Calculus I - Math 3A - Chapter 4 - Applications of the Derivative

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1 Berkele Cit College Just for Practice Calculus I - Math 3A - Chapter - Applications of the Derivative Name Identrif the critical points and find the maimum and minimum value on the given interval I. ) f() = ; I = [-8, 0] ) ) f(r) = ; I =[-, 5] ) r + 3) r(θ) = cos θ; I = - π, π 3 3) ) f() = ; I =(-3, 5) ) Find all critical points and find the minimum and maimum value of the function on the given domain. 5) Domain: [-, 3] 5) ) Domain: [-, ] 8 6) Instructor: K. Pernell

2 7) Domain: [-3, -] 3 7) Use the Concavit Theorem to determine where the given function is concave up and where it is concave down. Also find all inflection points. 8) q() = ) 9) G() = ) 0) f() = ) Use the Montonicit Theorem to find where the function is increasing and where it is decreasing. ) h(z) = 7z - z3 ) ) h(t) = cos t, 0 t π ) Sketch the graph of a continuous function f on the given domain that satisfies all conditions. 3) f has domain [0, 6]; f(0) = 0; f(6) = ; increasing and concave down on (0, 6) 6 3)

3 Determine where the graph of the function is increasing, decreasing, concave up, concave down. Then sketch the graph. ) g() = 3-3 ) ) F() = ) Solve the problem. 6) Translate into the language of a derivative of the number of businesses with respect to time. N is the number of businesses in the downtown district after time t. The number of businesses downtown is decreasing at a slower and slower rate. 6) A) dn dt > 0, d N dt < 0 dn B) < 0, d N dt dt > 0 C) dn dt < 0, d N dt < 0 dn D) < 0, d N = k, k is a constant dt dt 7) Translate into the language of a derivative of distance with respect to time. s is the position of the car at time t. The speed of the car is porportional to the distance it has traveled. 7) A) ds = k, k is a constant dt ds B) = ks, k is a constant dt C) d s dt = ks, k is a constant ds D) = kt, k is a constant dt 3

4 The first derivative f ʹ is given. Find all values of that make the function a local minimum and a local maimum. 8) fʹ() = ( - 6)( + 7) 8) 9) fʹ() = ( + )( + 8) 9) Identif the critical points. Then use the test of our choice to decide which critical points give a local maimum value and which give a local minimum value. Give these values. 0) g() = 0) + Find, if possible, the (global) maimum and minimum values of the given function on the indicated interval. ) h(t) = cos t - π 3 on 0, 7π ) ) g() = on [, 3] ) Solve the problem. 3) If the price charged for a cand bar is p() cents, then thousand cand bars will be sold in a certain cit, where p() = 6 -. How man cand bars must be sold to maimize 3) revenue? ) Suppose c() = ,000 is the cost of manufacturing items. Find a production level that will minimize the average cost of making items. ) 5) A baseball team is tring to determine what price to charge for tickets. At a price of $0 per ticket, it averages 50,000 people per game. For ever increase of $, it loses 5,000 people. Ever person at the game spends an average of $5 on concessions. What price per ticket should be charged in order to maimize revenue? 5) 6) Find the number of units that must be produced and sold in order to ield the maimum profit, given the following equations for revenue and cost: R() = C() = ) 7) A compan is constructing an open-top, square-based, rectangular metal tank that will have a volume of 8 cubic feet. What dimensions ield the minimum surface area? Round to the nearest tenth, if necessar. 7) 8) The velocit of a particle, in feet per second, is given b v = t - 8t +, where t is the time (in seconds) for which it has traveled. Find the time at which the velocit is at a minimum. 8)

5 9) Supertankers off-load oil at a docking facilit shore point 5 miles offshore. The nearest refiner is 9 miles east of the docking facilit. A pipeline must be constructed connecting the docking facilit with the refiner. The pipeline costs $300,000 per mile if constructed underwater and $00,000 per mile if over land. 9) 5 mi 9 mi Locate point B to minimize the cost of construction. Sketch a graph of a function f that has the given properties. 30) (a) Defined for all real numbers (b) Increasing for -3 < < 3 (c) Decreasing for - < < -3 and 3 < < (d) Concave downward for 0 < < (e) Concave upward for - < < 0 (f) fʹ(-3) = fʹ(3) = 0 (g) Inflection point at (0, 0) 30) 5

6 3) (a) Defined for all real numbers (b) Increasing for -3 < < - and < < (c) Decreasing for - < < -3 and - < < (d) Concave upward for - < < - and < < (e) Concave downward for - < < (f) fʹ(-3) = fʹ(-) = fʹ() = 0 (g) Inflection point at (-, 0) and (, ) 3) LʹHopitalʹs rule does not help with the given limit. Find the limit some other wa. 3) lim ) 33) lim 0+ cot sin 33) Make an analsis using calculus and sketch the graph. 6 3) g() = + 3) 6

7 35) g() = ) ) f() = + cos, 0 π 36) Sketch a possible graph of f() using fʹ(). 37) fʹ() = ( - ) and f(0) = 0 37) The graphs of the first and second derivatives of a function = f() are given. Select a possible graph of f that passes through the point P. (NOTE: Vertical scales ma var from graph to graph.) 7

8 38) fʹ fʹʹ 38) P P A) B) C) D) Provide an appropriate response. 39) A marathoner ran the 6. mile New York Cit Marathon in.8 hrs. Did the runner ever eceed a speed of 9 miles per hour? 39) 0) A trucker handed in a ticket at a toll booth showing that in hours he had covered 5 miles on a toll road with speed limit 65 mph. The trucker was cited for speeding. Wh? 0) Decide whether the Mean Value Theorem applies to the given function on the given interval. ) h(t) = t( - t); [-, 5] ) 8

9 ) g() = 3/; [0, ] ) Use the Mean Value Theorem and find all possible values of c on the given interval. 3) f() = + 5 ; [6, 9] 3) Approimate the values of that gives the maimum and minimum values of the function on the indicated intervals. ) f() = ; [0, ] ) Find the indicated root of the given equation b using Newtonʹs method. 5) = 0 (between and ) 5) 6) = 0 (between 0 and ) 6) Evaluate the indefinite integral. 7) ( ) d 7) 8) + d 8) 3 9) t + t dt 9) 3 Find the general antiderivative F() + C for the function. 50) f() = 5 3/ 50) 5) f() = 8 + 6π5 5) 5) f() = ) 53) f() = ) 5) f() = 8-3 5) 55) f() = ) Find the particular solution that satisfies the given condition. 56) d = - 6; curve passes through (, 5) 56) d 9

10 57) d d = -3/; curve passes through (, 3) 57) 58) d d = ; = at = 0 58) 59) du dt = u 3(t - t3); u = 3 at = 0 59) 60) d d = +, > 0; curve passes through (, 0) 60) 3 Use LʹHo^pitalʹs rule to find the limit. cos 3-6) lim 0 6) 6) lim 0 sin (5) sin 6) 63) lim 0 sin 9 63) 6) lim π 3 cos - - π 3 6) 65) lim ) 66) lim 0 e - sin 7 66) 67) lim ) 68) lim ) 69) lim sin 5 69) 0

11 70) lim ) 7) lim ( ) 7) ) lim ) 73) lim ( ln ) 73) 0+ 7) lim 0 csc 7) ) lim e 75)

12 Answer Ke Testname: MATH3A_CH_APPLICATIONS_PRACTICE ) Critical points: -8, -9, 0; maimum value 8; minimum value 0 ) Critical points: -, 0, 5; maimum value ; minimum value 7 3) Critical points: - π, 0, π ; maimum value ; minimum value 3 ) Critical points: -, ; no maimum value; minimum value - 5) Critical points: -, -,, 3; maimum value: 5; minimum value: 6) Critical points: -, 0,, ; maimum value: 7; minimum value: 0 7) Critical points: -3, -, -; maimum value: 0; minimum value: - 8) Concave up on (0, ), concave down on (-, 0); inflection point (0, 3) 9) Concave up on (-, 0) (, ), concave down on (0, ); inflection points (0, ) and (, 7) 0) Concave up on (-, ), concave down on (-, -); inflection point (-, -) ) Increasing on [-3, 3], decreasing on (-, -3] [3, ) ) Increasing on [π, π], decreasing on [0, π] 3) )

13 Answer Ke Testname: MATH3A_CH_APPLICATIONS_PRACTICE 5) ) B 7) B 8) Local minimum at = -7 9) Local minimum at = -; local maimum at = -8 0) Critical points: -, ; local maimum f() = ; local minimum f(-) = - ) Maimum value h π 3 = ; minimum value h π 3 = - ) Maimum value g 5 = ; minimum value g(3) = g() = 0 3) 37 thousand cand bars ) items 5) $7.50 6) 3 units 7).6 ft b.6 ft. b.3 ft 8) sec 9) Point B is.7 miles from Point A. 30) 3) 3) 6 33) 3

14 Answer Ke Testname: MATH3A_CH_APPLICATIONS_PRACTICE 3) ) ) 3 3 -

15 Answer Ke Testname: MATH3A_CH_APPLICATIONS_PRACTICE 37) 38) B 39) Yes, the Mean Value Theorem implies that the runner attained a speed of 9. mph, which was her average speed throughout the marathon. 0) As the truckerʹs average speed was 08 mph, the Mean Value Theorem implies that the trucker must have been going that speed at least once during the trip. ) No ) Yes 3) c = ) Minimum f(.8858) ; maimum f() = 5) ) ) C 8) 6 3/ C 9) 3 t 3 + t 6 + C 50) 5 7 7/ + C 5) + 6π5 + C 5) C 53) / + C 5) 6 9 9/8-3 + C 55) C 56) =

16 Answer Ke Testname: MATH3A_CH_APPLICATIONS_PRACTICE 57) = 8/ ) = + 59) u = t - t ) = ) - 9 6) 5 63) 0 6) - 65) 8 66) ) ln 5 ln 7 68) 0 69) 5 70) 7) 7 7) 3 73) 0 7) 0 75) 0 6