Math 125 Practice Problems for Test #3

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1 Math Practice Problems for Test # Also stud the assigned homework problems from the book. Donʹt forget to look over Test # and Test #! Find the derivative of the function. ) Know the derivatives of all of the trigonometric functions. Find the derivative of with respect to, t, or θ, as appropriate. ) = e7-9 ) = e( + ) ) = e - e ) = ( - + ) e ) = eθ(sin θ - cos θ) 7) = sin e-θ7 Solve the problem. ) A certain radioactive isotope decas at a rate of % per ears. If t represents time in ears and represents the amount of the isotope left, use the condition that =.97 to find the value of k in the equation = ekt. 7) In a chemical reaction, the rate at which the amount of a reactant changes with time is proportional to the amount present, such that d = -.7, when t is measured in hours. If dt there are gof reactant present when t =, how man grams will be left after hours? Give our answer to the nearest tenth of a gram. Find the angle. ) sin- Find the derivative of with respect to, t, or θ, as appropriate. ) = ln 7 9) cos- 9) = ln ( - ) ) tan- - ) = ln ) sec- ) = ln Evaluate eactl. ) cos sin- ) = ln - ( + ) ) sin cos- ) = ln + Find the inverse of the function. ) f() = + 7 ) f() = + ) sec cos- Find the derivative of the function. ) Know the derivatives of all of the inverse trigonometric functions.

2 Find the derivative of with respect to. ) = tan- 7) = cos- ( + ) ) = sin- 9) = sin- (et) ) = tan- 7 ) Minimum h() Find the location of the indicated absolute etremum for the function. ) Minimum f() ) Maimum g() Find the absolute etreme values of each function on the interval. ) f() = - ; - ) f() = sin + π, 7π Find the derivative at each critical point and determine the local etreme values. ) = /( - ); 7) = ( - ) ) = - Determine whether the function satisfies the hpotheses of the Mean Value Theorem for the given interval. 9) f() = /, -, ) g() = /,, ) s(t) = t( - t), -, Find an relative etrema for the function. ) f() = ) f() = ) f() =

3 Determine the location of each local etremum of the function. ) f() = Determine the intervals where f is concave up or concave down. Find the -coordinate of the point of inflection. ) f()=-+ ) f() = ) f()=-9+ Find the critical numbers of f. Determine the intervals on which f is increasing or decreasing and find the local etrema. 7) f()=-- ) f()=++ 9) f()=+++ ) f()=-+ Find the Horizontal and Veritcal asmptotes, if an. ) f() = + 7) f() = + + ) f()=-+ Use the graph of the function f() to locate the local etrema and identif the intervals where the function is concave up and concave down. ) Find a function that satisfies the given conditions and sketch its graph. ) lim f() =, lim f() =, lim f() =. ± ) lim g() = -, lim g() =, -, lim g() = -. - lim g() = + )

4 Sketch the graph and show all local etrema and inflection points. ) f() = + ) f() = /( - 7) ) f() = + - ) f() = + ) f() = + cos, π ) f() = Solve the problem.

5 ) Using the following properties of a twice-differentiable function = f(), select a possible graph of f. D) Derivatives < >, < - =, < - < < <, < - <, = < < <, > - =, > > >, > A) 7) From a thin piece of cardboard in. b in., square corners are cut out so that the sides can be folded up to make a bo. What dimensions will ield a bo of maimum volume? What is the maimum volume? Round to the nearest tenth, if necessar. B) ) A compan is constructing an open-top, square-based, rectangular metal tank that will have a volume of ft. What dimensions ield the minimum surface area? Round to the nearest tenth, if necessar. 9) Find two numbers whose difference is and whose product is a minimum ) Find two positive numbers whose product is and whose sum in a minimum. 7) Find a positive number such that the sum of the number and its reciprocal is as small as possible. C) Solve the problem. 7) A rectangular field is to be enclosed on four sides with a fence. Fencing costs $ per foot for two opposite sides, and $7 per foot for the other two sides. Find the dimensions of the field of area ft that would be the cheapest to enclose. - -

6 Answer Ke Testname: REVIEWTFALL ) ) -9e7-9 ) + e( + ) ) e ) ( + ) e ) eθ sin θ 7) (-7θ e-θ7 ) cos e -θ7 ) 9) - ) ) - ln 7 ) - ( + )( - ) - - ) ( + ) ) f-() = - 7 ) f-() = - + ) -. 7) 7. g ) π 9) π ) -π ) π ) ) ) ) ) +

7 Answer Ke Testname: REVIEWTFALL 7) - - ( + ) - ) - 9) et - et 7 ) ( + 7) 7 ) = ) = ) No minimum ) Maimum value is at = ; minimum value is - 7 at = - ) Maimum value of at = ; minimum value of - at = π ) Critical Pt. derivative Etremum Value = = Undefined local ma minimum - 7) Critical Pt. derivative Etremum Value =. = -. local ma local min. -. ) Critical Pt. derivative Etremum Value min = undefined min = = local ma 9) No ) Yes ) No ) Approimate local maimum at.7; approimate local minimum at 99.7 ) Approimate local maimum at.7; approimate local minima at and. ) Approimate local maimum at.; approimate local minima at -.9 and.7 ) Local maimum at -; local minimum at - ) No local etrema 7), decreasing on -,, increasing on,, local minimum at ) -, ; increasing on -, - and, ; decreasing on -, ; local ma at -, local min at 9) -, increasing for all real numbers, no relative etrema ), ; decreasing on (-, ); increasing on (, ); local minimum at ) Local minimum at = +; local maimum at = -; concave up on (, ); concave down on (-, ) ) Local minimum at = ; local maimum at = - ; concave up on (, ); concave down on (-, ) 7

8 Answer Ke Testname: REVIEWTFALL ) Concave up for all ; no inflection points ) Concave up on (, ) ; Concave down on (-, ) ; inflection point at ) Concave up on (-, ) and (, ) ; Concave down on (, ); inflection points at and ) Asmptotes: = -, = ) Asmptote: = ) (Answers ma var.) Possible answer: f() =

9 Answer Ke Testname: REVIEWTFALL 9) (Answers ma var.) Possible answer: f() =, > -, < ) Minimum: (-,-) No inflection points ) Local minimum: (-,-) Local maimum: (,) Inflection point: (,), (-, - ),(, )

10 Answer Ke Testname: REVIEWTFALL ) Local ma:,, min:,- Inflection point:, ) Local ma: -,, min:, - Inflection point: (,) ) Min: (,) - Inflection points: -,,,

11 Answer Ke Testname: REVIEWTFALL ) Local minimum: π, π - ; local maimum: π, π + Inflection points: π, π and π, π - ) A 7). in. b. in. b. in.; 99. in. ).7 ft b.7 ft b. ft 9) -, 7), 7) 7). $ b. $7

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Chapter Practice Dicsclaimer: The actual eam is different. On the actual eam ou must show the correct reasoning to receive credit for the question. SHORT ANSWER. Write the word or phrase that best completes