clac final questions

The function f is defined by f(x)= 2x^3-4x^2+1. The application of the Mean Value Theorem to f on the interval 1<x<3 guarantees the existence of a value of c, where 1<c<3, such that f'(c)=

c. 10

Let f be the function given by f(x)=(x-4)(2x-3)/(x-1)^2. If the line y=b is a horizontal asymptote to the graph of then b=

c. 2

The table above gives values for the functions f and g and their derivatives at x=3. Let k be the function give by k(x)= f(x)/g(x), where g(x)≠0 . What is the value of k'(3)?

c. 2

An object moves along a straight line so that at any time t>0 its velocity is given by v(t)=2cos(3t). What is the distance traveled by the object from t=0 to the first time that it stops?

c. 2/3

Let f be the function with derivative defined by f'(x)=x^3 -4x. At which of the following values of x does the graph of f have a point of inflection?

c. 2/√3

If f(x)= 5-x/ x^3 + 2, then f'(x)=

c. 2x^3+ 15x^2- 2/(x^3 +2)^2

lim In(e^3x + x)/x = x-->inf

c. 3

The velocity v, in meters per second, of a certain type of wave is given by v(h)=3√x, where x is the depth, in meters, of the waste through which the wave moves.

c. 3/2√2

Lim x-->-3 x^2-9/x^2-2x-15 is

c. 3/4

If e^xy -y^2 = e-4, then at x=1/2 and y=2, dy/dx=

c. 4e/8-e

If 4∫1 f(x)dx= 8 and 4∫1 g(x)dx= -2, which of the following cannot be determined from the information given?

c. 4∫1 3f(x)g(x)dx

The points (3, 0), (x, 0), (x, 1/x^2), and (3, 1/x^2) 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?

c. 6

Let f be the function given by f(x)=9^x. If four sub intervals of equal length are used, what is the value of the right Riemann sum approximation for 2∫0 f(x)dx?

c. 60

Water is flowing into a tank at the rate r(t), where r(t) is measured in gallons per minute and t is measured in minutes. The tank contains 15 gallons of water at time t=0. Values of r(t) for selected values of t are given in the table above, Using a trapezoidal sum with the three intervals indicated by the table, what is the approximation of the number of gallons of water in the tank at time t=9?

c. 67

If the average value of a continuous function f on the interval [-2,4] is 12 what is 4∫-2 f(x)/8 dx?

c. 9

the figure above shows the graph of the function f. Which of the following statements are true? I. lim x->2- f(x)=f(2) II. lim x->6- f(x)= lim x->6+ f(x) III. lim x->6 f(x)= f(6)

c. I and II only

The graph of f', the derivative of the function f, is shown in the figure above. Which of the following statements must be true?

c. I and III only

Let f be the function defined above. Which of the following statements about f are true?

c. II only

At time t=0, a storage tank is empty and begins filling with water. For t>0 hours, the depth of the water in the tank is increasing at a rate of W(t) feet per hour. Which of the following is the best interpretation of the statement W'(2)>3

c. Two hours after the tank begins filling with water, the rate at which the depth of the water is rising is increasing at a rate greater than 3 feet per hour per hour.

Let f be the function given by f(x)= x-2/2Ix-2I . Which of the following is true?

c. f has a jump discontinuity at x=2

For which of the following pairs of functions f and g is lim f(x)/g(x) x-->inf infinite?

c. f(x)=3^x and g(x)=x^3

A particle moves along a straight line. the graph of the particle's velocity v(t) at time t is shown above, for 0<t<m, where j, k, l, and m are constants. The graph intersects the horizontal axis at t=0, t=k, and t=m and has horizontal tangents at t=j and t=l. For what values of t is the speed of the particle decreasing? (graph)

c. j<t<k and l<t<m

The graph of the functions f and g are shown in the figures above. Which of the following statements is false?

c. lim (f(x)g(x+1)) does not exist x-->1

Let f be the function given by 2cos(x)+ 1. What is the approximation found by using the line tangent to the graph of f at x=pi/2?

c. pi-2

Sand is deposited into a pile with circular base. the volume V of the pile is given by V=r^3/3, where 4r is the radius of the base, in feet. the circumference of the base is increasing at a constant rate of 5pi feet per hour. When the circumference of the base is 8pi feet, What is the rate of change of the volume of the pile, in cubic feet per hour.

c.20

Which of the following is the solution to the differential equation dy/dx= e^(y+x) with the initial condition y(0)= -In4?

c.y= -In(-e^x +5)

The graph of y=f(x) on the closed interval [0,4] is shown above. Which of the following could be the graph not y=f'(x)?

d.

If g is the function given by g(x)=1/3x^3 + 3/2x^2 -70x +5, on which of the following intervals is g decreasing?

d. (-10,7)

Let f be the function defined by f(x)= 2x^3 -3x^2-12x+18. On which of the following intervals is the graph of f both decreasing and concave up?

d. (1/2, 2)

Let f be the function defined by f(X)=x∫0 (2t^3-15t^2 +36t) dt . On which of the following intervals is the graph of y=f(x) concave down?

d. (2,3) only

For x>0, d/dx[2x∫0 In(t^3 +1)dt] =

d. (8x^3 +1)

∫x^2(x^3-1)^10 dx=

d. (x^3 -1)^11 / 33 +c

Let f be the function given by f(x)= (x-2)^2(x+3)/(x-2)(x+1). For which of the following values of x is f not continuous?

d. -1 and 2 only

∫ (x^2 +1)/(x^3 +3x -5)^3 dx

d. -1/6 * 1/(x^3 +3x-5)^2 +c

What are all the values of x for which 2∫x t^3 dt is equal to 0?

d. -2 and 2 only

The number of gallons of water in a storage tank at time t, in minutes, is modeled by w(t)= 25-t^2 for 0<t<5. At what rate, in gallons per minute, is the amount of water in the tank changing at time t=3 minutes?

d. -6

If f(x) =4x^-2 +1/4x^2 + 4, then f'(2)=

d. 0

A particle moves on the x-axis so that at any time t, 0<t<1, its position is given by x(t)=sin(2pi*t) + 2pi*t. For what value of t is the particle at rest?

d. 1/2

using the substitution u= sin(2x), pi/2 ∫ pi/6 [sin^5(x^2)Cos(2x)]

d. 1/2* 0∫√3/2 (u^5) du

For time t>0, the height h of an object suspended from a spring is given by h(t)= 16+ 7cos(pi*t/4). What is the average height of the object from t=0 to t=2?

d. 16 + 14/pi

4∫0 x/√x^2+9 dx=

d. 2

If f(x)= (2x^2 +5)^7, then f'(x)=

d. 28x(2x^2 +5)^6

for any real number x, lim h->0 [sin(2(x+h)) - sin(2x)]/h =

d. 2cos(2x)

If f(x)= x^2-4 and g is a differentiable function of x, what is the derivative of F(g(x))?

d. 2g(x)g'(x)

d/dx [2(sin√x)^2]=

d. 2sin√xcos√x / √x

A particle moves along a straight line so that at time t>0 its acceleration is given by a(t)=12t. At time t=0, the velocity of the particle is 2 and the position of the particle is 5. Which of the following is an expression of the position of the particle at time t>0?

d. 2t^3 +2t +5

d/dx(x^3*sec(2x))=

d. 2x^3*sec(2x)tan(2x) + 3x^2*sec(2x)

The graph of a function f is shown above. Which of the following expresses the relationship between 2∫0 f(x) dx, 3∫0 f(x) dx, and 3∫2 f(x) dx? (with graph)

d. 3∫2 f(x) dx< 3∫0 f(x) dx< 2∫0 f(x) dx

For 0<x<6, the graph of f', the derivative of f, is piece wise linear as shown above. If f(0)=1, What is the maximum value of f on the interval?

d. 4

If x+ [3y^(1/3)] = y, what is dy/dx at the point (2,8)?

d. 4/3

If y=5x√x^2 +1, then dy/dx at x=3 is

d. 45/√10 + 5√10

The velocity of a particle moving along the x-axis is given by v(t)= sin(2t) at time t. If the particle is at x=4 when t=0, what is the position of the particle when t= pi/2?

d. 5

lim x^2+x-6/ x^2-4 = x--> 2

d. 5/4

d/dx(x^5-5^x)=

d. 5x^4 - (In5)5^x

If y=(x/x+1)^5, then dy/dx=

d. 5x^4/(x+1)^6

a particle moves along the x-axis so that at time t>0, the acceleration of the particle is a(t)=15√t. T^he position of the particle is 10 when t=0, and position of the particle is 20 when t=1. What is the velocity of the particle at time t=0?

d. 6

If -10∫4 g(x) dx= -3 and 6∫4 g(x) dx= 5, then 6∫-10 g(x) dx=

d. 8

Let F be the function given by F(x)= x∫3 (Tan(5t)sec(5t)-1) dt. Which of the following is an expression for F'(x)?

d. Tan(5x)sec(5x)-1

Shown above is a slope field for which of the following differential equations?

d. dy/dx= y^2

Shown above is a slope field for which of the following differential equations? #3

d. dy/dx= y^2(4-y)/4

If y= f(x) is a solution to the differential equation dy/dx= e^x^2 with the initial condition f(0)=2, which of the following is true?

d. f(x)= 2 + x∫0e^t^2 dt

the figure above shows the graphs of the functions f and g. if h(x)=f(x)g(x), then h'(2)=

d. inf

If y= x^2-2x and u=2x+1, then dy/du=

d. x-1

Let h be the function defined by h(x)= x ∫pi/4 sin^2(t) dt. Which of the following is an equation for the line tangent to the graph of h at the point where x=pi/4?

d. y= 1/2(x- pi/4)

Which of the following is the solution to the differential equation dy/dx=5y^2 with the initial condition y(0)=3?

d. y= 3/1-15x

If f(x)= ae^-ax for a>0, then f'(x)=

e. -a^2 *e^-ax

d/dx(tan^-1 x+2√x) =

e. 1/(1+x^2) + 1/√x

2∫1 dx/2x+1

e. 1/2 (In5-In3)

∫x^2(x^3+5)^6 dx

e. 1/21(x^3+5)^7+c

Let f be a differentiable function such that f(0)=-5 and f'(x)<3 for all x. Of the following, which is not a possible value for f(2)?

e. 2

If y^3 +y = x^2, then dy/dx=

e. 2x/1+3y^2

d/dx (sin^3(x^2))=

e. 6xsin^2(x^2)cos(x^2)

For Which of the following does lim f(x)=0 (x-->inf) I. f(x)= Inx/x^99 II. f(x)= e^x/Inx III. f(x)= x^99/e^x

e. I and III only

∫(e^x + e) dx=

e. e^x +ex+ c

The function f is continuous for all real numbers, and the average rate of change of f on the closed interval [6,9] is -3/2. for 6<c<9, there is no value of c such that f'(c)=-3/2. of the following which must be true?

e. f is not differentiable on the open interval (6,9)

f(x)={ 3x+5 when x<-1 { -x^2 +3 when x>= -1 If f is the function defined above, then f'(-1) is?

e. nonexistent

Let f be the function defined above. For what value of b is f continuous at x=2?

e. there is no such value of b

Which of the following is an antiderivative of f(x)=√1+x^3 ?

e. x∫0 √1+t^3 dt

Which of the following is an equation of the line tangent to the graph of x^2 -3xy=10 at the point (1, -3)?

e. y+3= 11/3 (x-1)

The function y= g(x) is differentiable and increasing for all real numbers. On what intervals is the function y = g(x^3-6x^2) increasing?

a. (-inf,0] and [4, inf) only

Let f be the function defined b y f(x)= -3 + 6x^2 - 2x^3. What is the largest open interval on which the graph of f is both concave up and increasing?

a. (0,1)

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

if y=sinxcosx, then at x=pi/3, dy/dx=

a. -1/2

1∫-1 x^2 -x / x dx is

a. -2

The top of a 15-foot-long ladder rests against a vertical wall with the bottom of the ladder on level ground, as shown above. The ladder is sliding down the wall at a constant rate of 2 feet per second.l At what ate, in radians per second, is the acuter angle between the bottom of the ladder and the ground changing at the instant the bottom of the ladder is 9 feet from the base of the wall?

a. -2/9

If y=cos2x, then dy/dx=

a. -2sin2x

If f(x)= sin (x^2+ pi), then f'(√2pi)=

a. -2√2pi

What is the average rate of change of y=cos(2x) on the interval [0, pi/2]?

a. -4/pi

The function g is given by g(x)= 4x^3 +3x^2 -6x +1. What is the absolute minimum value of g on the closed interval [-2,1]?

a. -7

The graph of the piece wise linear function f is shown above. What is the value of 12∫0 f'(x)dx?

a. -8

The graph of a function f is shown above. If g is the function defined by g(x)= (x^2+1)/ f(x) , what is the value of g'(2)?

a. -8/9

lim x^3/e^3x is x->inf

a. 0

lim sinx/ e^x-1 is x->0

a. 1

for x>0, d/dx √x∫1 (1/1+t^2) dt =

a. 1/ (2√x * (1+x))

If f(x)= √x + 3/√x, then f'(4)=

a. 1/16

If f(x)=Inx, then lim f(x)-f(3)/x-3 is x-->3

a. 1/3

A particle moves along the x-axis with velocity given by v(t)= 3t^2-4 for time t>0. If the particle is position x=-2 at time t=0, what is the position of the particle at time t=3?

a. 13

The values of a continuous function f for selected values of x are given in the table above. What is the value of the left Riemann sum approximation to 50∫0f(x)dx using the sub intervals [0,25],[25,30], and [30,50]?

a. 290

If x^2 +xy -3y= 3, then at the point (2,1), dy/dx=

a. 5

2∫0 (x^3+1)^1/2*x^2 dx

a. 52/9

The graph of a function f is shown above. What is the value of 7∫0 f(x)dx?

a. 6

A particle moves along the y-axis so that at time t>0 its position is given by y(t)= t^3 -4t^2 +4t +3 . Which of the following statements describes the motion of the particle at time t=1?

a. The particle is moving down the y-axis with decreasing velocity.

Let f be the function with derivative given by f'(x)=-2x/(1+x^2)^2 . On what intervals is f decreasing?

a. [0, inf) only

f(x)={ x+b if x<&= 1 { ax^2 if x>1 Let f be the function given above. What are all the values of a and b for with f is differentiable at x=1?

a. a=1/2 and b=-1/2

Shown above is a slope field for which of the following differential equations? #5

a. dy/dx= xy-x

f(x)= { -x^2 +3 if x<=5 {-10x +28 if x>5 Let f be the function defined above. Which of the following statements about f is true?

a. f is continuous and differentiable at x=5

Let f be the function given by f(x)=kx/x^2 +1, where k is a constant.l for what values of k is any is f strictly decreasing on the interval (-1,1)

a. k<0

How many vertical asymptotes does the graph of y= x-2 / x^4 -16 have?

a. one

A particle moves along the x-axis so that at time t>0 its position is given by x(t)=12e^-t sint. What is the first time t at which the velocity of the particle is zero?

a. pi/4

The first derivative of the function f is given by f'(x)= 3x^4-12x^3 . What are the x-coordinated of the points of inflection of the graph of f?

a. x=3 only

x∫2 (3t^2-1) dt

a. x^3 -x-6

The equation y=2e^6x -5 is a particular solution to which of the following differential equations?

a. y' -6y -30=0

If f is the function given by f(x)=e^x/3 , which of the following is an equation of the line tangent to the graph of f at the point (3In4, 4)?

a. y-4=4/3 (x-3In4)

The graph of which of the following functions has exactly on horizontal asymptotes?

a. y= 1/(x^2 +1)

The region enclosed by the graphs of y=x^2 and y=2x is the base of a solid. For the solid, each cross section perpendicular to the y-axis is a rectangle whose height is 3 times its base in the xy plane. Which of the following expression gives the volume of the solid?

a.3* 4∫0(√y- (y/2) )^2 dy

4∫2 dx/5-3x =

b. - In7/3

lim Ix-3I/x-3 is x->3-

b. -1

The slope of the line tangent to the graph of y=In(1-x) at x=-1 is

b. -1/2

The table above gives selected values for a differentiable and decreasing function f and its derivative. If f^-1 is the inverse function of f, what is the value of (f^-1)' (2)?

b. -1/8

lim e^-1-h - e^-1 / h is h-->0

b. -1/e

If f is the function given by f(x)= 3x^2 -x^3, then the average rate of change of f on the closed interval [1,5] is

b. -13

The figure above shows the graph of the function g and the line tangent to the graph of g at x=-1. Let h be the function given by h(x)e^x*g(x). What is the value of h'(-1)?

b. -3/e

lim 10-6x^2/ 5+ 3e^x is x-> inf

b. 0

lim sin (pi/3 +h)-sin(pi/3) / h is h-->0

b. 1/2

Which of the following definite integrals has the same value as 4∫0 xe^x^2 dx

b. 1/2* 16∫0 e^u du

The base of a solid is the region in the first quadrant bounded by the y-axis, the x-axis , the graph of y=e^x, and the vertical line x=1. For this solid, each cross section perpendicular to the x-axis is a square. What is the volume of the solid?

b. 1/2*e^2- 1/2

Left f be the function defined by f(x)= 2x +e^x. If g(x)= f^-1(x) for all x and the point (0,1) is on the graph of f what is the value of g'(1)?

b. 1/3

∫ (1/3x+12)dx =

b. 1/3 InIx+4I+ c

What is the slope of the line tangent ot the graph of y= In(2x) at the point where x=4?

b. 1/4

Lim In(x+3)-In(5)/ x-2 is x->2

b. 1/5

What is the value of x at the minimum value of y= 3x^(3/4) -2x occurs on the closed interval [0,1]?

b. 1/8

The weight of a population of yeast is given by a differentiable function y, where y(t) is measured in grams and t is measure din days. the weight of the yeast population increasing according to the equation dy/dt= ky, where k is a constant. At time t=0, the weight of the yeast population is 120 grams and is increasing at the rate of 24 grams per day. which of the following is an expression for y(t)?

b. 120e^t/5

The table above gives the velocity v(t), in miles per hour, of a truck at selected times t, in hours. Using a trapezoidal sum with the three sub intervals indicated by the table, what is the approximate distance, in miles, the truck traveled from time t=0 to t=3?

b. 130

if dy/dx= 2-y, and y=1 when x=1 then y=?

b. 2- e^(1-x)

If dy/dt= -10e^-t/2 and y(0)= 20, what is the value of y(6)?

b. 20e^-3

If In(2x+y)= x+1, then dy/dx=

b. 2x +y -2

An isosceles right triangle with legs of length s has area A=1/2*s^2. At the instant when s= √32 centimeters, the area of the triangle is increasing at a rate of 12 square centimeters per second. At what rate is the length of the hypotenuse of the triangle increasing, in centimeters per second, at that instant?

b. 3

A person stands 30 feet from point P and watches a balloon rise vertically from the point, as shown in the figure above. The balloon is rising at a constant rate of 2 feet per second. What is the rate of change, in radians per second, of angle θ at the instant when the balloon is 40 feet above point p?

b. 3/125

lim √9x^4+1 / 4x^2+3 is x-->inf

b. 3/4

The function f is defined by f(x)= sinx + cosx for 0<x<2pi. What is the x-coordinate of the point of inflection where the graph of f changes from concave down to concave up?

b. 3pi/4

Which of the following is an antiderviative of 3 sec^2x+2?

b. 3tanx + 2x

The base of a solid is the region bounded by the x-axis and the graph of y=√1-x^2 . for the solid, each cross section perpendicular to the x-axis is a square. what i8s the volume of the solid?

b. 4/3

Let g be the function given by g(x)= x∫3 (t^2 -5t-14) dt. what is the point of inflection of the graph of g?

b. 5/2

∫(5e^2x + 1/x) dx

b. 5/2e^2x + InIxI+c

For a certain continuous function f, the right riemann sum approximation of 2∫0 f(x) dx with n subintervals of equal length is 2(n+1)(3n+2)/ n^2 for all n. What is the value of 2∫0 f(x) dx

b. 6

f(x)={ x^2sin(pix) for x<2 {x^2 +cx-18 for x>2 Let f be the function defined above, whee c is a constant. For what value of c is a constant. for what value of c, if any, is F continuous at x=2.

b. 7

If f(x)= x^3-x^2+x-1, then f'(2)=

b. 9

If 0<c<1, what is the area of the region enclosed by the graph of y=0, y=1/x, x=c, and x=1?

b. In(1/c)

A student attempted to solve the differential equation dy/dx=xy with initial condition y=2 when x=0. In which step, if any, does an error first appear? Step 1: ∫1/y dy= ∫xdx Step 2: InIyI= x^2/2 +c Step 3: IyI= e^(x^2/2) + c Step 4: since y =2 when x=0, 2=e^0 +c Step 5: y=e^(x^2/2) + 1

b. Step 3

A function f(t) gives the rate of evaporation of water, in liters per hour, from a pond, where t is measured in hours since 12 noon. which of the following gives the meaning of 10∫4 f(t)dt in the context described?

b. The total volume of water, in liters, that evaporated from the pond between 4 P.M. and 10 P.M.

For what value of b does the integral b∫1 x^2 dx equal n lim n->inf ∑(1+ 2k/n)^2 * (2/n) k=1

b. b=3 only

The table above gives selected values for a twice differentiable function f. Which of the following must be true?

b. f'(x)=8 for some value of x in the interval -1<x<5.

The graph of the function f is shown in the figure above. Which of the following could be the graph of f' the derivative of f?

b. neg quadratic above x axis

For what values of x does the graph of y=3x^5 +10x^4 have a point of inflection?

b. x=-2 only

If f''(x)=x(x+2)^2, then the graph of f is concave up for

b. x>0

∫ x^2 /4 dx=

b. x^3 /12 +c

Which of the following is an equation for the line tangent to the graph of y=3- (x∫-1 e^-t^3 dt)

b. y-3= -e(x+1)

Which of the following is the solution to the differential equation dy/dx= 2y/2x+1 with the initial condition y(0)=e for x>-1/2?

b. y= 2ex + e

Which of the following is the solution to the differential equation dy/dx= 2xy/ x^2+1 whose graph contains the point (0,1)?

b. y= x^2 +1

Let f be the function defined by f(x)= (3x+8)(5-4x)/(2x+1)^2 . which of the following is a horizontal asymptote of the graph of f?

b. y=-3

Let f be the function given by f(x)=x^3 +5x. For what value of x in the closed interval [1,3] does the instantaneous rate of change of f equal the average rate of change of f on the at interval?

b. √13/√3

If lim h->0 arcsin(a+h) - arcsin(a)/h =2, which of the following could be the value of a?

b. √3/2

Let R be the region bounded by the graphs of y=2x and y=4x-x^2. what is the area of R?

b.4/3

d/dx(x+1/ x^2 +1)=

c. (-x^2 -2x +1)/(x^2 +1)^2

The function f is given by f(x)= 4x^3-x^4. on what intervals is the graph of f concave up?

c. (0,2) only

The table above gives selected values for the differentiable function f. In which of the following intervals must thee be a number c such that f'(c)=2?

c. (8,12)

Let f be the function given by f(x)= x^3 -6x^2 + 8x -2

c. -1

Functions w, x, and y are differentiable with respect to time and are related by the equation w=x^2*y. If x is decreasing at a constant rate of 1 unit per minute and is increasing at a constant rate of 4 units per minute at what rate is w changing with respect to time when x=6 and y=20?

c. -96

Which of the following is an equation of the line tangent to the graph of y=cosx at x= pi/2?

c. -x + pi/2

Shown above is a slope field for the differential equation dy/dx= y^2(4-y^2). If y= g(x) is the solution to the differential equation with he initial condition g(-2)=-1, then lim x--> inf g(x) is

c. 0

Let f be the function defined by f(x)= 1/x. What is the average value of f on the interval [4,6]?

c. 1/2 In3/2

Let f be the function defined by f(x)= x^3 + x^2+. Let g(x)= f^-1(x), where g(3)=1. what is the value of g'(3)?

c. 1/6