AP Calc AB Final Exam (1998 1A - 2B 76-92)

integrate x^2•cos(x^3) dx =

(1/3)•sin(x^3) + C

integral: 0,x sin (t) dt =

1-cosx

Let f be a differentiable function such that f(3)=15, f(6)=3, f'(3)=-8 and f'(6)=-2. The function g is differentiable and g (x) = f^-1 (x) for all x. What is the value of g'(3)?

-1/2

Integral of (sin (2x) + cos (2x)) dx

-1/2cos(2x) + 1/2sin(2x) + C

What is the x^2 - 4 ---------------- lim as x approaches ∞ 2 + x -4x^2

-1/4

If integral: -5,2 f(x)dx = -17, and integral: 2,5 f(x)dx=-4, what is the value of the integral: -5,5 f(x)dx?

-13

(2x-1) (3-x) lim as x approaches ∞ ------------ (x-1) (x+3)

-2

f(x) = { cx + d for x≤2 { x^2 -cx for x>2 Let f be the function defined above, where c and d are constants. If f is differentiable at x=2, what is the value of c + d.

-2

Let f be the function defined by f(x)= x^3 + x. If g(x) = inverse f (x) and g(2) =1, what is the value of g'(2)? (told to be on test)

1/4

If y = (x^3 + 1)^2, then dy/dx =

6x^2(x^3 + 1)

.5 • integral e^(1/2) dt = (if confused number 6 on apr 2)

e^.5 + C

If f(x) = e^(2/x), then f'(x) =

(-2/x^2) • (e^2/x)

if f(x)= sin(e^-x), then f'(x)=

(-e^-x)(cos(e^-x))

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

(0, (2)^1/3)

the function f is given by f(x)= x^4 + x^2 -2. On which of the following intervals is f increasing?

(0,∞)

Using the substitution u= 2x + 1, integral: 0,2 √(2x + 1) dx is equivalent to

(1/2)•integral 1,5 √(u) du

integral: 0,1 e^-4x dx=

(1/4) - (e^-4 ÷ 4)

if f(x) = x(2x-3)^1/2, then f'(x) =

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

What is the slope of the line tangent to the curve 3y^2 - 2x^2 = 6 - 2xy at the point (3,2)?

(4/9)

integral: 1,e (x^2 -1)/x dx =

(e^2 / 2) - (3/2)

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

(x^2 + 2)^2 • (7x^2 - 6x + 2)

If f"(x)= x(x+1)(x-2)^2, then the graph of f has inflection points when x=

-1 and 0 only

5x^4 + 8x^2 Lim as x -> 0 --------------- 3x^4 -16x^2

-1/2

if f(x) = -x^3 + x + (1/x), then f'(-1)=

-3

The table above (#85 on 2008 76-92) gives values of a function and its derivative at selected values of x. If f' is continuous on the interval [-4,1], what is the value of integral: -4,-1 f'(x) dx?

-2.25

if f(x) = ln (x + 4 + e^-3x), then f'(0) is

-2/5

If f(x)= sin^2(3-x), the f'(0)=

-2sin(3)•cos(3)

The derivative g' of a function g is continuous and has exactly two zeros. Selected values of g' are given in the table above (18 on 2A 1-28). If the domain of g is the set of all real numbers, then g is decreasing on which of the following intervals?

-2≤x≤2 only. (on graph x=-2 and x=2 are zeros. and values between are negative)

In the xy-plane, the line x+y=k, where k is a constant, is tangent to the graph of y=x^2 +3x + 1. What is the value of k?

-3

What are all the values of k for which integral: -3,k x^2 dx =0?

-3

if f(x) = cos(3x), then f'(π/9) =

-3 √3 -------- 2

The radius of a sphere is decreasing at a rate fo 2 cm per second. At the instant when the radius of the sphere is 3cm, what is the rate of change, in square cms per second, of the surface area of the sphere? (S = 4πr^2)

-48π

What is the x-cordinate of the point of inflection on the graph y= 1/3x^3 + 5x^2 + 24 ?

-5

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

-5 ---- (3x + 2)^2

The slope of the tangent to the curve y^3x + y^2x^2=6 at (2,1) is

-5/14

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

-6x^2 • sin(x^3) • cos(x^3)

If G(x) is an antiderivative for f(x) and G(2)= -7, then G(4)=

-7 + integral: 2,4 f(t)dt

if x^2 +xy =10, then when x=2, dy/dx =

-7/2

integral of (1/x^2) dx =

-x^-1 +C

Integral: 0,π/4 sinx dx =

-√(2) ------- + 1 2

The average value of the function f(x) = e^(-x^2) on the closed interval [-1,1] is

.747

If f is a linear function and 0<a<b, then integral: a,b f"(x) dx =

0

The function f is continuous on the closed interval [0,2] and has values that are given in the table above (26 on 1998 #1-28 MC). The equation f(x)= 1/2 must have at least two solutions in the interval [0,2] if k=

0

A particle moves along a straight line. The graph os the particle's position at x(t) at time t is shown above (number 21 on 2008 #1-28) for 0<t<6. The graph has horizontal tangents at t=1 and t=5 and a point of inflection at t=2. For what values of t is the velocity of the particle increasing?

0 < t < 2

The graph of function f shown in the figure above (13 on 1998 #1-28) has a vertical tangent at the point (2,0) and horizontal tangents at the points (1,-1) and (3,1). For what values of x, -2<x<4, is f not differentiable?

0 and 2

A particle moves along a straight line with velocity given by v(t) = 7- (1.01)^(t^-2) at time t≥0. What is the acceleration of the particle at time t=3?

0.055

What is the average value of y = cos/ (x^2 + x+ 2) on the closed interval [1,-3]?

0.183

The function f is twice differentiable with f(2)=1, f'(2)= 4, and f"(2) =3. What is the value of the approximation of f(1.9) using the line tangent to the graph of f at x=2?

0.6

Let f and g be differentiable functions with the following properties: (i) g(x)>0 for all x (ii) f(0)=1 If h(x)=f(x)g(x) and h'(x)= f(x)g'(x), then f'(x) =

1

what is lim as h approches 0 cos (3π/2 + h) - cos (3π/2) ----------------------------- h

1

if sin (xy) =x, then dy/dx =

1 - y•cos(xy) ------------- x• cos(xy)

The first derivative of the function f is defined by f'(x)= sin(x^3 - x) for 0≤x≤2. On what intervals is f increasing?

1 ≤ x ≤ 1.691

Let f be the function with first derivative defined by f'(x)= sin(x^3) for 0≤x≤2. At what value of x does f attain its maximum value on the closed interval 0≤x≤2?

1.465

Let f be a function such that f'(x)<0 for all x in the closed interval [1,2], with the selected values shown in the table above (18 on apr 11). Which of the following must be true about f'(1.2)?

1.8 < f'(1.2) < 2.0

A particle moves along the x-axis so that its velocity at any time t≥0 is given by v(t) = 5te^-1. At t=0, the particle is at position x=1. What is the total distance traveled by the particle from t=0 to t-4?

1.821

intergral: 1,2 (1/x^2) dx =

1/2

integral of (x/x^2 - 4) dx =

1/2 • ln |x^2 - 4| + C

lim as x approaches ∞ x^3 - 2x^2 +3x -4 ------------------- 4x^3 - 3x^2 + 2x -1

1/4, since approaching infinity and leading powers of numerator and denominator are the same, just divide leading coefficients.

What is the area enclosed by the curves y=x^3 - 8x^2 + 18x -5 and y= x+5

11.833

The rate at which water is sprayed on a field of vegetabels is given by R(t)= 2√(1 + 5t^3), where t is in minutes and R(t) is in gallons per minute. During the time interval 0≤t≤4, what is the average rate of water flow, in gallons per minute?

14.691

What is the area of the region between the graphs of y=x^2 and y=-x from x=0 and x=2?

14/3

What is the instantaneous rate of change at x=2 of the function f by f(x)= (x^2 -2)/(x-1)

2

What is the slope of the line tangent to the curve y = arctan(4x) at the point at which x= 1/4?

2

The graph of the function f shown above (17 on 2008 #1-28) has horizontal tangents at x=2 and x=5. Let g be the function defined by g(x) = integral: 0,x • f(t) dt. For what values of x does the graph of g have a point of inflection?

2 and 5 only

If f (x) = x^2 + 2x, the d/dx (f(ln x))

2 lnx + 2 ---------- x

The graph of a piecewise-linear function f, for -1 ≤ x≤ 4, is shown above (number 2 on 1998 #1-28). What is the value of integral: -1 ,-4 f(x) dx?

2.5

the graph of y= 3x^4 - 16x^3 + 24x^2 + 48 is concave down for

2/3 < x < 2

The maximum acceleration attained on the interval 0≤t≤3 by the particle whose velocity is given by v(t)= t^3 - 3t^2 + 12t + 4 is

21

what is the average value of y= x^2•(x^3 + 1)^1/2 on the interval [0,2]?

26/9

Let g be a twice differentiable function with g' (x) > 0 and g"(x) > 0 for all real numbers x, such that g(4)=12 and g(5)= 18. Of the following, which is a possible value for g(6)?

27

A particle moves along the x-axis so that at any time t≥0 its velocity is given by v(t)= t^2•ln(t+2). What is the acceleration of the particle at time t=6?

29.453

if dy/dx= ky and k is a nonzero constant, then y could be

2e^kt

If y= x^2• sin(2x), then dy/dx

2x( sin2x + x•cos2x)

d/dx[ integral 0,x^2 sin(t^3) dt]

2x•sin(x^6)

A particle moves along the x-axis so that its position at time t is given by x(t)= t^2-6t+5. For what value of t is the velocity of the particle zero?

3

If the line tangent to the graph of the function f at the point (1,7) passes through the point (-2,-2), then f'(1)=

3

if F(x)= integral: 0,x (t^3 + 1)^1/2 dt, then F'(2)= (square root = x^1/2) if need clarification

3

Let f be a function with a second derivative given by f"(x)= x^2 • (x-3) • (x-6). What are the x-coordinates of the points of inflection of the graph of f?

3 and 6 only

The flow of oil, in barrels per hour, through a pipeline on July 9 is given by the graph shown above. (number 9 on 1998 #1-28 MC) Of the following, which best approximates the total number of barrels of oil that passed through the pipeline that day?

3,000

The function f is differentiable and has values as shown in the table above (83 on 2B 76-92). Both f and f' are strictly increasing on the interval 0≤x≤5. Which of the following could be the value of f'(3)?

30

if integral: 0,3 f(x) dx=6 and integral 3,5 f(x) dx=4, then integral: 0,5 (3+ 2f(x)) dx=

35

A spherical tank contains 81.637 gallons of water at time t=0 minutes. For the next 6 minutes, water flows out of the tank at the rate of 9•sin(√t+1) gallons per minute. How many gallons of water are in the tank at the end of the 6 minutes?

36.606

d/dx integral: 0,3x • cos (t) dt=

3cos(3x)

The graph os the derivative of a function f is shown in the figure above (number 84 on 2008 #76-92). 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?

4 only

Find the volume of the solid formed by rotating the region bounded by the graph of y=√(x+1), the y-axis, and the line y=3 about the line y=5.

41.888

The table above (89 on 2B 76-92) gives values of the differentiable functions f and g and their derivatives at x=1. If h(x)= (2 f(x) + 3)•(1+g(x)), then h'(x)=

44

A particle moves along the x-axis with the velocity given by v(t) = 3t^2 + 6t for time t ≥ 0. If the particle is at position x=2 at time t = 0, what is the position of the particle at time t =1 ?

6

intergral: 1,2 (4x^3 -6x) dx =

6

A bug begins to crawl up vertical wire at time t=0. The velocity of v of the bug at time t, 0≤t≤8, is given by the function whose graph is shown above. (number 8 on apr 2). uses same graph 9 on apr 2: what is the total distance the bug traveled from t=0 to t=8?

6 13

An object traveling in a straight line has position x(t) at time t. If the initial position is x(0)=2 and the velocity of the object is v(t)= (1 + t^2)^1/3, what is the position of the object at time t+3?

6.512

Find the volume of the solid formed by rotating about the x-axis the region enclosed by the graph of y=√(x+1), the x-axis, the y-axis and the line x=4.

71.209

if integral: a,b f(x)dx = a+ 2b, then integral: a,b (f(x) + 5) dx =

7b-a

The graph of f' , the derivative of f, is the line shown in he figure above (22 on 2A 1-28). If f(0) = 5, then f(1)=

8

if f(x)= tan(2x), then f'(π/6)=

8

(8 and 9 on apr 7) According to the graph, at what time t does the bug change direction? According to the graph, at what time t is the speed of the bug greatest?

8 (crosses the x-axis) 10 (greatest absolute value of y)

The base of a loudspeaker is determined by the two curves y= (x^2 / 10) and y= ( -x^2 /10) for 1≤x≤4, as shown in the figure above (85 on 2B 76-92). For this loudspeaker, the cross sections perpendicular to the x-axis are squares. What is the volume of the loud speaker?

8.184

The function f is continuous on the closed interval [2,4] and twice differentiable on the open interval (2,4). If f'(3)= 2 and f"(x) < 0 on the open interval (2,4), which of the following could be a table of values for f? (90 on 2008 76-92)

A, slope at x=3 is greater than slope from x=3 to x=4. Slope at x=3 is less than slope from x=2 to x=3. Thus concave down which means f" is negative!

The second derivative of the function f is given by f"(x) = x•(x-a)•(x-b)^2. The graph of f" is shown above (21 on 2A 1-28). For what values of x does the graph of f have a point of inflection?

A.) O and a only. (since graph is of second derivative, PoI is when it crosses x-axis)

At which of the five points on the graph in the figure at the right (#3 on apr 4) are dy/dx and d^2y/dx^2 both negative?

B

The graph of a function f is shown above. Which of the following could be the graph of f', the derivative of f? (number 11 on 2008 #1-28)

B, a parabola (makes a U)

The function f has the property that f(x), f'(x) and f"(x) are negative for all real values x. Which of the following could be the graph of f? (10 on 2A 1-28)

B, below x-axis so f'(x) is negative. concave down so f"(x) is negative

The table given (86 on 2008 76-92) gives selected values of the velocity, v(t), of a particle moving along the x-axis. At t=o, the particle is at the origin. Which of the following could be the graph of the position, x(t), of the particle for 0≤x≤4?

C, starts a origin, then below x-axis, crosses x-axis then between 1 and 2 and has a relative maximum around x=3 and then dips down again.

The graphs of five functions are shown below. Which function has a nonzero average value over the closed interval [-π,π]? (number 21 on apr 11)

E, makes a m with the middle point of the m at the origin.

The derivative of the function f is given by f'(x)= x^2 • cos(x^2). How many points of inflection does the graph of f have on the open interval (-2,2)?

Five

The function f is continuous for -2≤x≤2 and f(-2)=f(2)=0. If there is no c, where -2<c<2, for which f'(c)=0, which of the following statements must be true?

For some k, where -2 < k < 2, f'(k)= does not exists.

The first derivative fo the function f is given by f'(x)= x- 4e^-sin(2x). How many points of inflection does the graph of f have in the interval 0 < x < 2π?

Four (graph f'(x) and all the max/mins are points of inflection)

Let f be defined as follows, where a does not equal 0. { (x^2 -a^2)/(x-a) for x does not equal a f(x)= { 0 for x=a Which of the following are true about f? I. lim as x approches a f(x) exists II. f(a) exists III. f(x) is continuous at x=a

I and II only

The figure above (77 on 2008 #76-92) shows the graph of function f with domain 0≤x≤4. Which of the following statements are true? I. Lim as x approaches 2 from the left f(x) exists. II. lim as x approaches 2 from the right f(x) exists. III. lim as x approaches 2 f(x) exists.

I and II only

{ x +2 if x≤3 f(x) = { 4x-7 if x > 3 Let f be a function given above. Which of the following statements are true about f? I. lim as x approches x f(x) exits II. f is continuous at x=3 III. f is differentiable at 3

I and II only

A left Riemann sum, a right Riemann sum, and a trapezoidal sum are used to approximate the value of integral: 0,1 f(x)dx, each using the same number of subintervals. The graph of the function f is shown in the figure above. Which of the sums gives an underestimate of the value of integral: 0,1 f(x)dx? I. left sum II. right sum III. trap sum (number 80 on 2B 76-92)

I and III only

f(x) = { (x^2 - 4 / x-2 ) if x doesn't = 2 { 1 if x =2 Let f be the function defined above. Which of the following statements about f are true? I. f has a limit at x=2 II. f is continuous at x=2 III. f is differentiable at x=2

I only

The function f, whose graph consists of two line segments, is shown above. Which of the following are true for f on the open interval (a,b)? I. The domain of the derivative of f is the open interval (a,b). II. f is continuous on the open interval (a,b). III. The derivative of f is positive on the open interval (a,c). (SEE GRAPH ON APR 4, #1)

II and III only

The function f is continuous and differentiable on the closed intercal [3,7]. The table above (86 on 2B 76-92) gives selected values of f on this interval. Which of the following must be true? I. the min value of f on [3,7] is 12 II. there exists c, for 3 < c <7, such that f'(c)=0 III. f'(x) > 0 for 5<x<7

II only

For x≥0, the horizontal line y=2 is an asymptote for the graph of the function f. Which of the following statements must be true?

Lim as x approches ∞ f(x) =2

The graph of f is shown in the figure above (23 on 1998 #1-28MC). Which of the following could be the graph of the derivative of f?

Makes a sin graph from points (a,b)

The graph of the function f is shown above for 0≤x≤3. Of the following, which has the least value? (number 10 on 2008 #1-28)

Right Riemann sum approximation of integral: 1,3 • f(x) dx with 4 subintervals of equal length

The polynomial function f has selected values of its second derivative f" given in the table above. Which of the following statements must be true? (table is #14 on 2008 1-28)

The graph f changes concavity in the interval (0,2).

Which of the following statements about the function given by f(x) = x^4 - 2x^3 is true?

The graph of the function has two points of inflection and the function has one relative extremum.

For t≥0 hours, H is a differentiable function of t that gives the temperature, in C, at an arctic weather station. Which of the following is the best interpretation of H'(24)?

The rate at which the temperature is changing at the end of the 24th hour.

If f is a continuous function on the closed interval [a,b], which of the following must be true?

There is a number c in the closed interval [a,b] sich that f(c)≥ f(x) for all x in [a,b]. (extreme value theorem)

Two particles start at the origin and move along the x-axis. For 0≤t≤10, their respective position functions are given by x1 = sin(t) and x2 = (e^-2t) - 1. For how many values of t do the particles have the same velocity?

Three

The graph of f', the derivative os f, is shown above (number 76 on 2008 #76-92) for -2≤x≤5. On what interval is f increasing?

[-2,3]

The graph of f', derivative of f, is shown above (84 on 2B 76-92). On which of the following intervals if f decreasing?

[0,2] and [4,6]

The graph of a function f is shown above (13 on @2A 1-28). At which value of x is f continuous, but not differentiable?

a (its a kink)

The rate of change of the volume, V, of water in a tank with respect to time, t, is directly proportional to the square root of the volume. Which of the following is a differential equation that describes this relationship?

dV/dt = k√(V)

A rumor spreads among a population of N people at a rate proportional to the product of the number of people who have heard the rumor and the number of people who have not heard the rumor. If p denotes the number of people who have heard the rumor, which of the following differential equations could be used to model this situation with respect to time t, where k is a positive constant?

dp/dt = kp(N-p)

Shown above (number 27 on 2008 #1-28) is a slope field for which of the following differential equations?

dy/dx = xy + y

The graph of f', the derivative of the function f, is shown above (7 on 2A 1-28). Which of the following statements is true about f?

f is increasing for -2≤x≤0. (on graph since it is f' from -2 to 0 the line is above the x-axis, this fx is increasing there)

The graph of a twice-differentiable function f is shown in the figure above (17 on 1998 #1-28 MC). Which of the following is true?

f"(1)<f(1)<f'(1)

if f is continuous for a<x<b and differentiable for a<x<b, which of the following could be false?

f'(c)=0 for some c such that a<c<b.

The graph of the piecewise linear function f is shown in the figure above. If g(x) = integral -2,x • f(t) dt, which of the following values is the greatest? (number 9 on 2008 #1-28)

g(1)

The functions f and g are differentiable, and f(g(x))= x for all x. If f(3)= 8 and f'(3)=9, what are the values of g(8) and g'(8)?

g(8)= 3 and g'(8)= 1/9

if f(x) = { lnx for 0<x≤2 { x^2 ln2 for 2<x≤4 then lim as x approaches 2 of f(x) is

nonexistent

The figure above (87 on 2B 76-92) shows the graph of f', the derivative of the function f, on the open interval -7<x<7. If f' has 4 zeros on -7<x<7, how many relative maxima does f have on -7<x<7?

one (when the derivative line goes from + to - at a zero value)

A particle moves along the x-axis so that at time t≥0 its position is given by x(t)= 2t^3 - 21t^2 + 72t -53. At what time t is the particle at rest?

t = 3 and t=4

Let f be the function given by f(x) = 2xe^x. The graph of f is concave down when

x < -2

The solution to the differential equation dy/dx = x^3/y^2, where y(2) = 3, is

y = (3/4x^4 + 15)^1/3

Let f be the function defined by f(x)= 4x^3 -5x +3. Which of the following is an equation of the line tangent to the graph of f at the point where x= -1?

y = 7x + 11

A curve has a slope 2x +3 at each point (x,y) on the curve. Which of the following is an equation for this curve if it passes through the point (1,2)

y = x^2 + 3x -2

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

y= - [(2x^3 / 3) - 14]^1/2 it is a negative square root with everything in the root in the [ ]

What are all horizontal aysmptotes of the graph of 5 + 2^x y = ---------- in the xy-plane? 1 - 2^x

y= -1 and y=5

An equation of the line tangent to the graph of y=x + cosx at the point (0,1) is

y=x+1