphing evice to check that the answer to Eample 4 is plausible. Figure 3 shows the graphs of the function of Eample 4 an its erivative. Notice that when y grows rapily (near 2), y is large. An when y grows slowly, y is near 0..5 yª _4 4 y _.5 FIGURE 3 EXAPLE 4 Let y 2 2 V. Then y V EXAPLE 5 Fin an equation of the tangent line to the curve y e 2 at the point (, 2e). SOLUTION Accoring to the Quotient Rule, we have y 2 e e e e 2 e

5 SECTION 3.2 THE PRODUCT AND QUOTIENT RULES y= + _ FIGURE 4 y= e 2 So the slope of the tangent line at (, 2e) is y 0 This means that the tangent line at (, 2e) is horizontal an its equation is y 2e. [See Figure 4. Notice that the function is increasing an crosses its tangent line at (, 2e).] NOTE Don t use the Quotient Rule every time you see a quotient. Sometimes it s easier to rewrite a quotient first to put it in a form that is simpler for the purpose of ifferentiation. For instance, although it is possible to ifferentiate the function F 3 2 2s using the Quotient Rule, it is much easier to perform the ivision first an write the function as F before ifferentiating. We summarize the ifferentiation formulas we have learne so far as follows. TABLE OF DIFFERENTIATION FORULAS c 0 n n n e e cf cf ft ft tf f t f t t f tf ft t 2 f t f t 3.2 EXERCISES. Fin the erivative of y 2 3 in two ways: by using the Prouct Rule an by performing the multiplication first. Do your answers agree? 2. Fin the erivative of the function in two ways: by using the Quotient Rule an by simplifying first. Show that your answers are equivalent. Which metho o you prefer? Differentiate y e y e 2 F 3s s 3. f 3 2e 4. t s e 3 7. t 8. f t 2t 2 4 t 2 9. V Yu u 2 u 3 u 5 2u 2. Fy y 2 3 y 4y 5y 3 2. Rt t e t (3 st ) 3. y y 3 2 t 2 2 t 5. y 6. y t 4 3t 2 t 2 7. y r 2 2re r 8. y s ke s

6 8 8 CHAPTER 3 DIFFERENTIATION RULES 9. y v3 2vsv 2 0. z w 32 w ce w v 2. f t 2t 2 2. tt 2 st A 23. f 2 4. f B Ce 2 5. f c Fin f an f Fin an equation of the tangent line to the given curve at the specifie point. 26. f 3., 32. y e y 2,, t st t 3 e e a b c 27. f 4 e 2 8. f 52 e 29. f f 2 3 e Fin equations of the tangent line an normal line to the given curve at the specifie point y 2e, 0, y s,, e 4, (a) If f e, fin f an f. ; (b) Check to see that your answers to part (a) are reasonable by comparing the graphs of f, f, an f (a) If f 2, fin f an f. ; (b) Check to see that your answers to part (a) are reasonable by comparing the graphs of f, f, an f. 4. If f 2, fin f. 42. If t e, fin t n Suppose that f 5, f 5 6, t5 3, an t5 2. Fin the following values. (a) ft5 (b) ft5 (c) tf Suppose that f 2 3, t2 4, f 2 2, an t2 7. Fin h2. (a) h 5f 4t (b) h f t (c) h f t 4 5. If f e t, where t0 2 an t0 5, fin f If h2 4 an h2 3, fin h () h t f If f an t are the functions whose graphs are shown, let u f t an v f t. (a) Fin u. (b) Fin v5. y 3 5. (a) The curve y 2 is calle a witch of aria Agnesi. Fin an equation of the tangent line to this curve at the point (, 2 ). ; (b) Illustrate part (a) by graphing the curve an the tangent line on the same screen. 0 f g 3 6. (a) The curve y 2 is calle a serpentine. Fin an equation of the tangent line to this curve at the point 3, 0.3. ; (b) Illustrate part (a) by graphing the curve an the tangent line on the same screen (a) If f e 3, fin f. ; (b) Check to see that your answer to part (a) is reasonable by comparing the graphs of f an f. 48. Let P FG an Q FG, where F an G are the functions whose graphs are shown. (a) Fin P2. (b) Fin Q7. y F 3 8. (a) If f 2, fin f. ; (b) Check to see that your answer to part (a) is reasonable by comparing the graphs of f an f. 0 G

7 SECTION 3.3 DERIVATIVES OF TRIGONOETRIC FUNCTIONS If t is a ifferentiable function, fin an epression for the erivative of each of the following functions. (a) y t (b) y (c) y t t 50. If f is a ifferentiable function, fin an epression for the erivative of each of the following functions. (a) y 2 f (c) y 2 f (b) () y y f 2 f s 55. write q f p. Then the total revenue earne with selling price p is Rp pf p. (a) What oes it mean to say that f 20 0,000 an f ? (b) Assuming the values in part (a), fin R20 an interpret your answer. (a) Use the Prouct Rule twice to prove that if f, t, an h are ifferentiable, then fth f th fth fth. (b) Taking f t h in part (a), show that f 3 3 f 2 f 5. How many tangent lines to the curve y ) pass through the point, 2? At which points o these tangent lines touch the curve? 52. Fin equations of the tangent lines to the curve y that are parallel to the line 2y In this eercise we estimate the rate at which the total personal income is rising in the Richmon-Petersburg, Virginia, metropolitan area. In 999, the population of this area was 96,400, an the population was increasing at roughly 9200 people per year. The average annual income was $30,593 per capita, an this average was increasing at about $400 per year (a little above the national average of about $225 yearly). Use the Prouct Rule an these figures to estimate the rate at which total personal income was rising in the Richmon-Petersburg area in 999. Eplain the meaning of each term in the Prouct Rule. 54. A manufacturer prouces bolts of a fabric with a fie with. The quantity q of this fabric (measure in yars) that is sol is a function of the selling price p (in ollars per yar), so we can (c) Use part (b) to ifferentiate y e (a) If F f t, where f an t have erivatives of all orers, show that F f t 2f t ft. (b) Fin similar formulas for F an F 4. (c) Guess a formula for. F n 57. Fin epressions for the first five erivatives of f 2 e. Do you see a pattern in these epressions? Guess a formula for f n an prove it using mathematical inuction. 58. (a) If t is ifferentiable, the Reciprocal Rule says that Use the Quotient Rule to prove the Reciprocal Rule. (b) Use the Reciprocal Rule to ifferentiate the function in Eercise 8. (c) Use the Reciprocal Rule to verify that the Power Rule is vali for negative integers, that is, for all positive integers n. t t t 2 n n n N A review of the trigonometric functions is given in Appeni D. 3.3 DERIVATIVES OF TRIGONOETRIC FUNCTIONS Before starting this section, you might nee to review the trigonometric functions. In particular, it is important to remember that when we talk about the function f efine for all real numbers by f sin it is unerstoo that sin means the sine of the angle whose raian measure is. A similar convention hols for the other trigonometric functions cos, tan, csc, sec, an cot. Recall from Section 2.5 that all of the trigonometric functions are continuous at every number in their omains. If we sketch the graph of the function f sin an use the interpretation of f as the slope of the tangent to the sine curve in orer to sketch the graph of f (see Eer-