MATH 150/GRACEY PRACTICE FINAL. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

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1 MATH 0/GRACEY PRACTICE FINAL Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Choose the graph that represents the given function without using a graphing utilit. ) f() = + + A) B) ) C) D)

2 The graph of a function is given. Choose the answer that represents the graph of its derivative. ) ) A) B) C) D) Solve the problem.

3 3) The graphs below show the first and second derivatives of a function = f(). Select a possible graph f that passes through the point P. 3) f f P P A) B) C) D) 3

4 Provide an appropriate response. ) Which of the following integrals, if an, calculates the area of the shaded region? (-, ) (, ) 3 ) A) 0 d B) - - d C) - d D) d - Choose the graph that represents the given function without using a graphing utilit. ) f() = + 3 A) B) ) C) D)

5 The graph of a function is given. Choose the answer that represents the graph of its derivative. ) ) A) B) C) D)

6 Determine from the graph whether the function has an absolute etreme values on the interval [a, b]. 7) 7) A) Absolute minimum onl. B) Absolute maimum onl. C) No absolute etrema. D) Absolute minimum and absolute maimum.

7 The graph of a function is given. Choose the answer that represents the graph of its derivative. ) ) A) B) C) D)

8 Determine from the graph whether the function has an absolute etreme values on the interval [a, b]. 9) 9) A) Absolute minimum and absolute maimum. B) Absolute minimum onl. C) Absolute maimum onl. D) No absolute etrema. Solve the problem. ) If f() = + 3 and g() = - 7, find f(g()). A) - B) + C) - D) ) ) The accompaning figure shows the graph of = - shifted to a new position. Write the equation for the new graph. ) A) = - - B) = - + C) = -( + ) D) = -( - ) Find the domain and range of the function. ) g(z) = - z A) D: [0, ), R: (-, ) B) D: [-,], R: [0,] C) D: (-,), R: (-,) D) D: (-, ), R: (0, ) For the given function, simplif the epression f( + h) - f() h 3) f() = - 7 A) B) -9 C) D) -. ) 3)

9 Find the domain and range of the function. - ) g(z) = z + A) D: [, ), R: (-, ) B) D: (-,-), R: (0, ) C) D: [0, ), R: (-, ) D) D: (-, ), R: (-,0) ) Graph the function. Determine the smmetr, if an, of the function. ) = ) A) No smmetr B) Smmetric about the -ais C) Smmetric about the -ais D) No smmetr

10 Describe how to transform the graph of f into the graph of g. ) f() = and g() = ) A) Vertical scaling b a factor of C) Horizontal scaling b a factor of B) Horizontal scaling b a factor of D) Vertical scaling b a factor of Solve the problem. 7) The figure shown here shows a rectangle inscribed in an isosceles right triangle whose hpotenuse is units long. Epress the area A of the rectangle in terms of. 7) - A) A() = ( - ) B) A() = ( - ) C) A() = ( - ) D) A() = Find a formula for the function graphed. ) ) A) f() = - +, - -, - < 3 -, 3 < B) f() = +, - -, - < < 3 -, 3 C) f() = +, - -, - < 3 -, 3 < D) f() = +, - < -, - < 3 -, 3 < <

11 Solve the problem. 9) The accompaning figure shows the graph of = shifted to a new position. Write the equation for the new graph. 9) A) = ( - ) B) = ( + ) C) = - D) = + Find all vertical asmptotes of the given function. - 0) R() = A) = -7, = -30, = B) = -, = 7 C) = -, = 0, = 7 D) = -7, = 0, = 0) Find the limit, if it eists. ) lim ) A) B) - C) - D) Does not eist Find the limit. ) lim (-π/)- sec A) B) 0 C) - D) ) Find the slope of the curve for the given value of. 3) = +, = 3) A) slope is -39 B) slope is 0 C) slope is - D) slope is 3

12 Determine the limit b sketching an appropriate graph. ) lim f(), where f() = - - for < for 7 ) A) B) - C) -3 D) -9 Find the limit. ) lim A) B) 0 C) - 7 D) ) Determine the limit b sketching an appropriate graph. ) lim f(), where f() = -+ - < 0, or 0 < 3 = 0 0 < - or > 3 ) A) -0 B) Does not eist C) - D)

13 Find all horizontal asmptotes of the given function, if an ) h() = ) A) = 0 B) = 9 C) = D) no horizontal asmptotes Give an appropriate answer. ) Let lim f() = - and - lim g() = -. Find - lim [f() - g()]. - A) - B) - C) - D) - ) Find the limit. 9) lim -- - A) B) - C) - D) 0 9) Find the intervals on which the function is continuous. 30) = - A) continuous on the intervals (-, - ] and [, ) B) continuous everwhere C) continuous on the interval [, ) D) continuous on the interval [-, ] 30) Find all horizontal asmptotes of the given function, if an. 3) h() = ) A) = B) = C) = 0 D) no horizontal asmptotes Find the slope of the line tangent tangent to the graph a the given point. 3) = - -, = - A) m = - B) m = C) m = D) m = - 3) Solve the problem. 33) At time t 0, the velocit of a bod moving along the s-ais is v = t - 9t +. When is the bod moving backward? A) < t < B) t > C) 0 t < D) 0 t < 33) 3) Suppose that the velocit of a falling bod is v = ks (k a constant) at the instant the bod has fallen s meters from its starting point. Find the bod's acceleration as a function of s. A) a = ks3 B) a = ks3 C) a = ks D) a = ks 3) 3

14 Find the indicated derivative. 3) Find if = - cos. A) = - sin B) = - cos C) = cos D) = sin 3) Solve the problem. 3) At time t, the position of a bod moving along the s-ais is s = t3 - t + 3t m. Find the total distance traveled b the bod from t = 0 to t = 3. A) 3 m B) 9 m C) 3 m D) 7 m 3) The equation gives the position s = f(t) of a bod moving on a coordinate line (s in meters, t in seconds). 37) s (m) 37) 3 t (sec) When is the bod moving backward? A) < t < 3, < t <, 7 < t < 9 B) < t < 9 C) < t < 3, < t <, 7 < t < D) < t Solve the problem. 3) A heat engine is a device that converts thermal energ into other forms. The thermal efficienc, e, of a heat engine is defined b e = Q h - Qc, Qh where Qh is the heat absorbed in one ccle and Qc, the heat released into a reservoir in one ccle, is a constant. Find de dqh. 3) A) de dqh = - Q c Qh 3 B) de dqh = Q c Qh C) de dqh = Q c Qh 3 D) de dqh = - Q c Qh

15 39) The position(in feet) of an object oscillating up and down at the end of a spring is given b s = A sin k t at time t (in seconds). The value of A is the amplitude of the motion, k is a m measure of the stiffness of the spring, and m is the mass of the object. Find the object's velocit at time t. A) v = A k m cos k m t ft/sec B) v = A m k cos k m t ft/sec 39) C) v = - A k m cos k m t ft/sec D) v = A cos k m t ft/sec Use implicit differentiation to find d/d and d/d. 0) - = A) d d = - ; d d = - B) d d = - ; d d = - ( - ) 0) C) d d = - ; d d = ( - ) 3 D) d d = - ; d d = - Solve the problem. Round our answer, if appropriate. ) Bole's law states that if the temperature of a gas remains constant, then PV = c, where P = pressure, V = volume, and c is a constant. Given a quantit of gas at constant temperature, if V is decreasing at a rate of in. 3/sec, at what rate is P increasing when P = 0 lb/in. and V = 0 in.3? (Do not round our answer.) A) lb/in. per sec B) 30 lb/in. per sec C) lb/in. per sec D) 0 3 lb/in. per sec ) Solve the initial value problem. ) dr dt = t + sec t, r(-π) = ) A) r = t + cot t + - π B) r = + tan t - 3 C) r = t + tan t + - π D) r = t + tan t + - π Solve the problem. 3) The strength S of a rectangular wooden beam is proportional to its width times the square of its depth. Find the dimensions of the strongest beam that can be cut from a -in.-diameter clindrical log. (Round answers to the nearest tenth.) 3) " A) w =.9 in.; d =. in. B) w =.9 in.; d = 9. in. C) w = 7.9 in.; d =. in. D) w = 7.9 in.; d =. in.

16 Use differentiation to determine whether the integral formula is correct. ) d = - ( + )3 ( + ) + C A) Yes B) No ) Determine all critical points for the function. 3 ) f() = - 7 A) = -7 B) = 7 C) = 0 and = 7 D) = - and = 0 ) 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. ) ) A) Local minimum at = ; local maimum at = -; concave down on (-, ) B) Local minimum at = ; local maimum at = -; concave up on (-, ) C) Local minimum at = ; local maimum at = -; concave up on (0, ); concave down on (-, 0) D) Local minimum at = ; local maimum at = -; concave down on (0, ); concave up on (-, 0) Solve the problem. 7) Given the velocit and initial position of a bod moving along a coordinate line at time t, find the bod's position at time t. v = -t +, s(0) = A) s = - t + t + B) s = - t + t - C) s = t + t - D) s = -t + t + ) A compan is constructing an open-top, square-based, rectangular metal tank that will have a volume of 30 ft3. What dimensions ield the minimum surface area? Round to the nearest tenth, if necessar. A) 3.9 ft 3.9 ft ft B) 7.7 ft 7.7 ft 0. ft C) 3. ft 3. ft 3. ft D). ft. ft. ft 7) )

17 9) 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. A).7 in..7 in..7 in.; 7. in3 B).7 in..7 in. 3.3 in.;. in3 C) 3.3 in. 3.3 in. 3.3 in.; 37 in3 D) in. in.. in.;. in3 9) 7 0) Suppose that g is continuous and that g() d = and g() d = 3. Find A) -3 B) 3 C) -3 D) 3 7 g() d. 0) Use a finite approimation to estimate the area under the graph of the given function on the stated interval as instructed. ) f() = between = and = using a left sum with four rectangles of equal width. A) B) 9 C) D) ) Use the substitution formula to evaluate the integral. 0 3t ) - + t dt A) - 3 B) C) D) ) Evaluate the integral b using multiple substitutions. 3) + sin ( - ) sin ( - ) cos ( - ) d A) ( + cos ( - )) 3/ + C B) 3 + sin( - ) + C 3) C) 3 ( + sin ) 3/ + C D) 3 ( + sin ( - )) 3/ + C Evaluate the sum. ) k = 3 A) B) C) D) ) Find the length of the curve. ) = 3/ from = 0 to = ) A) 33 7 B) 33 3 C) 9 D) 33 7

18 Find the area of the shaded region. ) = sec 3 ) = cos A) + B) C) - D) - Evaluate the integral. 7) e- d A) - e - + C B) e - + C C) -e - + C D) - e - + C 7) Evaluate or simplif the given epression. ) sin(3 sin- ) A) 3-33 B) 3-3 C) D) ) 9) sin (tan- ) 9) A) + + B) - C) - D) + Evaluate the integral. e 0) dt t A) 0 B) e - C) ln D) 0) ) + + d ) A) ln 3 3 B) ln C) ln D) ln 7

19 Find the value of df-/d at = f(a). ) f() =, 0, a = 3 A) B) C) D) ) Solve the problem. 3) Let g() =. Find a function = f() so that (f g)() =. 3) A) f() = B) f() = C) f() = D) f() = ) If f() = 7-9 and g() = - 9 +, find g(f()). A) B) - C) 3 D) ) ) If f() =, g() =, and h() = + 0, find f(g(h())). ) A) + 0 B) + C) + D) + 0 Find the limit. + - ) lim 0 A) 0 B) / C) / D) Does not eist ) Use the graph to evaluate the limit. 7) lim 0 f() 7) A) - B) C) does not eist D) - Find the limit. ) lim ( + csc ) 0+ A) B) C) 0 D) Does not eist ) 9

20 9) lim 0 ( - cot ) A) - B) C) 0 D) Does not eist 9) Suppose that the functions f and g and their derivatives with respect to have the following values at the given values of. Find the derivative with respect to of the given combination at the given value of. f() g() f () g () 70) ) f() g(), = 3 A) B) C) D) Suppose u and v are differentiable functions of. Use the given values of the functions and their derivatives to find the value of the indicated derivative. 7) u() =, u () = -, v() = 7, v () = -3. 7) d (u - v) at = d A) -0 B) - C) 0 D) 3 Graph the equation and its tangent. 7) Graph = 3 + and the tangent to the curve at the point whose -coordinate is 0. 7) A) - B)

21 C) D) Solve the problem. 73) Find the optimum number of batches (to the nearest whole number) of an item that should be produced annuall (in order to minimize cost) if 300,000 units are to be made, it costs $ to store a unit for one ear, and it costs $0 to set up the factor to produce each batch. A) 0 batches B) batches C) batches D) batches 73) Find the point(s) at which the given function equals its average value on the given interval. 7) f() = ; [0, ] A) 3 B) C) 7 D) 7) Find the area of the shaded region. 7) 7) A). B) C) D) 7.

22 7) = - 3 7) A) = - B) C) D) Evaluate the integral. 77) 0 ln π e sin e d 77) A) B) - cos C) + cos D) - Find all vertical asmptotes of the given function. - 7) f() = - 3 A) = -, = B) = 0, = C) = 0, = - D) = 0, = -, = 7) Calculate the derivative of the function. Then find the value of the derivative as specified. 79) dr dt t =3 if r = - t 79) A) dr dt = ( - t)3/ ; dr dt t =3 = C) dr dt = - ( - t)3/ ; dr dt t =3 = - B) dr dt = - ( - t)3/ ; dr dt t =3 = - D) dr dt = ( - t)3/ ; dr dt t =3 = Evaluate the integral. 0) 3 + d A) - + -/ + C B) 3 + 3/ + C 0) C) + 3/ + C D) 3 + 3/ + C

23 Set up an integral for the length of the curve. ) = 3 tan, 0 π ) A) C) π/ + 9 sec d B) 0 π/ + 3 sec d D) 0 π/ 0 π/ sec d - 9 sec d Evaluate the integral. ) 3 d A) 77 ln B) 7 ln + C C) ln D) 77 ) Find all points where the function is discontinuous. 3) 3) A) = -, = B) = -, = 0, = C) None D) = 0 Solve the problem. ) The number of gallons of water in a swimming pool t minutes after the pool has started to drain is Q(t) = 0(0 - ). How fast is the water running out at the end of minutes? A) 0 gal/min B) gal/min C) 0 gal/min D) 00 gal/min ) Evaluate the integral b using multiple substitutions. ) (3-7) sin (3-7) cos (3-7) d ) A) cos (3) + C B) sin (3-7) + C C) sin (3-7)+ C D) sin (3-7) + C Solve the problem. ) A fisherman is about to reel in a -lb fish located ft directl below him. If the fishing line weighs oz per foot, how much work will it take to reel in the fish? Round our answer to the nearest tenth, if necessar. A) ft lb B) 7. ft lb C) ft lb D) ft lb ) 3

24 Answer Ke Testname: M0_FINAL_PRACTICE ) C ) D 3) A ) D ) B ) B 7) C ) A 9) A ) A ) A ) B 3) C ) D ) D ) A 7) B ) C 9) A 0) C ) A ) D 3) D ) A ) D ) C 7) B ) A 9) A 30) A 3) B 3) A 33) A 3) C 3) C 3) B 37) A 3) A 39) A 0) C ) B ) D 3) B ) A ) B ) C 7) A ) A 9) A 0) A

25 Answer Ke Testname: M0_FINAL_PRACTICE ) C ) C 3) D ) B ) D ) C 7) D ) B 9) A 0) A ) D ) D 3) C ) C ) B ) C 7) D ) A 9) D 70) A 7) C 7) B 73) C 7) D 7) C 7) A 77) C 7) C 79) A 0) C ) B ) A 3) D ) D ) C ) B

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. D) D: (-, 0) (0, )

Midterm Practice Test MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the domain and graph the function. ) G(t) = t - 3 ) 3 - -3 - - 3 - - -3