AP Calculus Exam Review

Calc P(x) = 11ln((x+1)/6). M(x) and P are inverses. What is M'(10)?

1.353

f(x)= 5 + (2x-1)^3*(x+1). What is equation of the tangent line to f at x=1?

y = 13x-6

If the integral of (1/2*tan(7x/24)= k*ln(abs value of cos(7x/24) + c, where k is a constant, what is the value of k?

-12/7

Calc A particle moves along the x-axis so that its position at time t>0 is given by x(t) and dx/dt = -3t^4 +5t^2 +12/10t. The acceleration of the particle is zero when t=

0.967

Calc f(x) = 4x/17 +2/7. g(x) is the integral from 0 to x of f(t). On what open interval is g negative and increasing?

(-1.214, 0)

Calc 5.5 f(x)= -2ln10 + 1/7 *sqrtx + 2ln(5x +10). g and f are inverses and the point (4/5, k) is on g. What is g'(4/5)?

1.258

Calc Stacie moves so that v(t)= 3.9-e^x +2ln(x+.1) From 0 to 5 seconds, what is the total amount of time Stacie moves toward the right?

1.447

Using the substitution u = (6x^4 - 24x) the integral from -1 to 2 of (2x^3 -2)/cuberoot(6x^4 -24x) is equal to

1/12* the integral from 30 to 48 of (du/cuberoot(u)

f(-x) = f(x) for all x. If the integral from 0 to 3 of (3(fx))dx=6 then the integral from -3 to 3 of (f(x) +1)dx =

10

Find the limit as x approaches 11 of (g(x) - 20 + ln(34-3x)/(22+9x-x^2). g(11)=20, g'(11)= -7.

10/13

If g and f are inverses, what is the value of g'(7)? f(11)=7, f'(11)= 3/10

10/3

Molecule Mike bought a new car, he took this car for a test drive prior to sealing the deal for the car. When Mike drove the car off the car dealer's lot and begun to travel, the odometer read 5 inches. The car dealer's lot is located on the x-axis and is 7 units to the left of the origin. Mike drives so that his velocity is -t/5 + 1. If Mike drives at this velocity from t=0, when he leaves the car dealer's lot, to t=11, exactly what number will is on his odometer at t=11?

111/10

Calc 5.5 Approximate the value of f(5.26) using the equation of the tangent line to f(x)= 21 -5lnx + 4cos(3x/2) at x=5

12.616

H(x) = the integral from 4 to (1/9*x^2) of f(9*sqrt(t))dt. f(9)= 5. f'(9)= 1/11. What is H''(3)?

128/99

f(x) = 6x - ln(2x+4). If g(x) is f(x)'s inverse and the point (k, -1/2) is on g, where is the value of g'(k)?

3/16

Calc 5.5 The integral of (3x^2 -16x + 23)/(x-3) dx =

3/2 *x^2 -7x +2ln(x-3) + c

Calc 5.5 f(x) = g(g(x/5)). g is composed of line segments where g(-2)= 3 and g(2)=9. What is f'(0)

9/100

Calc 5.5 g'(x) = the integral from -4 to x of f(t)dt. f is composed of 5 line segments where f(-8)=2, f(-4)=2, f(0)= -2, f(2)=2, f(6)=0, and f(10)=2. Which of the following is true? I. g'(0)= g'(2) II. g'(-8) - 8 = g'(2) III. at x=6, the graph of g is increasing and has a point of inflection IV. at x=0, the graph of g changes from concave down to concave up

I

the integral from pi/2 to x^2 of (-sinx)dx =

cos(x^2)

If f(x) = (x-1)^2 + 2, then rank f(1) to f'(1) to f''(1) greatest to least.

f'(1) < f''(1) = f(1)

f(x)=(2/x). What is the average value of f from 4 to 6?

ln(3/2)

u = x-1, the integral of (x/sqrt(x-1) equals

the integral of (sqrt(u) + 1/(sqrt(u))

If u=sin(2x), the integral from pi/4 to pi/6 of (sin^5(2x)*cos(2x) is equal to

(-1/2)* the integral from sqrt3/2 to 1 of u^5

Calc The rate of change of the weight of a baby lizard is modeled by dw/dt = (-1/8)*(t^2 -21t -10). On what open interval of time, if any exist, is the weight of the baby increasing at a decreasing rate?

(10.499, 21.465)

If A,B,C,D are constants, the slope of the line tangent to the graph of y= Aln(abs value(B-Cx^2)/(1 +Dx^3) at x= -1 is

(2AC/B-C) + (-3AD/1-D)

Calc What is the slope of the tangent line to the graph of y= 8/13 - arctan(x) when it crosses the x-axis?

-0.667

Calc h(x)= f(g(x)). g(x)= 3+lnx. f(x)= .25x^2 -7x + 5. What is h'(5)?

-0.939

Which of the following integrals has the same value as the integral of (-pi/2) to (pi/6) of (cos^5(x)*sin(x)? (u-sub)

-1* the integral from 0 to sqrt3/2 of u^5

Calc For time t>0, Jenny moves along the x-axis so that v(t) = (t^4 -33t^2 +16)/(8t+35). Find the acceleration of Jenny the first time she changes direction

-1.106

Calc Jenny moves so that v(t) = (t^4 -36t^2 +16)/(8t+35). x(0)=-3. What is x(4)?

-11.540

Calc 5.5 g(x)= 4x + the integral from 0 to x of f(t)dt. f consists of line segments where f(0)= 11/3, f(4)=0, f(6)= -4, and f(7+1/3)=0. What is g''(2)?

-11/12

Calc Stacie moves along the x-axis so that her velocity is 4.2 - e^t +2ln(t+.1). Over the interval 0<t<5, what is Stacie's average velocity?

-23.866

If e^(2xy) - y^2 = e^2 -25, then at x=1/5 and y=5, dy/dx =

-25e^2/(e^2-25)

If the integral of ((6-3x)(5x+2)/(x^2))dx = Aln(abs value (x)) + Bx + C/x + K where A,B,C,K are constants, find the value of A+B+C

-3

The integral from 1 to -2 of (3/2 * (1-8x)^7)dx equals what using u-sub?

-3/16* the integral from -7 to 17 of u^7du

f(x) - the integral of 0 to x of g(t) where g consists of line segments. g(-6)=3, g(-2)=0. What is f''(-4)?

-3/4

P and f are inverses. f(3)= -3. f'(3)= -4/7. What is P'(-3)?

-7/4

y=sin(2x) - .5ln(3x), then the third derivative =

-8cos(2x) - (1/x^3)

Calc T(m) = 75- m*(sqrt(8m)). At what rate is the temperature of the cup changing at time m= 5?

-9.486

Calc Patty's velocity= 10ln((x+1)/6). x(1)=12, what is her position at x(8)?

0.464

V and g are inverses. If g(4)=0, and g'(4)= 31/14, what is V'(0)?

14/31

The function f is defined on the interval -7 < x < k where k > 0 and f(k)=0=f(-7)=f(0), f(-5) = -4, f(1)=5. Find the value of k for which the average value of f over the entire interval is 5/8

147/15

If f is differentiable at x=3, then what is the value of a+b?. f is a piecewise fnc. (x<0, f(x) = e^(11x -33), (0<x<3, f(x) = ln(10-3x)^4), (x>3, f(x) = 3b-2ax)

18

Estimate the instantaneous rate at the which the temperature of the oven is changing at time t=14. H(12)= 134. H(16)= 152

18/4

Calc u= sec(x/2). The integral from pi/3 to pi/2 of (sec^2(x/2)*tan(x/2))dx =

2*the integral from 2/sqrt3 to sqrt2 of udu

Calc 5.5 If f(x) = (ln(abs value of (e + e^2x))) - arctan(3x), then f''(3/4)=

2.041

T(x) represents the line tangent to the graph of f(x) when x=e, with f(x)= 17 - (the integral from 2e to 2x of (-1/x) dt). Find the value of T(4e)

20

dy/dx = (7/2sqrt(x)). Let y = f(x) be the particular solution to the differential equation with the initial condition f(4)=13. Find the value of f(9).

20

Calc If f''(x) = 13 +3^2x -6e^x and f'(-1)=5, then f'(1)=

20.943

Estimate the average temperature of the oven between t=6 and t=14 using a right Riemann sum with two subintervals of equal length. H(6)= 146. H(8)= 164. H(10)= 184. H(12)=206. H(14)= 230.

207

For time t>0, the height h of an object is suspended from a spring given by h(t)= (20/pi) + cos(pi/4*t). What is the average height of the object from t=0 to t=2?

22/pi

If f is the piecewise function given by (for x<0, f(x)= -1), (for 0<x<3, f(x) = x+1), (for x>3, f(x)= (x-3)^2 +2), then the integral from -4 to 6 of the integral of f(x) is

25/2

Calc 5.5 g(x) = 3x + the integral from 0 to x of f(t). f is composed of line segments where f(0)=16/3, f(4)=0, f(6)= -4, and f(7+1/3)=0. What is g(7+1/3)

26

f(x) is the integral of -2 to x of g(t)dt where g consists of 5 line segments. f(-2)=0, f(0)=8, f(4)=0, f(7)= -1, f(9)=0. What is f(9)?

43/2

If the integral from 1 to e^3 of (7-3x)/x = c+ke^3 where c and k are constants and e is the natural base. Find the value of 2c + k

45

Water is pumped into the tank at a rate of R(t)= 2t+3 gallons per minute. If the tank initially contained 11 galloons, how much water was in the tank after 5 minutes?

51

Calc T(m) = 75- m*sqrt(4m). What is the average temperature of the cup between m=2 and m=8?

51.618

If the average value of the differentiable func from 2 to 9 is 2/3, what is the value of the integral from 2 to 9 of (4*f(x))?

56/3

Calc Jenny moves so that v(t) = (t^4 -39t^2 +16)/(8t+35). x(0)=3. From t=4 to t=10, at what exact times is Jenny experiencing at a speed of 5/7?

6.078 and 6.334

dy/dx = 7(2sqrt(x)). y =f(x) is the particular solution to the differential eq with the initial condition f(4)=5. Use the equation of the tangent line to the graph of f at x=4 to estimate f(4.25)

87/16

What is the eq of the line tangent to y= -5 + the integral from 2 to 2x of (e^-x)dx at the point x=1?

y = -2xe^2 -2e^2 -5

Calc Patty's velocity = 12ln((1+x)/6). What is total distance traveled from t=1 to t=8?

29.423

If f is the function given by f(x) = the integral from 1 to x^2 of (5/(t+1)), then f'(2) =

4

f(x) is the piecewise fnc where (x<1, f(x)= 5x-4), (1<x<e, f(x)=ln(5x-4), 9 x>e, f(x)=x^(x-e). Which of the following is true? I. f is differential at x=1 II. the limit of f(x) as x approaches 1 from the left = the limit of f as x approaches 1 from the left III. the limit of f' as x approaches 1 from the left = the limit of f' as x approaches 1 from the right IV. the limit of f as x approaches 1 from the right = the limit of f as x approaches e from the right

III

f(x) is the piecewise function where (for x<3, f(x) = 10/3), (for 3<x<4, f(x)= (-x/3 +13/3), (for x>4, f(x) = 3). If the integral from 0 to 9 of f(x) is approximated from left Riemann (L), right Riemann (R), and trapezoidal (T) with three subintervals of equal width, rank them from least to greatest.

R<actual area<T<L

T(x) = sqrt(f(x)). f(0) = 4. f'(0)= -4/3. What is T'(0)?

-1/3

The function f is defined on the interval -5<x<13 where f(-5)=f(0)=f(13)=0, f(-3)= -4, f(1)=3. g'(x)= f(x). If g(1) = -11, what is the value of g(-3)?

-13/2

Calc If f'(x) = 3 + (4/x^2) + xln(x), and f(11)=13, then f(2) =

-130.072

Calc 5.5 g'(x) = the integral from 1 to x of f(t)dt and f(x)= 1/2 + ln(x/3 +2e). If it exists, find the x-coordinate where the graph of g changes from concave down to concave up.

-14.490

If the integral from 2 to 7 of f(x)=-3 and the integral from 4 to 7 of g(x)=5 and the integral from 4 to 2 of g(x)=9, then the integral from 2 t 7 of (f(x) + 5g(x)) equals

-23

Calc 5.5 g(x) = 2x + the integral from 0 to x of f(t). f(-3) = 7. f(0)= 14/3. f is composed of line segments. What is g(-3)?

-23.5

M(x) = f(x)*g(x). If f(2)= -3, g(2)=4, f'(2)= -1/2, g'(2)= 2/3, then what is M'(2)?

-4

Calc 5.5 The limit as x approaches 0 of (2e^5x -10x -2)/(cosx -lne - the integral from 0 to 4x of sin(2t)dt) =

-50/33

The integral from 2 to 6 of f(u)= 17/31, the integral from 8 to -22 of f(u) is 7/2, the integral from 2 to -40 of f(u) is 10/3, the integral of 8 to -8 of f(u) is 9/4, the integral of 2 to -22 of f(u) is 11/4, the integral of 8 to 2 of f(u) is 3/4, the integral of 2 to -8 of f(u) is 3/2. What is the value of the integral of 2 to 6 of (4x*f(10-.5x^2)dx

-9

f(x) is the integral of 0 to x of g(t) where g consists of line segments. g(-5)=2, g(-2)=0, g(0)=6. What is f(-5)?

-9

Calc 5.5 If f''(x) = 4/(3x+2) and f'(1)=5 and f(1)=10, then f(2)=

15.337

h(x) is the integral from -1 to x of f(t). f(x) is two line segments and a semicircle. f(-1)=1, f(1)=5, f(5)=0=f(13). What is the value of h(13)?

16 +8pi

f(x) = the integral from 0 to x of g(u)du. Evaluate the integral from -2 to 2 of g(4-2x)dx. Area under the curve g(x): 0 to 1 =6, 1 to 2= 10, 2 to 3=11, 3 to 4=5, 4 to 5=13, 5 to 6= -7, 6 to 7=8, 7 to 8=15.

19/2

Calc 5.5 An above ground swimming pool can hold maximum of 7500 cubic feet of water. At time t=0, the pool contains 2000 cubic feet of water. During the time interval 0<t<12, water is added to the pool at the rate of R(t) cubic feet per hour. R(0)=5, R(4)=47, R(6)=59, R(7)=61, R(12)=57. During the time interval 7<t<12, water leaks from the pool at the rate of L(t) cubic feet per hour, where L(t)=53arctan(x/100 - 1/25). Find the total amount of water, measured in cubic feet, that is in the pool at time t= 12 hours, if the amount of water added to pool is estimated using a right Riemann sum with four subintervals.

2637.442

Calc 5.5 g(x)= 4x + the integral from 0 to x of f(t)dt. f consists of line segments where f(0)= 17/3, f(4)=0, f(6)= -4, and f(7+1/3)=0. What is g'(7)?

3

If the integral from 1 to 4 of f(x) is -30 and the integral from 2 to 3 of f(x) is 10 and the integral from 8 to 3 of f(x) is 4, then the integral from 2 to 8 of (3 + 2f(x)) is

30

The tank contains 5 liters of oil at time t=3 hours. Oil is being pumped into the tank at rate R(t) in liters per hour. R(3) = 2, R(5)= 4, R(9)=5, R(10)=9. Find a trapezoidal summation with the three subintervals indicated by the data above to approximate the number of liters of oil that were pumped into the tank between 3 hours and 10 hours.

31

The tank contains 5 liters of oil at time t=3 hours. Oil is being pumped into the tank at rate R(t). R(3)=2, R(5)=4, R(9)=3, R(10)=7. Using a right Riemann sum with three subintervals and data above, what is the approximation of the number of liters of oil that in the tank at time t= 10 hours?

32

m(x)=e^2x*f(3x). If m'(4) = ke^8, then what is the value of k?. f(12)=4, f'(12)=11

41

What is the total distance that Patricia travels from t=4 to t=10. At t=0, Patricia is 8 units to the left of the origin. The area under the curve of g: 3 to 4= -2, 4 to 5=11, 5 to 6= -6, 6 to 7= 7, 7 to 8= -4, 8 to 9= 12, 9 to 10= -3, 10 to 11=2

43

f(5)= -2. f'(x) has the horizontal asymptote y=0, f'(12)=17, f'(13)=12, f'(14)=8. The integral of 15 to 5 of f''(15-(1/5*u))du =

45

What is the area of the region in the first quadrant bounded by the x and y axes, the graph of y = e^(x/4) and the vertical line x=4?

4e-4

Calc Jenny moves so that v(t) = (t^4 -33t^2 +16)/(8t+35). x(0)=-3. During the time period 0 to 13, at what exact time is Jenny furthest to the left?

5.701

If f(x)= the integral from e to x^5 of (1/t) then f'(6) =

5/6

Calc 5.5 For f'(x): (-8<x<0, f'(x) = g(x)) and (0<x<8, f'(x) = 3e^(x/8) -7). The graph of f' has x-intercepts at x= -4 and x=8ln(7/3). The graph of g on -8<x<0 is a semicircle, with a diameter of 8. It is given that f(-4)=25. What is f(8)?

6.805

If the integral from 5 to 9 of (.5g(x))dx= 24 and the integral from 6 to 5 of g(x)dx=7, then the integral from 6 to 9 of (g(x) + g'(x) dx = g(6)=1, g(9)=7, g'(6)=1/3, g'(9)=1/11

61

A particle moves along the x-axis with velocity given by v(t) = 3t^2 -4 for time t>0. If the particle is at position x = -15 at time t=3, what is the position of the particle at time t=5?

75

f(x) is the integral from 0 to x of g(t) where g consists of line segments. g(0)=9, g(4)=0. What is f'(2)?

9/2

If R(x)= g(f(x)), f(1)= -2, g(1)=3, f'(1)= -3/4, g'(-2)= -3/2, and g'(1)= 1/4, then R'(1)=

9/8

g'(x) consists of two line segments where g'(0)=2b, g'(a)=b, g'(3a)=0. Which of the following is true? I. g(x) is continuous (0, 3a) II. g(x) if differential (0, 3a) III. g(x) is increasing (0, 3a) IV. the limit as x approaches a of g' = g'(a)= DNE V. g''(2a) = b/2 VI. g(0) + .5(ab) = g(a)

I, II, III

The function f is defined on the interval A < x < B, where A and B are constant. It is known that f(A)>0 and f(B)>0. f'(x)>0 and f''(x)<0. If the integral from A->B of f(x) is approximated by a left Riemann Sum = L, a right Riemann Sum = R, and a trapezoidal sum = T, each with 3 subintervals of equal width, then rank the areas from greatest to least.

L<T<actual area<R

Calc 5.5 g'(x) = the integral from 1 to x of f(t)dt. f(x) = -3x/8 + 3/2. , If, they exist, find the x-coordinates of the relative minimum and maximum of g(x).

Relative min = 1 Relative max = 7

g'(x) is the integral of 1 to x of (2x/3 - 2)dt, is the following true? The graph of g has a relative minimum at x=5 and a point of inflection at x=3.

True

m(x)= the integral from x to pi/3 of sin^2(t)dt. What is the eq for the line tangent to the graph of m at x=pi/3

y = -3x/4 + pi/4