AP Calculus Semester Exam

if fx = the sqr root of x^2 -4 and gx= 3x-2, then the derivative of f(g(x)) at x=3 is

7/5^1/2

let f be a function defined for all real numbers x such that F' >0 and F''>0. Which of the following could be a table of values for F?

x Fx 1 -3 2 -1 3 3 4 19

if f'>0 for all real numbers x and the integral from 4 to 7 f(t)dt=0, which of the following could be a table of values for the function f?

x f(x) 4 -4 5 -2 7 5

for -1.5<x<1.5, let f be a function with first derivative given by f'x= e^(x^(4)-2x^(2)+1)-2. Which of the following are all intervals on which the graph of f is concave down?

(-1.5,-1) and (0,1)

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

-a^2e^-ax

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

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

if fx= 4x^-2+1/4x^2=4, then f'2=

0

really fun waterslide graph.the figure above shows the graph f', for -6<x<8. which best describes the graph of f on the same interval?

1 relative minimum, 1 relative maximum, 3 points of inflection

the velocity of a particle moving along the x-axis is given by v(t)=sin2t at time t. IF the particle is at x=4 when t=0, what is the position of the particle when t=pi/2

5

a differentiable function f has the property that f'x <3 for 1<x<8 and f(5)=6. What is true?

I and II only

the graph of f' is shown above. which of the following statements must be true? bell graph

I and III only

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

x=-2 only

for x>0, d/dx time the integral from 1 to rad x of 1/1-t^2dt=

1/(2radx)(1+x)

a particle moves along the x-axis. the velocity of the particle at time t is 6t-t^2. What is the total distance traveled by the particle from time t=0 to t=3?

18

the graph above gives the velocity in ft/sec of a car for 0<t<8, where t is the time in seconds. Of the following, which is the best estimate of the distance traveled by the car from t=0 until the car comes to a complete stop?

210ft

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

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

fx=7x-3+lnx, then f(-1)=

8

let f be the function given by fx = 300x-x^3, on what intervals is the function f increasing

[-10,10]

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

(-infinity,0] and [4,infinity)

if sin(1/x^2 +1) is an antiderivative for fx, then the integral from 1 to 2 of fxdt =

-.281

what is the limit as x approaches 3 from the left of the abs value of x-3 over x-3

-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?

-1 only

if (x+2y)dy/dx=2x-y, what is the value of d^2y/dx^2 at the point (3,0)?

-10/3

the graph of f'' is shown above for -2<x<4, what are all intervals on which the graph of the function f is concave down?

-2<x<-1 ND 1<X<3

the figure above shows the graph of f. If fx= the integral from 2 to x of gtdt, which of the following could be the graph of y=g(x)?

the graph with a line at y=1

the function is differentiable and increasing for all real numbers x, and the graph of f has exactly one point of inflection. Of the following, which could be the graph of f'?

the graph with the sharp point

lwr f be the function defined by the fx= the sqr root of the absolute value of x-2. what statement is true?

f is continuous but not differentiable at x=2

let f b a function that is continuous on the closed interval [2,4] with f(2)=10 and f(4)=20. Which of the following is guaranteed by the Intermediate Value Theorem?

f(x)=13 has at least one solution in the open interval (2,4)

let g be a function with the first derivative fiven by g'x = the integral from 0 to x of e^-t^2dt. What must true on the interval 0<x<2?

g is increasing, and the graph of g is concave up.

the graph of the differentiable function f is shown in the figure above. let h be defined by hx= the integral of ftdt. Which of the following correctly orders h(2), h'(2), and h''(2)?

h''2<h'2<h2

let f be a polynomial function with values of f' at selected values of x given in the table above. which of the following must be true for -2<x<6?

the graph of f has at least two points of inflection

if f is the function defined above, then f'(-1) is

nonexistent

if xsinx=y, then dy/dx=

sinx=xcosx

a particle moves along the x axis with its position at time t given by x(t)=(t-a)(t-b), where a and b are constants and a does not equal b, for which of the following values of t is the particle at rest?

t= (a+b)/2

the function f is defined by fx= (x)/x+2, what points x,y on the graph have the property that the line tangent to f at x,y has slope 1/2?

(0,0) and (-4,2)

a particle moves along a line so that its velocity is given by vt=-t^3+2t^2+2^-t for t>0. For what values of t is the speed of the particle increasing?

(0.177, 1.256) and (2.057, infinity)

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?

(1/2, 2)

let f and g be continuous functions such that the integral from 0 to 6 fxdx=9, the integral from 3 to 6 fxdx=5 and the integral from 0 to 3 gxdx=7. What is the value of the integral from 0 to 3 of .5fx-3gx dx?

-19

for t>0, the position of a particle moving along the x axis is given by x(t)= sint-cost. What is the acceleration of the particle at the point where the velocity is first equal to 0?

-2^1/2

let g be the function given by gx=(x^2)e^(kx) where k is a constant. for what value of k does g have a critical point at x=2/3?

-3

the graph f' is shown in the figure above. the function f has a local maximum at x=?mountain looking graph crosses the x axis at x=1

-3

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 y 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?

-96

if f'= the square root of (x^4+1)+x^3-3x, then f has a local maximum at x=

.350

the table above gives selected values of a function f. THe function is twice differentiable with f'' >0. What is the value of f'(3)

.7

a particle moves along a line so that its acceleration for t>0 is given by a(t)=(t+3)/(t^(3)+1)^1/2. if the particles velocity at t=0 is 5, what is the velocity of the particle at t=3?

.713

a particle moves along the x axis so that its velocity at time t>0 is given by (t^(2)-1)/(t^(2) +1). what is the total distance traveled by the particle from t=0 to t=2?

.927

a particle moves on the x-axis so that at anytime t, 0<t<1, its postion is given by x(t)=sin(2pit)-2pit. For what value of t is the particle at rest?

1/2

using the substitution u=sin2x, the integral from pi/6 to pi/2 of sin^52xcos2xdx is equivalent to

1/2 times the integral from rad 3/2 to 0 of u^5du

what is the integral from 1 to 2 of dx/2x+1

1/2(ln5-ln3)

lim ln(4+h)-ln4/h h->0

1/4

what is the slope of the line tangent to the graph ln2x at the point where x=4

1/4

a particle moves along the x axis. the velocity of the particle at time t is given by v(t), and the acceleration of the particle at time t is given by a(t). which of the following gives the average velocity of the particle from t=0 to t=8

1/8!v(t)dt from 0 to 8

let f be the function defined by fx= lnx/x. what is the absolute maximum of f?

1/e

the function f is defined by the piecewise fx= 2 for x<3 and x-1 for x>3, what is the value of the integral from 1 to 5 of fxdx

10

a tank contains 50 liters of oil at time t = 4 hours. OIl is being oumoed into the tank at a rate r(t). Do a right Riemann sum with three subintervals and solve for the number of liters of oil that are in the tank at t=15 hrs

114.9

water is pumped into a tank at a rate of r(t)=30(1-e^-.16t) gallons per minute. if the tank contained 800 gallons of water when the pump was turned on, how much water to the nearest gallon is in the tank after 20 minutes?

1220 gallons

using the substitution u= the sqr root of x, integrate the big equation using e

2 times the integral from 1 to 4 of e to the u du

a person whose height is 6 feet is walking away from the base of a streetlight along a straight path at a rate of 4 ft/sec. IF the height of the streetlight is 15 feet, what is the rate at which the person's shadow is lengthening?

2.667 ft/sec

the graph of (e^tanx)-2 crosses the x-axis at one point in the interval [0,1]. What is the slope of the graph at this point?

2.961

origami graph. for 0<x<6 the graph of f' is piecewise linear as shown above. if f(0)=1, what is the maximum value of f on the interval?

4

let f be the function defined above by the piecewise of fx = (2x+1)(x-2)/x-2 and fx=k. for what value of k is f continuous at x=2?

5

y=(x^3 - cosx)^5, then y' =

5(x^3 -cosx)^4(3x^2+sinx)

The regions A,B, and C in the figure above are bounded by the graph of the function f and the x-axis. The area of region A is 14, the area of region b is 16, and the area of region c is 50. What is the average value of f on the interval [0,8]

6

let f be the function given by f(x)=9^x. If four subintervals of equal length are used, what is the value of the right Riemann sum for the integral from 0 to 2 of fxdx?

60

the figure above shows the graph of the function f. which of the following statements are true? broken piecewise named ellie

I and II only

a particle moves along a straight line so that at time t>0 the position of the particle is given by s(t), the velocity is given by v(t), and the acceleration is given by a(t). Which of the following expressions gives the average velocity of the particle on the interval [2,8]?

[s(8)-s(2)]/6

the graph of f is shown, for which of the following values of x is f'x positive and increasing?

e

graph that looks like a bird. the graph is differentiable. if hx= the integral from 0 to x ftdt, what is true?

h(6)<h'(6)<h''(6)

the graph of the function is shown above, which of the following statements it false? discontinuous graph of f

limfx x->4 exists

the integral of secxtanxdx=

secx+c

let f be a function such that the integral from 6 to 12 f(2x)dx =10. Which of he following must be true?

the integral from 12 to 24 f(t)dt=20

let f be the function defined above, for what value of b is f continuous at x=2?

there is no such value of b

what is the integral from 2 to x of (3t^2-1)dt

x^3-x-6

the line y=5 is a horizontal asymptote to the graph of which of the following functions?

y=(20x^2-x)/1+4x^2

which of the following is the solution to the differential equation dy/dx =2sinx with the initial condition y(pi)=1?

y=-2cosx-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 to the graph of f?

y=-3