Basic Differentiation Rules and Rates of Change. The Constant Rule

Transcription

1 460_00.q //04 4:04 PM Page 07 SECTION. Basic Differentiation Rules an Rates of Change 07 Section. The slope of a horizontal line is 0. Basic Differentiation Rules an Rates of Change Fin the erivative of a function using the Constant Rule. Fin the erivative of a function using the Power Rule. Fin the erivative of a function using the Constant Multiple Rule. Fin the erivative of a function using the Sum an Difference Rules. Fin the erivatives of the sine function an of the cosine function. Use erivatives to fin rates of change. The Constant Rule In Section. ou use the limit efinition to fin erivatives. In this an the net two sections ou will be introuce to several ifferentiation rules that allow ou to fin erivatives without the irect use of the limit efinition. The erivative of a constant function is 0. The Constant Rule Figure.4 f() = c NOTE In Figure.4, note that the Constant Rule is equivalent to saing that the slope of a horizontal line is 0. This emonstrates the relationship between slope an erivative. THEOREM. The Constant Rule The erivative of a constant function is 0. That is, if c is a real number, then c 0. Proof Let f c. Then, b the limit efinition of the erivative, c f f f c c EXAMPLE Using the Constant Rule Derivative a. 7 0 b. f 0 f 0 c. st st 0. k, k is constant 0 EXPLORATION Writing a Conjecture Use the efinition of the erivative given in Section. to fin the erivative of each function. What patterns o ou see? Use our results to write a conjecture about the erivative of f n. a. f b. f c. f. f 4 e. f f. f

2 460_00.q //04 4:04 PM Page CHAPTER Differentiation The Power Rule Before proving the net rule, it is important to review the proceure for epaning a binomial. The general binomial epansion for a positive integer n is n n n n nn n... n. is a factor of these terms. This binomial epansion is use in proving a special case of the Power Rule. THEOREM. The Power Rule If n is a rational number, then the function f n is ifferentiable an n n n. For f to be ifferentiable at 0, n must be a number such that n is efine on an interval containing 0. 4 = Proof If n is a positive integer greater than, then the binomial epansion prouces n n n 0 nn n n n n... n n 0 0 n nn n n... n n n n n. This proves the case for which n is a positive integer greater than. You will prove the case for n. Eample 7 in Section. proves the case for which n is a negative integer. In Eercise 75 in Section.5 ou are aske to prove the case for which n is rational. (In Section 5.5, the Power Rule will be etene to cover irrational values of n. ) When using the Power Rule, the case for which n is best thought of as a separate ifferentiation rule. That is, 4 The slope of the line is. Figure.5. Power Rule when n This rule is consistent with the fact that the slope of the line is, as shown in Figure.5.

3 460_00.q //04 4:04 PM Page 09 SECTION. Basic Differentiation Rules an Rates of Change 09 EXAMPLE Using the Power Rule a. b. c. f g Derivative f) g In Eample (c), note that before ifferentiating, Rewriting is the first step in man ifferentiation problems. was rewritten as. f() = 4 Given: Rewrite: Differentiate: Simplif: EXAMPLE Fining the Slope of a Graph (, ) (, ) Fin the slope of the graph of f 4 when a. b. 0 c.. (0, 0) Note that the slope of the graph is negative at the point,, the slope is zero at the point 0, 0, an the slope is positive at the point,. Figure.6 Solution The slope of a graph at a point is the value of the erivative at that point. The erivative of f is f 4. a. When, the slope is f 4 4. Slope is negative. b. When 0, the slope is f Slope is zero. c. When, the slope is f 4 4. Slope is positive. See Figure.6. EXAMPLE 4 Fining an Equation of a Tangent Line f() = (, 4) 4 = 4 4 The line 4 4 is tangent to the graph of f at the point, 4. Figure.7 Fin an equation of the tangent line to the graph of f when. Solution To fin the point on the graph of f, evaluate the original function at., f, 4 Point on graph To fin the slope of the graph when, evaluate the erivative, f, at. m f 4 Slope of graph at, 4 Now, using the point-slope form of the equation of a line, ou can write m Point-slope form 4 4 Substitute for, m, an Simplif. See Figure.7.

4 460_00.q //04 4:04 PM Page 0 0 CHAPTER Differentiation The Constant Multiple Rule THEOREM.4 The Constant Multiple Rule If f is a ifferentiable function an c is a real number, then cf is also ifferentiable an cf cf. Proof cf cf cf 0 0 c c 0 cf f f f f Definition of erivative Appl Theorem.. Informall, the Constant Multiple Rule states that constants can be factore out of the ifferentiation process, even if the constants appear in the enominator. cf c f cf f c c f c f c f EXAMPLE 5 a. b. c.. e. 4t ft 5 Using the Constant Multiple Rule Derivative ft t 4 5 t 4 5 t t 4 5 t 8 5 t 5 5 The Constant Multiple Rule an the Power Rule can be combine into one rule. The combination rule is D c n cn n.

5 460_00.q //04 4:04 PM Page SECTION. Basic Differentiation Rules an Rates of Change EXAMPLE 6 Using Parentheses When Differentiating Original Rewrite Differentiate Simplif a. b. c The Sum an Difference Rules THEOREM.5 The Sum an Difference Rules The sum (or ifference) of two ifferentiable functions f an g is itself ifferentiable. Moreover, the erivative of f g or f g is the sum (or ifference) of the erivatives of f an g. f g f g f g f g Sum Rule Difference Rule Proof A proof of the Sum Rule follows from Theorem.. (The Difference Rule can be prove in a similar wa.) f g f g f g 0 0 f f 0 0 f g f g f g f f g g g g 0 The Sum an Difference Rules can be etene to an finite number of functions. For instance, if F f g h, then F f g h. EXAMPLE 7 Using the Sum an Difference Rules a. f 4 5 Derivative f 4 b. g 4 g 9

6 460_00.q //04 4:04 PM Page CHAPTER Differentiation FOR FURTHER INFORMATION For the outline of a geometric proof of the erivatives of the sine an cosine functions, see the article The Spier s Spacewalk Derivation of an b Tim Hesterberg in The College Mathematics Journal. To view this article, go to the website sin cos Derivatives of Sine an Cosine s In Section., ou stuie the following limits. sin lim 0 an cos lim 0 0 These two limits can be use to prove ifferentiation rules for the sine an cosine functions. (The erivatives of the other four trigonometric functions are iscusse in Section..) THEOREM.6 Derivatives of Sine an Cosine s sin cos cos sin = increasing = 0 π = π = sin = π = 0 ecreasing increasing positive negative positive π π = cos π The erivative of the sine function is the cosine function. Figure.8 Proof sin sin sin 0 0 cos 0 cos sin 0 cos sin cos cos sin sin sin sin 0 Definition of erivative cos sin sin cos 0 0 cos sin sin cos cos This ifferentiation rule is shown graphicall in Figure.8. Note that for each, the slope of the sine curve is equal to the value of the cosine. The proof of the secon rule is left as an eercise (see Eercise 6). EXAMPLE 8 Derivatives Involving Sines an Cosines = sin = sin a. b. sin sin sin Derivative cos cos cos c. cos sin = sin = sin a sin a cos Figure.9 TECHNOLOGY A graphing utilit can provie insight into the interpretation of a erivative. For instance, Figure.9 shows the graphs of a sin for a,,, an. Estimate the slope of each graph at the point 0, 0. Then verif our estimates analticall b evaluating the erivative of each function when 0.

7 460_00.q //04 4:04 PM Page SECTION. Basic Differentiation Rules an Rates of Change Rates of Change You have seen how the erivative is use to etermine slope. The erivative can also be use to etermine the rate of change of one variable with respect to another. Applications involving rates of change occur in a wie variet of fiels. A few eamples are population growth rates, prouction rates, water flow rates, velocit, an acceleration. A common use for rate of change is to escribe the motion of an object moving in a straight line. In such problems, it is customar to use either a horizontal or a vertical line with a esignate origin to represent the line of motion. On such lines, movement to the right (or upwar) is consiere to be in the positive irection, an movement to the left (or ownwar) is consiere to be in the negative irection. The function s that gives the position (relative to the origin) of an object as a function of time t is calle a position function. If, over a perio of time t, the object changes its position b the amount s st t st, then, b the familiar formula Rate istance time the average velocit is Change in istance Change in time s t. Average velocit EXAMPLE 9 Fining Average Velocit of a Falling Object If a billiar ball is roppe from a height of 00 feet, its height s at time t is given b the position function s 6t 00 Position function where s is measure in feet an t is measure in secons. Fin the average velocit over each of the following time intervals. a., b.,.5 c.,. Richar Megna/Funamental Photographs Time-lapse photograph of a free-falling billiar ball Solution a. For the interval,, the object falls from a height of s feet to a height of s feet. The average velocit is s feet per secon. t b. For the interval,.5, the object falls from a height of 84 feet to a height of 64 feet. The average velocit is s feet per secon. t c. For the interval,., the object falls from a height of 84 feet to a height of feet. The average velocit is s feet per secon. t. 0. Note that the average velocities are negative, inicating that the object is moving ownwar.

8 460_00.q //04 4:04 PM Page 4 4 CHAPTER Differentiation s P Secant line Tangent line Suppose that in Eample 9 ou wante to fin the instantaneous velocit (or simpl the velocit) of the object when t. Just as ou can approimate the slope of the tangent line b calculating the slope of the secant line, ou can approimate the velocit at t b calculating the average velocit over a small interval, t (see Figure.0). B taking the limit as t approaches zero, ou obtain the velocit when t. Tr oing this ou will fin that the velocit when t is feet per secon. In general, if s st is the position function for an object moving along a straight line, the velocit of the object at time t is t = t The average velocit between t an t is the slope of the secant line, an the instantaneous velocit at t is the slope of the tangent line. Figure.0 t st t st vt st. t 0 t Velocit function In other wors, the velocit function is the erivative of the position function. Velocit can be negative, zero, or positive. The spee of an object is the absolute value of its velocit. Spee cannot be negative. The position of a free-falling object (neglecting air resistance) uner the influence of gravit can be represente b the equation st gt v 0 t s 0 Position function where s 0 is the initial height of the object, v 0 is the initial velocit of the object, an g is the acceleration ue to gravit. On Earth, the value of g is approimatel feet per secon per secon or 9.8 meters per secon per secon. EXAMPLE 0 Using the Derivative to Fin Velocit ft Velocit is positive when an object is rising, an is negative when an object is falling. Figure. NOTE In Figure., note that the iver moves upwar for the first halfsecon because the velocit is positive for 0 < t <. When the velocit is 0, the iver has reache the maimum height of the ive. At time t 0, a iver jumps from a platform iving boar that is feet above the water (see Figure.). The position of the iver is given b st 6t 6t Position function where s is measure in feet an t is measure in secons. a. When oes the iver hit the water? b. What is the iver s velocit at impact? Solution a. To fin the time t when the iver hits the water, let s 0 an solve for t. 6t 6t 0 Set position function equal to 0. 6t t 0 Factor. t or Solve for t. Because t 0, choose the positive value to conclue that the iver hits the water at t secons. b. The velocit at time t is given b the erivative st t 6. So, the velocit at time t is s 6 48 feet per secon.

9 460_00.q //04 4:04 PM Page 5 SECTION. Basic Differentiation Rules an Rates of Change 5 Eercises for Section. In Eercises an, use the graph to estimate the slope of the tangent line to n at the point,. Verif our answer analticall. To print an enlarge cop of the graph, go to the website (a) (b) (, ) (, ) In Eercises 8, fin the slope of the graph of the function at the given point. Use the erivative feature of a graphing utilit to confirm our results.. See for worke-out solutions to o-numbere eercises. Original 4 f Rewrite Differentiate Point, Simplif. (a) (b) (, ) (, ). ft 5t 5, f 7 5 0, f 5 7. f 4 sin 8. gt cos t, 8 0, 5, 0 0, 0, In Eercises 4, fin the erivative of the function f f 5 0. g 4. f. g. f t t t 6 4. t t 5. g st t t 4 8. f 9. sin cos 0. gt cos t. cos. 5 sin. sin 4. In Eercises 5 0, complete the table Original 5 Rewrite 5 cos Differentiate Simplif In Eercises 9 5, fin the erivative of the function. 9. f f 4. gt t 4 t 4. f h f f f hs s 45 s 50. f t t t 4 5. f 6 5 cos 5. In Eercises 5 56, (a) fin an equation of the tangent line to the graph of f at the given point, (b) use a graphing utilit to graph the function an its tangent line at the point, an (c) use the erivative feature of a graphing utilit to confirm our results. Point 5. 4, 0 54., 55. f 4, 56., 6 f cos

10 460_00.q //04 4:04 PM Page 6 6 CHAPTER Differentiation In Eercises 57 6, etermine the point(s) (if an) at which the graph of the function has a horizontal tangent line sin, 0 < 6. cos, 0 < In Eercises 6 66, fin k such that the line is tangent to the graph of the function f k 4 9 f k 4 7 f k f k Writing About Concepts Line Use the graph of f to answer each question. To print an enlarge cop of the graph, go to the website A B C D (a) Between which two consecutive points is the average rate of change of the function greatest? (b) Is the average rate of change of the function between A an B greater than or less than the instantaneous rate of change at B? (c) Sketch a tangent line to the graph between C an D such that the slope of the tangent line is the same as the average rate of change of the function between C an D. 68. Sketch the graph of a function f such that > 0 for all an the rate of change of the function is ecreasing. In Eercises 69 an 70, the relationship between f an g is given. Eplain the relationship between an g. 69. g f g 5f E f f f Writing About Concepts (continue) In Eercises 7 an 7, the graphs of a function f an its erivative are shown on the same set of coorinate aes. Label the graphs as f or an write a short paragraph stating the criteria use in making the selection. To print an enlarge cop of the graph, go to the website f Sketch the graphs of an 6 5, an sketch the two lines that are tangent to both graphs. Fin equations of these lines. 74. Show that the graphs of the two equations an have tangent lines that are perpenicular to each other at their point of intersection. 75. Show that the graph of the function f sin oes not have a horizontal tangent line. 76. Show that the graph of the function f 5 5 oes not have a tangent line with a slope of. In Eercises 77 an 78, fin an equation of the tangent line to the graph of the function f through the point 0, 0 not on the graph. To fin the point of tangenc, on the graph of f, solve the equation f f 78. f 0, 0 4, Linear Approimation Use a graphing utilit, with a square winow setting, to zoom in on the graph of f 4 to approimate f. Use the erivative to fin f. 80. Linear Approimation Use a graphing utilit, with a square winow setting, to zoom in on the graph of f 4 f 4 0, 0 5, 0 to approimate f4. Use the erivative to fin f4.

11 460_00.q //04 4:04 PM Page 7 SECTION. Basic Differentiation Rules an Rates of Change 7 8. Linear Approimation Consier the function f with the solution point 4, 8. (a) Use a graphing utilit to graph f. Use the zoom feature to obtain successive magnifications of the graph in the neighborhoo of the point 4, 8. After zooming in a few times, the graph shoul appear nearl linear. Use the trace feature to etermine the coorinates of a point near 4, 8. Fin an equation of the secant line S through the two points. (b) Fin the equation of the line tangent to the graph of f passing through the given point. Wh are the linear functions S an T nearl the same? (c) Use a graphing utilit to graph f an T on the same set of coorinate aes. Note that T is a goo approimation of f when is close to 4. What happens to the accurac of the approimation as ou move farther awa from the point of tangenc? () Demonstrate the conclusion in part (c) b completing the table. T f4 4 f 4 f4 T4 f4 T4 8. Linear Approimation Repeat Eercise 8 for the function f where T is the line tangent to the graph at the point,. Eplain wh the accurac of the linear approimation ecreases more rapil than in Eercise 8. True or False? In Eercises 8 88, etermine whether the statement is true or false. If it is false, eplain wh or give an eample that shows it is false. 8. If f g, then f g. 84. If f g c, then f g. 85. If, then. 86. If, then. 87. If g f, then g f. 88. If f n, then f n n. 0.5 In Eercises 89 9, fin the average rate of change of the function over the given interval. Compare this average rate of change with the instantaneous rates of change at the enpoints of the interval. 89. f t t 7,, 90. f t t, , f, 9. f sin,, Vertical Motion In Eercises 9 an 94, use the position function st 6t v 0 t s 0 for free-falling objects. 9. A silver ollar is roppe from the top of a builing that is 6 feet tall. (a) Determine the position an velocit functions for the coin. (b) Determine the average velocit on the interval,. (c) Fin the instantaneous velocities when t an t. () Fin the time require for the coin to reach groun level. (e) Fin the velocit of the coin at impact. 94. A ball is thrown straight own from the top of a 0-foot builing with an initial velocit of feet per secon. What is its velocit after secons? What is its velocit after falling 08 feet? Vertical Motion In Eercises 95 an 96, use the position function st 4.9t v 0 t s 0 for free-falling objects. 95. A projectile is shot upwar from the surface of Earth with an initial velocit of 0 meters per secon. What is its velocit after 5 secons? After 0 secons? 96. To estimate the height of a builing, a stone is roppe from the top of the builing into a pool of water at groun level. How high is the builing if the splash is seen 6.8 secons after the stone is roppe? Think About It In Eercises 97 an 98, the graph of a position function is shown. It represents the istance in miles that a person rives uring a 0-minute trip to work. Make a sketch of the corresponing velocit function. 97. s 98. Distance (in miles) Think About It In Eercises 99 an 00, the graph of a velocit function is shown. It represents the velocit in miles per hour uring a 0-minute rive to work. Make a sketch of the corresponing position function. 99. v 00. Velocit (in mph) (0, 0) (4, ) (0, 6) (6, ) Time (in minutes) Time (in minutes) t t Distance (in miles) Velocit (in mph) (0, 0) s Time (in minutes) v 0, 6 (0, 6) (6, 5) (8, 5) Time (in minutes) t t

12 460_00.q //04 4:04 PM Page 8 8 CHAPTER Differentiation 0. Moeling Data The stopping istance of an automobile, on r, level pavement, traveling at a spee v (kilometers per hour) is the istance R (meters) the car travels uring the reaction time of the river plus the istance B (meters) the car travels after the brakes are applie (see figure). The table shows the results of an eperiment. Driver sees obstacle Reaction time R Driver applies brakes Braking istance Car stops Spee, v Reaction Time Distance, R Braking Time Distance, B (a) Use the regression capabilities of a graphing utilit to fin a linear moel for reaction time istance. (b) Use the regression capabilities of a graphing utilit to fin a quaratic moel for braking istance. (c) Determine the polnomial giving the total stopping istance T. () Use a graphing utilit to graph the functions R, B, an T in the same viewing winow. (e) Fin the erivative of T an the rates of change of the total stopping istance for v 40, v 80, an v 00. (f) Use the results of this eercise to raw conclusions about the total stopping istance as spee increases. 0. Fuel Cost A car is riven 5,000 miles a ear an gets miles per gallon. Assume that the average fuel cost is $.55 per gallon. Fin the annual cost of fuel C as a function of an use this function to complete the table C C/ Who woul benefit more from a one-mile-per-gallon increase in fuel efficienc the river of a car that gets 5 miles per gallon or the river of a car that gets 5 miles per gallon? Eplain. 0. Volume The volume of a cube with sies of length s is given b V s. Fin the rate of change of the volume with respect to s when s 4 centimeters. 04. Area The area of a square with sies of length s is given b A s. Fin the rate of change of the area with respect to s when s 4 meters. B Velocit Verif that the average velocit over the time interval t 0 t, t 0 t is the same as the instantaneous velocit at t t 0 for the position function 06. Inventor Management The annual inventor cost C for a manufacturer is where Q is the orer size when the inventor is replenishe. Fin the change in annual cost when Q is increase from 50 to 5, an compare this with the instantaneous rate of change when Q Writing The number of gallons N of regular unleae gasoline sol b a gasoline station at a price of p ollars per gallon is given b N f p. (a) Describe the meaning of f.479. (b) Is f.479 usuall positive or negative? Eplain. 08. Newton s Law of Cooling This law states that the rate of change of the temperature of an object is proportional to the ifference between the object s temperature T an the temperature T a of the surrouning meium. Write an equation for this law. 09. Fin an equation of the parabola a b c that passes through 0, an is tangent to the line at, Let a, b be an arbitrar point on the graph of, > 0. Prove that the area of the triangle forme b the tangent line through a, b an the coorinate aes is.. Fin the tangent line(s) to the curve 9 through the point, 9.. Fin the equation(s) of the tangent line(s) to the parabola through the given point. (a) 0, a (b) a, 0 Are there an restrictions on the constant a? In Eercises an 4, fin a an b such that f is ifferentiable everwhere.. 4. st at c. C,008,000 Q f a, b, f cos, a b, 5. Where are the functions an ifferentiable? 6. Prove that 6.Q > < 0 0 cos sin. f sin f sin FOR FURTHER INFORMATION For a geometric interpretation of the erivatives of trigonometric functions, see the article Sines an Cosines of the Times b Victor J. Katz in Math Horizons. To view this article, go to the website