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.
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() -7 - - - - - - - - - - - - ) 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() =. - - + 9 + ) f() = - - + 7-7
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() =. ± - + - - - - - - - - - - - - - - 9) lim g() = -, lim g() =, -, lim g() = -. - lim g() = + ) - - - - - - - - - - - - - -
Sketch the graph and show all local etrema and inflection points. ) f() = + ) f() = /( - 7) - - - ) f() = + - ) f() = + ) f() = + cos, π ) f() = - + - Solve the problem.
) 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. - - - - - - - 7) 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. - -
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 ) ) ) ) ) +
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 -.777 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
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: = -, = - - - - - 7) Asmptote: = - - - - ) (Answers ma var.) Possible answer: f() = -. - - - - - - - - - -
Answer Ke Testname: REVIEWTFALL 9) (Answers ma var.) Possible answer: f() =, > -, < ) - - - - - - - - - - - - - - Minimum: (-,-) No inflection points ) Local minimum: (-,-) Local maimum: (,) Inflection point: (,), (-, - ),(, ) - - - - - 9
Answer Ke Testname: REVIEWTFALL ) Local ma:,, min:,- Inflection point:,- - - - - ) Local ma: -,, min:, - Inflection point: (,) - - - - - ) Min: (,) - Inflection points: -,,,.7.. - - - -. -. -.7
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). ft @ $ b. ft @ $7