Math 2

Let f be a function such that f(1)=2. At each point (x,y) on the graph of f, the slope is given by dydx=5xy−x2−y2−5. Which of the following statements is true? A f has a relative minimum at x=1 B f has a relative maximum at x=1 C f has neither a relative minimum nor a relative maximum at x=1 D There is insufficient information to determine whether f has a relative minimum, a relative maximum, or neither at x=1

A

Let f be function such that f(-1) = 1. At each point (x,y) on graph of f, slope is given by dy/dt = -x2 - xy + y2 - 1. Which of following statements is true? A f has relative minimum at x=-1 B f has relative maximum at x=-1 C f has neither relative minimum nor relative maximum at x=-1 D There is insufficient information to determine whether f has a relative minimum, a relative maximum, or neither at x=−1

A

Let R be the region between the graph of y=e−2x and the x-axis forx≥3. The area of R is A 1/2e6 B 1/e6 C 2/e6 D π/2e6 E infinite

A 1/2e6

The absolute maximum values of f(x)=x^3-3x^2+12 on the closed interval [−2, 4] occurs at x = A 4 B 2 C 1 D 0 E -2

A 4

If ∫(1 to x) f(t)dt = 20x/sqrt of (4x2 + 21) - 4, then ∫(1 to ∞) f(t)dt is A 6 B 1 C -3 D -4 E divergent

A 6

Let f(x)= ∫(-2 to x2-3x) e^t2 dt. At what value of x is f(x) a minimum? A For no value of x B 1/2 C 3/2 D 2 E 3

C 3/2

The function f has a first derivative given by f′(x)=x(x−3)2(x+1). At what values of x does f have a relative maximum? A -1 only B 0 only C -1 and 0 only D -1 and 3 only E -1, 0, and 3

A -1 only

limx→π/2 3cosπ/2x-π is A -3/2 B 0 C 3/2 D nonexistent

A -3/2

∫(1 to ∞) x2/(x3+2)2 dx is A -1/2 B 1/9 C 1/3 D 1 E divergent

B 1/9

The graph of f′, the derivative of f, is shown in the figure above. The function f has a local maximum at x= A -3 B -1 C 1 D 3 E 4

C 1

The function f is defined by f(x)=x2e^−x2. At what values of x does f have a relative maximum? A -2 B 0 C 1 only D -1 and 1

D -1 and 1

limx→2 (x2 + x -6)/(x2 - 4) is A -1/4 B 0 C 1 D 5/4 E nonexistent

D 5/4

∫(0 to 3) dx/(1-x)2 A -3/2 B -1/2 C 1/2 D 3/2 E divergent

E divergent

A rectangle with one side on the x-axis has its upper vertices on the graph of y = cos x, as shown in the figure above. What is the minimum area of the shaded region? A 0.799 B 0.878 C 1.140 D 1.439 E 2.000

B 0.878

∫(1 to ∞) 1/xP dx and ∫(0 to 1) 1/xP dx both diverge when p = A 2 B 1 C 1/2 D 0 E -1

B 1

The figure above shows the graph of f′ , the derivative of function f, for −6<x<8 . Of the following, which best describes the graph of f on the same interval? A 1 relative minimum, 1 relative maximum, and 3 points of inflection B 1 relative minimum, 1 relative maximum, and 4 points of inflection C 2 relative minima, 1 relative maximum, and 2 points of inflection D 2 relative minima, 1 relative maximum, and 4 points of inflection E 2 relative minima, 2 relative maxima, and 3 points of inflection

A

The graph of f', the derivative of f, is shown in the figure above. Which of the following describes all relative extrema of f on the open interval (a,b)? A One relative maximum and two relative minima B Two relative maxima and one relative minimum C Three relative maxima and one relative minimum D One relative maximum and three relative minima E Three relative maxima and two relative minima

A

Let g be the function given by g(x)=x2ekx, where k is a constant. For what value of k does g have a critical point at x=2/3? A -3 B -3/2 C -1/3 D 0 E There is no such k

A -3

The function f is defined on the closed interval [0,8]. The graph of its derivative f' is shown above. At what value of x does the absolute minimum of f occur? A 0 B 2 C 4 D 6 E 8

A 0

∫(0 to ∞) x/(1+x2)2 dx is A 1/2 B 1 C π/2 D divergent

A 1/2

The figure above shows the graph of f′, the derivative of the function f, on the open interval -7 < x < 7. If f′ has four zeros on -7 < x < 7, how many relative maxima does f have on -7 < x < 7? A One B Two C Three D Four E Five

A One

The graphs of the derivatives of the functions f, g, and h are shown above. Which of the functions f, g, or h have a relative maximum on the open interval a<x<b ? A f only B g only C h only D f and g only E f, g, and h

A f only

The function defined by f(x)=x3−3x2 for all real numbers x has a relative maximum at x = A -2 B 0 C 1 D 2 E 4

B 0

A particle moves along the x-axis so that its acceleration at any time t is a(t)=2t−7. If the initial velocity of the particle is 6, at what time t during the interval 0≤t≤4 is the particle farthest to the right? A 0 B 1 C 2 D 3 E 4

B 1

If f′(x) = cos x and g′(x) = 1 for all x, and if f(0)=g(0)=0, then limx→0 f(x)/g(x) is A π/2 B 1 C 0 D -1 E nonexistent

B 1

If a and b are positive constants, then limx→∞ ln(bx+1)/ln(ax2+3)= A 0 B 1/2 C 1/2ab D 2 E Infinity

B 1/2

∫(1 to ∞) xe^-x2 dx is A -1/e B 1/2e C 1/e D 2/e E divergent

B 1/2e

Let f be the function defined by f(x)=lnx/x. What is the absolute maximum value of f ? A 1 B 1/e C 0 D -e E f does not have an absolute maxima value

B 1/e

Let f be the function defined by f(x)=3x+2e−3x, and let g be a differentiable function with derivative given by g′(x)=4+1x. It is known that limx→∞g(x)=∞. The value of limx→∞f(x)g(x) is A 0 B 3/4 C 1 D nonexistent

B 3/4

The third derivative of the function f is continuous on the interval (0,4). Values for f and its first three derivatives at x=2 are given in the table above. What is limx→2 f(x)/(x-2)^2? A 0 B 5/2 C 5 D 7 E Limit does not exist

B 5/2

Let f be twice-differentiable function. Selected values of f' and f'' are shown in table above. Which of following statements are true? I. f has neither relative minimum nor relative maximum at x=1 II. f has relative maximum at x=1 III. f has relative maximum at x=4 A I only B II only C III only D I and III only

B II only

Let f be a differentiable function with a domain of (0,10). It is known that f′(x), the derivative of f(x), is negative on the intervals (0,2) and (4,6) and positive on the intervals (2,4) and (6,10). Which of the following statements is true? A f has no relative minima and three relative maxima. B f has one relative minimum and two relative maxima. C f has two relative minima and one relative maximum. D f has three relative minima and no relative maxima.

C

limx→0 ex-1/x is A infinity B e-1 C 1 D 0 E ex

C 1

Let f be the function with first derivative defined by f′(x)=sin(x3) for 0≤x≤2. At what value of x does f attain its maximum value on the closed interval 0≤x≤2? A 0 B 1.162 C 1.465 D 1.845 E 2

C 1.465

limx→0 (ex - cosx - 2x)/(x2 - 2x) is A -1/2 B 0 C 1/2 D 1 E nonexistent

C 1/2

∫(1 to ∞) x/(1+x2)2 is A -1/2 B -1/4 C 1/4 D 1/2 E divergent

C 1/4

The function f given by f(x)=2x3−3x2−12x has a relative minimum at x= A -1 B 0 C 2 D (3-sqrt of 105)/4 E (3+sqrt of 105)/4

C 2

∫(0 to 4) 1/(sqrtx(1+sqrtx)) dx is A ln 3 B ln 5 C 2 ln 3 D divergent

C 2 ln 3

The graph of f′, the derivative of the function f, is shown above for 0≤x≤10. The areas of the regions between the graph of f′ and the x-axis are 20, 6, and 4, respectively. If f(0)=2, what is the maximum value of f on the closed interval 0≤x≤10? A 16 B 20 C 22 D 30 E 32

C 22

The graph of the derivative of a function f is shown in the figure above. The graph has horizontal tangent lines at x = −1, x = 1, and x = 3 . At which of the following values of x does f have a relative maximum? A -2 only B 1 only C 4 only D -1 and 3 only E -2, -1, and 4

C 4 only

The points (3, 0), (x,0) , (x, 1/x2) , and (3, 1/x2) are the vertices of a rectangle, where x≥3, as shown in the figure above. For what value of x does the rectangle have a maximum area? A 3 B 4 C 6 D 9 E No such value of x

C 6

The graph of f is shown in the figure above. If g(x)=∫(a to x) f(t)dt, for what value of x does g(x) have a maximum? A a B b C c D d E Cannot be determined from info given

C c

An antiderivative of ex/ex−1 is ln|ex−1|. Which of the following statements about the integral ∫(-2 to 2) ex/(ex-1) dx is true? A Integral equals e2/e2-1 - e^-2/e^-2 - 1 B Integral equals lnIe2-1I - lnIe^-2 - 1I C Integral diverges because lim x->0- ex/ex-1 does not exist D Integral diverges because lim x->0- lnIex - 1I does not exist

D

The function f is defined on the closed interval [0, 1] and satisfies f(0)=f(12)=f(1). On the open interval (0, 1), f is continuous and strictly increasing. Which of the following statements is true? A f attains both a minimum value and a maximum value on the closed interval [0, 1]. B f attains a minimum value but not a maximum value on the closed interval [0, 1]. C f attains a maximum value but not a minimum value on the closed interval [0, 1]. D f attains neither a minimum value nor a maximum value on the closed interval [0, 1].

D

Let g be a continuously differentiable function with g(1)=6 and g′(1)=3. What is limx→1 (∫(1 to x) g(t)dt)/(g(x)-6)? A 0 B 1/2 C 1 D 2 E limit does not exist

D 2

limx→0 (2x6 + 6x3)/(4x5 + 3x3) is A 0 B 1/2 C 1 D 2 E nonexistent

D 2

limx→0 x2/1-cosx is A -2 B 0 C 1 D 2 E nonexistent

D 2

The maximum acceleration attained on the interval 0≤t≤3 by the particle whose velocity is given by v(t)=t3-3t2+12t+4 is A 9 B 12 C 14 D 21 E 40

D 21

For 0≤x≤6, the graph of f′, the derivative of f, is piecewise linear as shown above. If f(0)=1, what is the maximum value of f on the interval? A 1 B 1.5 C 2 D 4 E 6

D 4

The volume of a cylindrical tin can with a top and a bottom is to be 16π cubic inches. If a minimum amount of tin is to be used to construct the can, what must be the height, in inches, of the can? A 2 cube root of 2 B 2 sqrt of 2 C 2 cube root of 4 D 4 E 8

D 4

limh→0 (e^(2+h) - e2)/h = A 0 B 1 C 2e D e2 E 2e2

D e2

The function f given by f(x) = 9x2/3+ 3x − 6 has a relative minimum at x = A -8 B -cube root of 2 C -1 D -1/8 E 0

E 0

Consider all right circular cylinders for which the sum of the height and circumference is 30 centimeters. What is the radius of the one with maximum volume? A 3 cm B 10 cm C 20 cm D 30/π2 cm E 10/π cm

E 10/π cm

The figure above shows the graph of f', the derivative of a function f, for 0≤x≤2. What is the value of x at which the absolute minimum of f occurs? A 0 B 1/2 C 1 D 3/2 E 2

E 2

A rectangular area is to be enclosed by a wall on one side and fencing on the other three sides. If 18 meters of fencing are used, what is the maximum area that can be enclosed? A 9/2 m2 B 81/4 m2 C 27 m2 D 40 m2 E 81/2 m2

E 81/2 m2

If f(x) = 1/3x3 - 4x2 + 12x -5 and the domain is the set of all x such that 0 < x < 9, then the absolute maximum value of the function f occurs when x is A 0 B 2 C 4 D 6 E 9

E 9

Let f be a function defined and continuous on the closed interval [a,b]. If f has a relative maximum at c and a<c<b, which of the following statements must be true? I. f'(c) exists. II. If f'(c) exists, then f'(c)=0 III. If f''(c) exists, then f′′(c)≤0 A II only B III only C I and II only D I and III only E II and III only

E II and III only

Let f be the function with first derivative given by f′(x)=(3−2x−x2)sin(2x−3). How many relative extrema does f have on the open interval −4<x<2? A Two B Three C Four D Five E Six

E Six

limx→∞ x3/e3x is A 0 B 2/9 C 2/3 D 1 E Infinite

A 0