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1 Chapter Applications o Dierentiation Review Eercises See CalcChat.com or tutorial help and worked-out solutions to odd-numbered eercises. Finding Etrema on a Closed Interval In Eercises, ind the absolute etrema o the unction on the closed interval..,,.,.,,. h,.,., 9,. g cos,. sin, Using Rolle s Theorem In Eercises 9, determine whether Rolle s Theorem can be applied to on the closed interval [a, b. I Rolle s Theorem can be applied, ind all values o c in the open interval a, b such that c. I Rolle s Theorem cannot be applied, eplain wh not. 9.,.,.,. sin, Using the Mean Value Theorem In Eercises, determine whether the Mean Value Theorem can be applied to on the closed interval [a, b]. I the Mean Value Theorem can be applied, ind all values o c in the open interval a, b such that c I the Mean Value Theorem cannot be applied, eplain wh not..,... b a. b a. cos,,.,, 9. Mean Value Theorem Can the Mean Value Theorem be applied to the unction,,,,,,,,,,,,, on the interval,? Eplain.. Using the Mean Value Theorem,, 9, (a) For the unction A B C, determine the value o c guaranteed b the Mean Value Theorem on the interval,. (b) Demonstrate the result o part (a) or on the interval,. Intervals on Which Is Increasing or Decreasing In Eercises, identi the open intervals on which the unction is increasing or decreasing... h.. g.. Appling the First Derivative Test In Eercises, (a) ind the critical numbers o (i an), (b) ind the open interval(s) on which the unction is increasing or decreasing, (c) appl the First Derivative Test to identi all relative etrema, and use a graphing utilit to conirm our results ht t t..... Finding Points o Inlection In Eercises, ind the points o inlection and discuss the concavit o the graph o the unction Using the Second Derivative Test In Eercises, ind all relative etrema. Use the Second Derivative Test where applicable.... h, sin cos, g cos sin,, g sin, 9 g cos,, tan,, 9 g. ht t t >,, Copright Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed rom the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall aect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time i subsequent rights restrictions require it.

2 Review Eercises 9.. h cos, Think About It In Eercises and, sketch the graph o a unction having the given characteristics..., > or < > or < < < or > < or < or > does not eist. > or < < < or < or > < or > or < < 9. Writing A newspaper headline states that The rate o growth o the national deicit is decreasing. What does this mean? What does it impl about the graph o the deicit as a unction o time?. Inventor Cost The cost o inventor C depends on the ordering and storage costs according to the inventor model C Q s r. Determine the order size that will minimize the cost, assuming that sales occur at a constant rate, Q is the number o units sold per ear, r is the cost o storing one unit or one ear, s is the cost o placing an order, and is the number o units per order.. Modeling Data Outlas or national deense D (in billions o dollars) or selected ears rom 9 through are shown in the table, where t is time in ears, with t corresponding to 9. (Source: U.S. Oice o Management and Budget) (a) Use the regression capabilities o a graphing utilit to ind a model o the orm or the data., D at bt ct dt e t D t D (b) Use a graphing utilit to plot the data and graph the model. (c) For the ears shown in the table, when does the model indicate that the outla or national deense was at a maimum? When was it at a minimum? For the ears shown in the table, when does the model indicate that the outla or national deense was increasing at the greatest rate?. Modeling Data The manager o a store recorded the annual sales S (in thousands o dollars) o a product over a period o ears, as shown in the table, where t is the time in ears, with t corresponding to. (a) Use the regression capabilities o a graphing utilit to ind a model o the orm or the data. (b) Use a graphing utilit to plot the data and graph the model. (c) Use calculus and the model to ind the time t when sales were increasing at the greatest rate. Do ou think the model would be accurate or predicting uture sales? Eplain. Finding a Limit cos In Eercises, ind the it... cos 9.. Horizontal Asmptotes In Eercises, use a graphing utilit to graph the unction and identi an horizontal asmptotes... g. h. Analzing the Graph o a Function In Eercises, analze and sketch a graph o the unction. Label an intercepts, relative etrema, points o inlection, and asmptotes. Use a graphing utilit to veri our results t 9 S S at bt ct d. sin Copright Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed rom the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall aect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time i subsequent rights restrictions require it.

3 Applications o Dierentiation Chapter.. Using Newton s Method In Eercises, approimate the zero(s) o the unction. Use Newton s Method and continue the process until two successive approimations dier b less than.. Then ind the zero(s) using a graphing utilit and compare the results.. Maimum Area A rancher has eet o encing with which to enclose two adjacent rectangular corrals (see igure). What dimensions should be used so that the enclosed area will be a maimum?.... Finding Point(s) o Intersection In Eercises 9 and 9, appl Newton s Method to approimate the -value(s) o the indicated point(s) o intersection o the two graphs. Continue the process until two successive approimations dier b less than.. [Hint: Let h g. 9. g 9. sin g. Maimum Area Find the dimensions o the rectangle o maimum area, with sides parallel to the coordinate aes, that can be inscribed in the ellipse given b. g g 9. Minimum Length A right triangle in the irst quadrant has the coordinate aes as sides, and the hpotenuse passes through the point,. Find the vertices o the triangle such that the length o the hpotenuse is minimum.. Minimum Length The wall o a building is to be braced b a beam that must pass over a parallel ence eet high and eet rom the building. Find the length o the shortest beam that can be used.. Maimum Length Find the length o the longest pipe that can be carried level around a right-angle corner at the intersection o two corridors o widths eet and eet.. Maimum Length A hallwa o width eet meets a hallwa o width 9 eet at right angles. Find the length o the longest pipe that can be carried level around this corner. [Hint: I L is the length o the pipe, show that L csc 9 csc where is the angle between the pipe and the wall o the narrower hallwa.]. Maimum Volume Find the volume o the largest right circular cone that can be inscribed in a sphere o radius r. r r. Maimum Volume Find the volume o the largest right circular clinder that can be inscribed in a sphere o radius r. Comparing and d In Eercises 9 and 9, use the inormation to evaluate and compare and d. Function -Value Dierential o 9.. d. 9. d. Finding a Dierential In Eercises 9 and 9, ind the dierential d o the given unction. 9. cos Volume and Surace Area The radius o a sphere is measured as 9 centimeters, with a possible error o. centimeter. (a) Use dierentials to approimate the possible propagated error in computing the volume o the sphere. (b) Use dierentials to approimate the possible propagated error in computing the surace area o the sphere. (c) Approimate the percent errors in parts (a) and (b). 9. Demand Function A compan inds that the demand or its commodit is p where p is the price in dollars and is the number o units. Find and compare the values o p and dp as changes rom to. Copright Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed rom the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall aect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time i subsequent rights restrictions require it.

4 A Answers to Odd-Numbered Eercises. (a).9 (b).. (a). (b).9. (a) ± (b).% in.. (a) ±. cm (b) about.9% 9. (a) ±. in. (b) ±. in. (c).%;.%.. mi; About.%. (a) % (b) sec. min. t., d d Calculator: , d d.99 Calculator:.99.. The value o d becomes closer to the value o as decreases.. ; d d.... True 9. True Review Eercises or Chapter (page ). Maimum:, ;. Maimum:, ; Minimum:, Minimum:. Maimum:,, ;. Maimum:,.; Minimum:, Minimum:.,. 9.. Not continuous on,. 9. is not dierentiable at.. 9. No; The unction has a discontinuit at, which is in the interval,.. Increasing on Decreasing on,, ;. Increasing on,, Decreasing on,, ;. Increasing on, ; Decreasing on,. (a) Critical number: (b) Increasing on, ; Decreasing on, (c) Relative minimum:, 9 (, ) 9. (a) Critical number: t (b) Increasing on, ; Decreasing on, (c) Relative minimum:,. (a) Critical number: ; Discontinuit: (b) Increasing on, ; Decreasing on, and (c) Relative minimum:,,. (a) Critical numbers: (b) Increasing on Decreasing on (c) Relative minimum: Relative maimum: and., ); Concave upward:, ; Concave downward:, ). No point o inlection; Concave upward:, 9.,,, ; Concave upward:, ; Concave downward:,,,. Relative minimum: 9,. Relative maima:,,, ; Relative minimum:. Relative maimum:,, ; Relative minimum:,. 9. Increasing and concave down. (a) (b) (, ()) D.t.t.t.t. (, ()), ;, (, ),, ;, p, (c) Maimum in ; Minimum in 9 Copright Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed rom the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall aect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time i subsequent rights restrictions require it.

5 Answers to Odd-Numbered Eercises A =. (, = (, (, ) (, ), ) ). t and t 9.,,,,,.. t. r..,.,.9.., ; d. d cos sin d 9. (a) (b) (c) About.%; About.% = ±. cm ±. cm P.S. Problem Solving (page ). Choices o a ma var. a = a = a = a = (a) One relative minimum at, or a (b) One relative maimum at a = a =, or a < (c) Two relative minima or a = a < when ±a I a <, then there are three critical points; i a, then there is onl one critical point.. All c, where c is a real number. Proo. The bug should head towards the midpoint o the opposite side. Without calculus, imagine opening up the cube. The shortest distance is the line PQ, passing through the midpoint as shown. ) =, ) ( ( P (, ) (, ) (, ) (,.9) (, ) = (, ) 9. a, b, c. Proo. Greatest slope: Least slope:,, ;. Proo. Proo; Point o inlection:, 9. (a) P (b) Chapter Section. (page ). Proo. t C. C Original Integral Rewrite Integrate Simpli. d d C C 9. d. C. C. C. C 9. C. C. C. sin cos C. t csc t C 9. tan cos C. tan C. Answers will var. Sample answer:.. ht t t 9... (a) Answers will var. (b) Sample answer:. (a) 9 (b) (c) () = + () = 9 () P() d C (, ) C Q Copright Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed rom the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall aect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time i subsequent rights restrictions require it.