MAC 2311 Chapter 3 Review Materials (Part 1) Topics Include Max and Min values, Concavity, Curve Sketching (first and second derivative analysis)

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1 MAC Chapter Review Materials (Part ) Topics Include Ma and Min values, Concavit, Curve Sketching (first and second derivative analsis) MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Identif the open intervals where the function is changing as requested. ) Increasing f() ) A) (-, ) B) (-, ) C)(-, ) D) (-, ) ) Decreasing f() ) A) (-, -) B) (-, -) C)(-, -) D) (0, -)

2 ) Increasing h() ) A) (-, -) B) (-,-), (, ) C)(-, ) D) (-, ) ) Decreasing f() ) A) (0, -) B) (-, ) C)(-, -) D) (0, ) 5) Increasing f() 5) A) (-, 0), (, ) B) (0, ) C)(-, -), (0, ) D) (-, -), (, )

3 Suppose that the function with the given graph is not f(), but f (). Find the open intervals where f() is increasing or decreasing as indicated. 6) Increasing 6) A) (0, ) B) (-, ) C)(, ) D) (-, -), (, ) 7) Increasing 5 7) A) (-, -), (, ) B) (0, ) C)(, ) D) (-, )

4 8) Increasing 8) A) (-, ) B) (-, ) C)(-, -),(-, ), (, ) D) (-,-),(, ) Find all the critical numbers of the function. 9) = A) 5 9 B) 7 C) 8 D) - 5 9) 0) f() = A) -, B) - C), - D) 6 0) ) f() = ( + ) /5 A) 5 B) - C) D) 0 ) ) = /5-6/5 ) A) - 5, 0 B) 0, 6 C) 5 D) 6 Find the open interval(s) where the function is changing as requested. ) Increasing; = 7-5 A) (-, ) B) (-, 7) C)(-5, 7) D) (-5, ) ) Find all the critical numbers of the function. ) f() = A)-, B)- C)6 D), - ) 5) f() = ( + ) /5 A) B) 5 C) 5 D)- 5)

5 6) = /5-9/5 6) A)- 5, 0 B) 9 C)0, 9 D) 5 Find the open interval(s) where the function is changing as requested. 7) Increasing; f() = - + A) (0, ) B) (-, 0) C)(-, ) D) (, ) 8) Increasing; f() = + A) (-, 0) B) (-, ) C)(0, ) D) (, ) 7) 8) 9) Decreasing; f() = - A) -, - B), C)(-, ) D) -, 9) 0) Increasing; = + 9 A) (0, ) B) (-, 0) C)(-, ) D) none 0) Solve the problem. ) Suppose the total cost C() to manufacture a quantit of insecticide (in hundreds of liters) is given b C() = Where is C() decreasing? A) (0, 800) B) (8, 800) C)(8, 0) D) (0, 800) ) Suppose a certain drug is administered to a patient, with the percent of concentration in the 6t bloodstream t hr later given b K(t) =. On what time interval is the concentration of the drug t + increasing? A) (0, 6) B) (6, ) C)(0, ) D) (, ) ) ) Find the location and value of all relative etrema for the function. ) ) A) Relative minimum of - at - ; Relative maimum of at - ; Relative minimum of at. B) Relative minimum of 0 at - ; Relative maimum of - at ; Relative minimum of at. C)Relative minimum of - at - ; Relative maimum of at - ; Relative minimum of 0 at. D) Relative minimum of - at - ; Relative maimum of - at ; Relative minimum of at. 5

6 ) ) A) Relative maimum of at -. B) Relative maimum of at - ; Relative minimum of 0 at. C)Relative minimum of 0 at. D) None 5) 5) A) None B) Relative maimum of at. C)Relative minimum of at ; Relative maimum of - at -. D) Relative minimum of - at -. 6) 6) A) Relative minimum of 0 at 0. B) None C)Relative maimum of 5 at - ; Relative maimum of at. D) Relative maimum of 5 at - ; Relative minimum of 0 at 0 ; Relative maimum of at. 6

7 7) 7) A) Relative minimum of at. B) Relative maimum of - at -. C)Relative maimum of - at - ; Relative minimum of at. D) None Suppose that the function with the given graph is not f(), but f (). Find the locations of all etrema, and tell whether each etremum is a relative maimum or minimum. 8) 8) A) Relative minimum at - B) Relative maimum at -; relative minimum at C)Relative maima at - and D) Relative minimum at -; relative maimum at 9) 9) A) Relative minimum at 0 B) Relative maima at - and C) Relative maimum at 0 D) No relative etrema 7

8 0) 6 5 0) A) Relative minimum at 0 B) Relative maimum at -; relative minimum at C) No relative etrema D) Relative maimum at 0 ) 5 h() ) A) Relative maimum at B) Relative minima at - and C)Relative minimum at D) Relative minimum at Find the -value of all points where the function has relative etrema. Find the value(s) of an relative etrema. ) f() = + - A) Relative minimum of - at 0. B) Relative maimum of - at -. C)Relative minimum of 0 at -. D) Relative minimum of - at -. ) ) f() = - + A) Relative maimum of 0 at ; Relative minimum of - at -. B) Relative maimum of at 0. C) No relative etrema. D) Relative maimum of at 0; Relative minimum of - at. ) 8

9 ) f() = A) Relative minimum of at 0. B) Relative minimum of 0 at -. C)Relative maimum of 8 at -; Relative minimum of at 0. D) No relative etrema. ) 5) f() = - A) Relative maimum of 0 at. B) Relative minimum of - at 0. C)Relative maimum of - at 0. D) No relative etrema. 5) 6) f() = / - / A) Relative maimum of 0 at 0; Relative maimum of - at - B) Relative minimum of of - at C)Relative maimum of 0 at 0; Relative minimum of - at and - D) No relative etrema. 6) Solve the problem. 7) The annual revenue and cost functions for a manufacturer of grandfather clocks are approimatel R() = and C() = ,000, where denotes the number of clocks made. What is the maimum annual profit? A) $55, B) $65, C) $85, D) $75, 8) Find the number of units,, that produces the maimum profit P, if C() = and p = A) units B) 56 units C) units D) units 7) 8) 9) S() = , 0 is an approimation to the number of salmon swimming upstream to spawn, where represents the water temperature in degrees Celsius. Find the temperature that produces the maimum number of salmon. A) C B) 8 C C) C D) 0 C 9) Find f"() for the function. 0) f() = A) 0 B) C)8 D) 8 + 0) ) f() = / - 6/ A) / - -/ B).5/ +.5-/ C) -/ + -/ D).5-/ +.5-/ ) ) f() = + A) 8 / + / B) / - / C) 8 / - / D) / + / ) 9

10 ) f() = A) ( - 7)/ B) - 0 ( - 7)/ C) 0 ( - 7)/ D) - 9 ( - 7)/ ) Find the requested value of the second derivative of the function. ) f() = ; Find f (6). A) -575 B) 57 C) 576 D) 580 ) 5) f() = ; Find f (0). A) 9 B) 0 C) -8 D) 8 5) Find the indicated derivative of the function. 6) f () of f() = A) 8 B) 8 +6 C)6 D) ) 7) f()() of f() = A) 80-8 B) C)70-96 D) ) Find the open intervals where the function is concave upward or concave downward. Find an inflection points. 8) 8) A) Concave upward on (0, ); concave downward on (-, 0); inflection points at (-, 0), (-, 0), and 7, 0 B) Concave upward on (-, ); concave downward on (-, ); inflection point at (, -) C)Concave upward on (0, ); concave downward on (-, 0); inflection point at (0, -) D) Concave upward on (-, ); concave downward on (-, ); inflection points at (-, 0) and (, -) 0

11 9) 9) A) Concave upward on (-, -); concave downward on (-, ); inflection point at (-, ) B) Concave upward on (-, ); concave downward on (, -); inflection point at (-, ) C) Concave upward on (-, ); concave downward on (-, -); no inflection points D) Concave upward on (-, -); concave downward on (-, ); no inflection points Find the largest open intervals where the function is concave upward. 50) f() = + + A) (-, ) B) None C)(-, ) D) (-,-) 50) 5) f() = A), B) -, C) -,- D) -, 5 5) 5) f() = A) None B) (-, ), (, ) C)(-, ) D) (, ) 5) 5) f() = + A) (-,-) B) (-,-),(-, ) C)None D) (, ) 5) Find an inflection points given the equation. 5) f() = + A) B) Inflection point at (,-) C) Inflection point at (-,-) D) Inflection point at (-,-) 8 55) f() = + A) Inflection points at (0, 0), (-, -), (, ) B) C)Inflection points at (0, 0), -, -,, D) Inflection points at (-, -), (, ) 5) 55)

12 Suppose that the function with the given graph is not f(), but f (). Find the open intervals where the function is concave upward or concave downward, and find the location of an inflection points. 56) 56) A) Concave upward on (-, 0); concave downward on (0, ); inflection point at 0 B) Concave upward on (-, -) and (, ); concave downward on (-, ); inflection points at -0 and 0 C)Concave upward on (-, -) and (, ); concave downward on (-, ); inflection points at - and D) Concave upward on (-, ); concave downward on (-, -) and (, ); inflection points at - and 57) ) A) Concave upward on (-, 0); concave downward on (0, ); inflection point at 0 B) Concave upward on (-, -) and (, ); concave downward on (-, ); inflection points at -0 and 0 C)Concave upward on (-, -) and (, ); concave downward on (-, ); inflection points at - and D) Concave upward on (-, ); concave downward on (-, -) and (, ); inflection points at - and Decide if the given value of is a critical number for f, and if so, decide whether the point is a relative minimum, relative maimum, or neither. 58) f() = ; = 8 A) Critical number, relative minimum at (8, -) B) Not a critical number C) Critical number but not an etreme point D) Critical number, relative maimum at (8, -) 58)

13 59) f() = ( - 6)( - ); = 59) A) Critical number, relative minimum at, B) Critical number, relative maimum at, C)Not a critical number D) Critical number but not an etreme point 60) f() = ; = 0 A) Critical number but not an etreme point B) Critical number, relative minimum at (0, ) C)Not a critical number D) Critical number, relative maimum at (0, ) 60) 6) f() = - - 6; = 6) A) Critical number, relative minimum at 5,- B) Critical number but not an etreme point C)Critical number, relative maimum at 5,- D) Not a critical number The function gives the distances (in feet) traveled in time t (in seconds) b a particle. Find the velocit and acceleration at the given time. 6) s = t +, t = 6) A) v = - 5 ft/s, a = 5 ft/s B) v = 5 ft/s, a = - 5 ft/s C)v = 5 ft/s, a = - 5 ft/s D) v = - 5 ft/s, a = 5 ft/s 6) s = t + t + 6t + 7, t = A) v = 5 ft/s, a = 6 ft/s B) v = 6 ft/s, a = ft/s C)v = 6 ft/s, a = 5 ft/s D) v = ft/s, a = 6 ft/s 6) 6) s = t - 5, t = A) v = - ft/s, a = 5 8 ft/s B) v = ft/s, a = ft/s 6) C)v = ft/s, a = ft/s D) v = 5 8 ft/s, a = - ft/s Sketch the graph and show all etrema, inflection points, and asmptotes where applicable.

14 65) f() = ) A) Rel min: (, 0) B) No etrema Inflection point: (0, 0) C)Rel ma: (0, 0), Rel min: (-6, 6) Inflection point: (-, 08) D) Rel ma (-,-), Rel min: (-,-5) Inflection point: -,

15 66) f() = ) A) Rel min: (-8, -0) 00 B) Rel min: (-, -80) C)Rel min: (, -80) D) Rel min: (8, -0)

16 67) f() = /( - ) 00 67) A) Rel ma: (-, 96 ), Rel min: (, -96 ) Inflection point: (0,0) B) Rel ma: (0,0), Rel min: ± 8, - Inflection point: ±, C)Rel min: (0, 0) 00 D) No etrema Inflection point: (0, 0)

17 68) f() = + 68) A) Rel min: (0, 0) B) Rel min: 0, C)Rel min: 0, D) Rel min: (0, 0) Inflection points: - 6,, 6,

18 69) f() = 5-69) - - A) Rel ma: (0, ) B) Rel ma: 0, C)Rel min: (0, ) D) Rel min: 0,

19 70) f() = ) A) Rel ma:, - 9 B) Rel ma: -, C)Rel min: -, 9 D) Rel min:,

20 7) f() = - 5 7) A) Rel ma: 0, - 5 B) No etrema Inflection point: (0, 0) C) No etrema Inflection point: (0, 0) D) Rel min: 0, Sketch the graph and show all local etrema and inflection points. 0

21 7) = + sin, 0 π 7) 6 π π π π A) Local minimum: (0, 0) Local maimum: (π, π) B) Local minimum: (0, 0) Local maimum: (π, π) Inflection point: (π, π) 6 6 π π π π π π π π C)Local minimum: (0, 0) Local maimum: (π, π) D) Local minimum: (0, 0) Local maimum: (π, π) Inflection point: (π, π) 6 6 π π π π π π π π Solve the problem. 7) The percent of concentration of a certain drug in the bloodstream hours after the drug is 5 administered is given b K() =. At what time is the concentration a maimum? + 9 A) 0.5 hr B) 5 hr C) hr D) 0.9 hr 7)

22 7) Find the point of diminishing returns (, ) for the function R() = , 0 0, where R() represents revenue in thousands of dollars and represents the amount spent on advertising in tens of thousands of dollars. A) (,,500) B) (5.6,,9.6) C)(56.9, -9,7.8) D) (,,59) 7)

23 Answer Ke Testname: CHAPTER (PART ) ST AND ND DER ANALYSIS, CURVE SKETCHING ) A ) C ) B ) B 5) C 6) D 7) D 8) C 9) C 0) A ) B ) B ) A ) A 5) D 6) C 7) D 8) A 9) D 0) A ) C ) C ) A ) B 5) C 6) A 7) D 8) B 9) D 0) D ) D ) D ) D ) A 5) C 6) C 7) A 8) A 9) C 0) C ) D ) C ) D ) C 5) D 6) C 7) C 8) C 9) D 50) C

24 Answer Ke Testname: CHAPTER (PART ) ST AND ND DER ANALYSIS, CURVE SKETCHING 5) A 5) D 5) D 5) A 55) C 56) C 57) C 58) B 59) C 60) D 6) A 6) D 6) C 6) C 65) D 66) B 67) A 68) D 69) D 70) B 7) C 7) D 7) C 7) D