Set 3: Multiple-Choice Questions on Differentiation

Transcription

1 Set 3: Multiple-Choice Questions on Differentiation 65 Set 3: Multiple-Choice Questions on Differentiation In each of Questions -5 a function is given. Choose the alternative that is the derivative, dy, of the function. dx. y = (4x + )( -x) 3 -(-x) ( -x) ( + 8x) ( -x) ( - 6x) 3(-x) (4x + ) ( - x) (6x + 7). -x y = 3x+I 7 (3x + ) 6x-5 (3x+ (3x + ) 7 (3x + ) 7-6x (3x + ) 3. y= x -x)3/ -x x I(3-x)3/ 3 4. y= (5x+ 30 (5x + ) 0 --(5x (5x + r 4 (5x + ) 4 30 (5x + ) 4 5. y = 3 x 3-4x - x 3 - x-l/ x - u. S/3 _ 8x3/ V,-JD-, ' - x-;3 -

2 66 Chapter 3: Differentiation In questions 6-3, differentiable functionsf and g have the values shown in the table. 6. If A= f + g, thena'(3) = f f' g g' If B = f g, then B () = If D = _!_, then D'(l) = g l If H(x) = -I f(x), then H'(3) = 4 0. If K(x) = (f) (x), then K'(O) = -3 l If M(x) =f(g(x)), thenm'(l) = If P(x) = f(x 3 ), then P' () = If S(x) =f~'(x), then S'(3) =

3 Set 3: Multiple-Choice Questions on Differentiation 67 In Questions 4- find y'. 4. y =,!x - \- -v x x+- xfx x- + 4x x,lx y =,/ x + x - x+l y 4y(x + ) x+l (x + x -)3; none of these 6. y= I-x (- -x -x (- none of these 7. ex y=ln-- ex - ex x ex - ex - ex 0 ex - -- ex - 8. y = taff x x +4

4 68 Differentiation y= + tan sec secx tanx+ 0 sec- x tanx secx + tanx secx + tanx 0. ex y= e ' + 0 (ex+ ) 4 l ) y= x + l l+-- +l + +I + + ) none of these of g atx= a? I. g is continuous II. g IS ~u.,-«~u.,,~~ ~ g s mcreasmg I HI only I, I and III only and III 3. is false? statements y fis atx = a fhas a asymptote at x = a a discontinuity at x = a f has a removable discontinuity atx= a

5 Set 3: Multiple-Choice Questions on Differentiation The functionfwhose graph is shown hasf' = 0 at x = y only and5 4and 7, 4, and 7, 4, 5, and A differentiable functionfhas values shown. Estimate f' (.5). 8.4 f(x) Water is poured into a conical reservoir at a constant rate. If h(t) is the rate of change of the depth of the water, then h is constant linear and increasing linear and decreasing nonlinear and increasing nonlinear and decreasing In Questions 7-33, find :. 7. y=x sm (x-:t=o). x sm--x cos- -- cos- x cos-. x sm--cos- -cos- 8. y= -- sinx -csc x cot x 4 cos x -4 csc x cot x cosx,vsinx -csc x

6 70 Chapter 3: Differentiation 9, y = C' COS x -e-x(cos x + sin x) e-x(sin x- cos x) e-x sin x -e-x(cos x + sin x) -e-x sin x 30. y = sec Fx sec./xtan./x tan,v x sec tan sec Fx tan Fx sec Fx tan Fx 3. y = ln 3 3 ln 3 ln x 3x ln x + ln 3 x 3(ln + ) none of these 3. l+x y= - 4x (- 4x (- ) x 4 -x 33. y = sin- x --JI- x -J- x l+x x -J-x ~l+x Use the graph to answer Questions It consists of two line segments and a semicircle. y 34. f'(x)=oforx= only only 4 only and4 and6 4

7 Set 3: Multiple-Choice Questions on Differentiation f' (x) does not exist for x = only and 6 only and,, and J'(5) = -3,,/3 37. At how many points on the interval [-5,5] is a tangent to y = x + cos x parallel to the secant line? none 3 more than From the values off shown, estimatef'() f(x) In each of Questions 39-4 a pair of equations is given which represents a curve parametrically. Choose the alternative that is the derivative :. *39. x = t- sin t and y = -cost sint -cost sin t -cos t sint cos t- -x -cos t y t- sin t *40. = COS 3 0 and y = sin 3 0 tan 3 0 -cot 0 cot 0 -tan 0 -tan 0 *4. = l - e-t and y = t + e-t e-r e-'- et+ l e' - e-, et - - *4. = - -t and y = - ln(l - t) (t < ) (- t) t- l -t t + Inx * An asterisk denotes a topic covered only in Calculus BC.

8 7 Chapter 3: Differentiation In each of Questions 43-46, y is a differentiable function of x. Choose the alternative that is the derivative :. 43. x 3 xy + y 3 = 3x x-3y 3x - -3y y-3x 3y-x 3x +3y-y 3x +3y COS ( + y) = 0 csc (x + y) - csc (x + y) -sinx siny sin(x+y) 45. sin x cos y - = 0 -cotx -coty cosx siny -csc y cos -cosx siny 46. 3x - xy + 5y = 3x+ y x-5y y-3x 5y-x 3x+ 5y 3x+4y none of these Individual instructions are given in full for each of the remaining questions of this set. *47. If x = t - and y = t4 - t3, then when t =, dzy is dx I If /(x) = x 4-4x 3 + 4x -, then the set of values of x for which the derivative equals zero is {,} {O} {O, -, -} {O,,} {-, +} 49. Iff(x) = 6-fx, then/"(4) is equal to * An asterisk denotes a topic covered only in Calculus BC. -

9 Set 3: Multiple-Choice Questions on Differentiation If ftx) = In x3, then/"(3) is none of these 5. If a point moves on the curve x dy. - 5, then, at (0, 5), -, s dx 5 nonexistent dy 5. If y = a sin ct+ b cos ct, where a, b, and care constants, then dt is ac (sin t + cos t) -c y -ay -y a c sin ct - b c cos ct 53. Iff(u) = sin u and u = g(x) = x -9, then (f O g)'(3) equals none of these 54. If f(x) = S and 5 00 = 5.06, which is closest tof'(l)? Iff(x) = (x-, then the set of x's for whichf'(x) exists is all reals all reals except x = and x = - all reals except x = - all reals except x = - and x = - 3 all reals except x = 56. If y = e'(x - ) then y"(o) equals none of these 57. If y =, then the derivative of y with respect to x is x +l x (x + ) 58. Iff(x) = and g(x) = ~, then the derivative off(g(x)) is +I,- --vx -x -(x + ) + n- (x + (x + ) ~(x+l)

10 74 Chapter 3: Differentiation *59. If = e 8 COS 0 and y = e 8 sin 0, then, when 0 =, : is 0 nonexistent - *60. If x =cost and y = cos t, then!; (sin t -:t= 0) is x- 4 cost 4 4y -4-4 cot t 6. If y = + x, then the derivative of y with respect to (x + l)(x - ) 6. lim h-,() ( 3-x h. ls x+l ( x) none of these - is I-x x nonexistent 63. lim h-+0 8+h -. ls h 0 l lim ls h-+0 h 0 e e nonexistent 65. lim x-+0 cos x-. ls none of these 66. The functionf(x) = x 3 on [-8, 8] does not satisfy the conditions of the mean-value theorem because f(o) is not defined ( C) f' ( -) does not exist f' (0) does not exist f(x) is not continuous on [-8, 8] f(x) is not defined for x < Iff(a) =f(b) = 0 andf(x) is continuous on [a, b], then f(x) must be identically zero f'(x) may be different from zero for all x on [a, b] there exists at least one number c, a< c < b, such thatf'(c) = 0 f'(x) must exist for every x on (a, b) none of the preceding is true * An asterisk denotes a topic covered only in Calculus BC.

11 Set 3: Multiple-Choice Questions on Differentiation If f(x) = x3-6x, at what point on the interval O ~ x ~./3, if any, is the tangent to the curve parallel to the secant line? - -Ji 0 nowhere 69. If his the inverse function off and iff(x) =, then h'(3) = Suppose y = f(x) = x 3-3x. If h(x) is the inverse function off, then h' (-) equals Suppose y = f(x) and x = J- (y) are mutually inverse functions. lff(l) = 4 and d = -3 ~ at x =, then dy at y = 4 equals Let y = f(x) and x = h(y) be mutually inverse functions. If f' () = 5, then what is the value of ~ at y =? dy It cannot be determined from the information given. *73. lim x->= ex equals 0 50! 00 none of these 74. Suppose lim g(x)-g(o) =. It follows necessarily that x->0 g is not defined at x = 0 g is not continuous at x = 0 The limit of g(x) as x approaches O equals g'(o) = g'(l)=o 75. If sin (xy) = x, then dy = ~ sec ( xy ) sec (xy) sec (xy)-y + sec (xy) sec (xy) - * An asterisk denotes a topic covered only in Calculus BC.

12 76 Chapter 3: Differentiation Use this graph of y = f (x) for Questions 7 6 and 77. y 76. f' (3) is most closely approximated by (D}.8 ls at = -3.3 O Use the following definition of the symmetric difference quotient for f' (x 0 ) Questions 78-8: for small values of h, for 78. f' (x 0 ) equals the exact value of the derivative at x = x 0 only when/ is linear whenever f is quadratic if and only ifj'(x 0 ) exists whenever lhl < none of these 79. To how many places is the symmetric difference accurate when it is used to approximate f'(o) forf(x) = 4x and h = 0.08? 3 4 more than To how many places isf'(x 0 ) accurate when it is used to approximatef'(o) forf(x) = 4x and h = 0.00? 3 4 more than 4

13 Set 3: Multiple-Choice Questions on Differentiation The value of f'(o) obtained using the symmetric difference quotient withf(x) = lxl and h = 0.00 is - 0 ± indeterminate 8. If! f(x) = g(x) and h(x) = sin x, then ; f(h(x)) equals g(sin x) COS g(x) g'(x) cos x g ( sin x) sin x. g(sin x) 83. r sin x. lffi -- ls x-> sin3x 84. lim -.-- is x-.0 sm4x nonexistent -cosx 85. lim is x->0 nonexistent 00 none of these tan nx 86. lim is x...,.o re 0 t r mx smx--t= is is 0 is oo oscillates between - and is none of these 88. Letf(x) = 3x -x3. The tangent to the curve is parallel to the secant through (0,) and (3,0) for x equal only to only to.44 only to.77 to and.804 to.44 and. 77

14 78 Chapter 3: Differentiation Questions 89 through 96 are based on the following graph of f(x), sketched on -6 ~ x ~ 7. Assume the horizontal and vertical grid lines are equally spaced at unit intervals. y =f(x) ~ I I \ I t V \ I/ ' ] fi /7 / - V \ I \ I "" I On the interval I< x <,f(x) equals -x- -x-3 -x 4 -x + x- 90. Over which of the following I. (-6,-3) II. I only II only I and II only I and III only II and III only 9. How many points of discontinuity on the interval -6 < x < 7? none For -6 < x -3,f' (x) equals Which of the following statements about the graph ofj'(x) is false? It consists of 6 horizontal segments. It has four jump discontinuities. f' (x) is discontinuous at each x in the set f' (x) ranges from -3 to. On the interval-i< x < l,f'(x) =,,,5}.

15 Set 3: Multiple-Choice Questions on Differentiation 79 *94. The graph in the xy-plane represented by x = 3 + sin t and y = cos t -, for -t :,;;;;; t :,;;;;; t, is a semicircle half of an ellipse a circle an ellipse a hyperbola 95. ll m _se_c x_c_os_x l x->0 equa s 0 none of these 96. The table gives the values of a functionfthat is differentiable on the interval [0,]: f(x) The best approximation off'(0.0) according to this table is At how many points on the interval [a,b] does the function graphed satisfy the Mean Value Theorem? y a b none 3 4 * An asterisk denotes a topic covered only in Calculus BC.

16 80 Chapter 3: Differentiation Answers for Set 3: Differentiation. C. D 4. E 6. A 8. B. A. E 4. C 6. C 8. D 3. B 3. C 43. C 63. B 83. B 4. B 4. A 44. A 64. B 84. C 5. E 5. D 45. D 65. B 85. E 6. B 6. E 46. B 66. E 86. D 7. B 7. D 47. E 67. B 87. C 8. E 8. A 48. E 68. A 88. E 9. D 9. A 49. E 69. B 89. A 0. C 30. D 50. A 70. C 90. E. A 3. E 5. D 7. A 9. E. B 3. B 5. B 7. E 9. D 3. D 33. C 53. C 73. D 93. B 4. D 34. C 54. D 74. D 94. B 5. A 35. E 55. E 75. C 95. C 6. E 36. B 56. D 76. B 96. C 7. C 37. D 57. A 77. D 97. D 8. E 38. C 58. B 78. B 9. A 39. A 59. E 79. B 0. D 40. D 60. B 80. E Many of the explanations provided include intermediate steps that would normally be reached on the way to a final algebraically simplified result. You may not need to reach the final answer. NOTE: the formulas or rules in parentheses referred to in the explanations are given on pages 46 and 47.. C.. A. 3. B. 4. B. By the product rule, (5), y' = (4x + )[3(- x)(-)] + (- x)3 4 = (l-x)(-x x) = (- x) (-6x) By the quotient rule, (6)., (3x+l)(-)-(-x)(3) -7 y = = (3x + ) (3x + ) Since y = (3 - x)u, by the power rule, (3). y'=_!_(3-3xrl/ (-)=-.,.J3-x Since y = (5x + r 3,y' = -6(5x + r4(5).

17 Answers for Set 3: Differentiation 8 5. E. 6. B. (f + g)'(3) = f'(3) + g'(3) = 4 + (-) 7. B. (f g)'() = f() g'() + g() J'() = 5(-) + (3) 8. E. ( _!_)' ()=- - -.g'(l)=-..!_(-3) g [g()] 3 9. D. 0. C., ( _) (0)= g(o) f'(o)- f(o) g'(o) = 5()-(-4) g [g(o)j 5. A M'(l) = f'(g(l)) g'(l) = f'(3)g'(l) = 4(-3). B. [f(x 3 )]' = f'(x 3 )-3x, so P'(l) = J'(l 3 ) 3! = 3 3. D. f(s(x)) = x implies that f'(s(x) S'(x) =. So, S'( 3 ) _ = j'(s(3)) f'(f- (3)) = j'(l) 4. D. Rewrite: y = xv ~ xlf ; so y' = x- + { x A Rewrite: y = (x + x- ) (Use rule (3).) 6. E. Use the quotient rule:,--- l 0 -x v'i-x --x -;c=- I-x + x y = '----- = ~--- l- x -x =--- ( x )3/ 7. C. Since y = Inex - In(ex -), y =x-in(ex -), 8. E. Use formula (8): y' =

18 8 Chapter 3: Differentiation 9. A. 0. D. Use formulas (3), (), and (9):, secxtanx+sec x secx(tanx+secx) y = = secx + tanx By the quotient rule, secx + tanx ++e-x)- - + e-x 4. D. Since y = In x + f In (x + ),, l x x + y = +- -,-=. x x" + x(x +I). E. 3. C. 4. A. 5. D. Since g' (a) exists, g is differentiable and thus continuous; g' (a)> 0. Near a vertical asymptote the slopes must approach ±oo. There is only one horizontal tangent. Using the symmetric difference quotient, f'(l.5) = f(l.6) - f(l.4) = ~ E. 7. D. 8. A. 9. A. 30. D. 3. E. 3. B. Since the water level rises more slowly as the cone fills, the answer is or. Furthermore, at every instant the portion of the cone containing water is similar to the entire cone; the volume is proportional to the cube of the depth of the water. The rate of change of depth (the derivative) is therefore not linear. I ( l ) ( ) y = x cos- - + sm - x. Since y = _!_ csc x, y' = (--csc x cot x. ). y' = e-x (-sinx) + cosx(-e-x). Use formulas (3) and (): y' = sec Sec tan~{ ). x(3 In x). y' = + ln 3 x. The correct answer s 3 ln x + ln 3 x. y'= (- x )(x)-(l + x )(-x) (l-x) 33. C. y'= -x l.(-x) ~-x

19 Answers for Set 3: Differentiation C. 35. E. 36. B. The only horizontal tangent is at x = 4. Note that f' () does not exist. The graph has corners at x = and x = ; the tangent line is vertical at x = 6. Consider triangle ABC: AB = ; radius AC= ; thus, BC = -J3 and AC has m = tangent line is perpendicular to the radius. A B 37. D. The graph of y = x + cos xis shown in window [-5,5] x [-6,6]. The average rate of change is represented by the slope of secant segment AB. There appear to be 3 points at which tangent lines are parallel to AB. 38. C.!'() = A. dy _ dt _ sin t dx - dx - -cos t. dt 40. D. 4. E. 4. C. 43. C. dy dy - de - 3sin 8cose dx - dx - e sine. de dy =.JfJ_ = -e-t = e' _. dx!i±_ e- dt dy =-- dx h Since and - = -- t en dt -t dt (l-t) ' Let y' be! ; then 3x dy =-t=.!.. dx (xy' + y) + 3yy' = O; y'(3y - x) = y-3x

20 84 Chapter 3: Differentiation 44. A. 45. D. 46. B. 47. E. 48. E. 49. E. 50. A. - sin (x + y)(l + y') = O; l-sin(x+y) = y'. sin(x+ y) I I COS cosx+ smy y = ;y = O smy 6x-(xy' + y) + loyy' = O; y'(0y-x) = y-6x. d ' 4t 3-6t d y 4t = = t -3t(t # O); -=--. Replace t by. dx t dx t f'(x) = 4x 3 - x + Sx = 4x(x- l)(x ). f'(x) = 8x-u;f"(x) = -4x-3 = - 4 ;f"(4) = - i f(x) = 3 In x; f' (x) = - ; f"(x) =. Replace x by D. 5. B. x + yy' = 0; y' = y" =- Y ' y' = ac cos ct- be sin ct; y" = -ac sin ct- bc cos ct.. At (0,5), y" = - 5 -o C. 54. D. 55. E. 56. D. 57. A. 58. B. 59. E. 60. B. l(f o g)' at x = 3 equalsf'(g(3)), g'(3) equals cos u (at u = 0) times x (at = 3) = 6 = 6. I 5LOOl ]6-5 f () = 0.00 = Heref'(x) equals -x x - I. (x - y I : e" + ex( - ) = xex y"=xex+exandy"(0)=0 + =. dy dy ~ _ dy _ x -=- Smcey -x +, -- dx dx dx x dx l Note thatf(g(x)) = x+j. When simplified, dy = cose+sine. dx cos0-sin0 Since (if sin t-::/. 0) dy = - sint = -4sint cost and dx = sint, dt dt then dy = 4 cos t. Then, dx

21 Answers for Set 3: Differentiation 85 dy dy 6. A. dx x+l = = de~-\) de-i ) -x dx ---~~ NOTE: Since each of the limits in Questions 6 through 65 yields an indeterminate form of the type %, we can apply L'Hopital's rule in each case, getting identical answers. 6. C. 63. B. 64. B. 65. B. 66. E. 67. B. 68. A. The limit given is the derivative of f(x) = x 6 at x =. The given limit is the definition for f' (8), where f (x) = This isf'(e), wheref(x) = ln x. I f (x) = -,--3 3x+ The limit is the derivative off(x) = cos x at x = O;f'(x) = - sin x. Sincef'(x) =,f'(o) is not defined;f'(x) must be defined on (-8,8). Sketch the graph off(x) = - Ix!; note thatf(-) =f(l) = 0 and thatfis continuous on [-,]. Only holds. Note thatf(o) = J(-3) = 0 and thatf'(x) exists on the given interval. By the MVT, there is a number c in the interval such thatf'(c) = 0. c =, then 6c - 6 = 0. (- is not in the interval.) 69. B. Since the inverse, h, off(x) = _!_ is h(x) = _!_, h'(x) = -. Replace x by C. 7. A. 7. E. Since f'(x) = 6x - 3, therefore h'(x) = _ 3 ; also, f(x), or x 3-3x, equals -, by observation, for x =. So h'(-) or _ 3 (when x = ) l equals 6 _ 3-3. Use (: J,." = (ti, wheref(x 0 ) = Yo _\-O (!t,=(il=~3 dx dy l dx.. However, to evaluate dy for a particular value of y, we need to dx evaluate at the corresponding value of x. This question does not say dx what value of x yields y =.

22 86 Chapter 3: Differentiation 73. D. 74. D. 75. C. After 50(!) applications of L'Hopital's rule we get ~~ ;~!, which "equals" oo. A perfunctory examination of the limit, however, shows immediately that the answer is oo. In fact, lim ~ for any positive integer n, no matter how x~oo xn large, is oo. The given limit is the derivative of g(x) at x = 0. cos(xy )(xy' + y) = ; xcos(xy )y' = - ycos(xy );, - ycos(xy) y - - xcos(xy) 76. B. 77. D. 78. B. 79. B. 80. E. 8. B. The tangent line appears to contain (3,-.6) and (4,-.8). f' (x) is least at the point of inflection of the curve, at about 0.7. Note that (x+h)-(x-h) = x, the derivative of f(x) = x. You should be h able to confirm that the symmetric difference quotient works for the general quadratic g(x) = ax +bx+ c. 40.os _ 4 -o.os By calculator,f'(o) = and --- = _ Now = Note that any line determined by two points equidistant from the origin will necessarily be horizontal. -h h 8. D. Note that! f(h(x)) = f'(h(x)) h'(x) = g(h(x)). h'(x) = g(sinx). cosx. NOTE: In Questions 83 through 87 the limits are all indeterminate forms of the type 0 0. We have therefore applied L'Hopital's rule in each one. The indeterminacy can also be resolved by introducing sin a which approaches as a approaches 0. The a latter technique is presented in square brackets.

23 Answers for Set 3: Differentiation B. 84. C. lirn sinx = lirn cos x =. =. x-->0 x-->0.... (sinx) [Usmg sin x = sm x cos x yields m - cos x = =.] l. sin3x. m -- = m x-->0 sin4x x-->0 3cos3x 3-3 =-=- 4cos4x 4 4 x->0 [We rewrite sin 3 x as sin4x tion approaches 3 4.] sin3x 4x 3 As x ~ 0, so d o 3x an d 4 x; the frac- 3x sin4x E. lim l-cosx = lim sinx =0. sin ~ [We can replace - cos x by sin.::, getting lim = lim -->0 l. j smlirn sin.:: -f = 0. l.] x-->0 = x-->() 86. D. lim tantx = lim (sectx)-t = t = n. x-->0 x-;o tan tx sin tx sin tx 0 h [ --= = n ; as x (or nx) approaches, t e ongmal COS t t COS fraction approaches n f = n.] 87. C. The limit is easiest to obtain here if we rewrite: lim x sin _!_ x-->= sin(l/ x) l -oo -oo (/ x) - -. = lim x-->= 88. E. Sincef(x) = 3' -x,f'(x) = 3x ln 3-3x Further,Jis continuous f' is differentiable on (0,3). So the MVT applies. We therefore [0,3] and c such that f' ( c) = f < 3 )-f (O) =!.. Using a graphing calculator, we key in 3 3 3' ln We can either graph and use the [root] option (twice; 3 where Y has x-intercepts) or use the [solve] option (estimating solutions at= and=.5). The graphics calculator shows that c may be either.44 or.77. These are the x-coordinates of points on the graph off(x) at which the tangents are parallel to the secant through points (0, ) and (3,0) on the curve.

24 88 Chapter 3: Differentiation 89. A. The line segment passes through (,-3) and (,-4). f'(x) J' I I , ~ Use the graph ofj'(x) for Questions 90 through E. 9. E. 9. D. 93. B. 94. B. f'(x) = 0 when the slope off(x) is O; that is, when the graph off is a horizontal segment. The graph ofj'(x) jumps at each comer of the graph off(x), namely, atx equal to-3, -,,, and 5. On the interval (-6,-3),f(x) = %(x + 5). Verify that all choices but are true. The graph ofj'(x) has five (not four) jump discontinuities. Since x - 3 = sin t and y + = cos t, we have (x-3)+ (y+ ) =4 This is the equation of a circle with center at (3,-) and radius. In the domain given, -t ~ t ~ n, the entire circle is traced by a particle moving counterclockwise, starting from and returning to (3, -3). 95. C. Using L'Hopital's rule,. sec x cos x. sec x tan x + sin x m = m ~o x -o x. sec3 x+secxtan x+cosx l+l O+l - Im - x-> C. The best approximation tof'(0.0) is f(0. 0)-f(O.lO)

25 Answers for Set 3: Differentiation D. y a b The average rate of change is represented by the slope of secant segment AB. There appear to be 3 points at which the tangent lines are parallel to AB.