AP Calc AB Final

Suppose the derivative of f has the graph shown above. Which of the following could be the graph of f?

(tangent graph rotated)

If y=xe^x , then dⁿy/dxⁿ, the nth derivative of y with respect to x, equals ... ? (note n=n)

(x+n)e^x

The amount of a radioactive substance decreases according to the equation dy/dt = ky and where k is a constant and time, t, is measured in days. If half of the present amount of the substance will decrease in 20 days, what is the value of k?

-0.035

Suppose that f(x) is an even function and let ∫₀¹ f(x)dx=5 and ∫₀⁷ f(x)dx=1. What is ∫−₇⁻¹ f(x)dx?

-4

Let f be the function given by, f(x)=∫(from 0.1 to x)sin(1/t)dt, 0.1<x<0.4. At which of the following values of x does f have a relative maximum value?

0.106 and 0.318

The average value of the function f(x)=e⁻ⁿsin x on the closed interval [1,π] is (note n=x)

0.129

If f is an antiderivative of (tan²x)/(x²+1) such that f(1)=½, then f(0) =

0.155

A missile rises vertically from a point on the ground 75,000 feet from a radar station. If the missile is rising at a rate of 16,500 feet per minute at the instant when it is 38,000 feet height, what is the rate of change, in radians per minute, of the missile's angle of elevation from the station at this instant?

0.175

Let f be the function given by f(x)=x³-6x²+7x+3. The tangent line to the graph at x = 4 is used to approximate f(4.2). What is the error in this approximation?

0.248

The mass, m(t), in grams, of a tumor t weeks after it begins growing is given by m(t)=teⁿ/800. What is the average rate of change, in grams per week, during the fifth week of growth? (n=t)

0.655

A particle with velocity at any given time t given by v(t)=2e²ⁿ moves in a straight line. How far does the particle travel during the time interval when its velocity increases from 2 to 4? (note n=t)

1

Let f be the function defined above. In order for f(x) to be continuous at x = 0, the value of k must be...? f(x)={(sinx/x for x≠0)/(k for x = 0)}

1

With respect to time, t, the rate at which sin²x is increasing at x=π/4is k times the rate at which x is increasing. What is the value of k?

1

Let f be the function given by f(x)=x²lnx. For what value of x is the slope of the line tangent to the graph of f at (x,f(x)) equal to 2?

1.305

The density of a thin metal rod one meter long at a distance of x meters from one end is given by f(x)=1+(1+x)2grams per meter. What is the mass, in grams, of this rod?

1.667

Let f(x)=x³-7x²+25x-39and let g be the inverse function of f. What is the value of g'(0)?

1/10

An ice field is melting at the rate M(t) = 4 -(sin t)³ acre-feet per day, where t is measured in days. How many acre-feet of this ice field will melt from the beginning of day 1 (t = 0) to the beginning of day 4 (t = 3)?

10.667

Let f be a differentiable function on the closed interval [0,7]. The graph of f '(x) is shown above. If f(2) =10, which of the following best approximates the maximum value of f(x)? (insert graph later)

110

The amount, A(t), of a certain item produced in a factory is given by A(t)=4000 + 48(t-3) -4(t-3)³ where t is the number of hours of production since the beginning of the workday at 8:00 am. At what time is the rate of production at its maximum?

11:00am

Let f and g be differentiable functions such that f(1)=4, g(1)=3, f '(3)=-5, f '(1)=-4, g '(1)=-3, and g '(3)=2. If h(x) =f(g(x)) then h '(1)=

15

A tank is being filled with water at the rate of 300 √(t) gallons per hour with t>0 measured in hours. If the tank is originally empty, how many gallons of water are in the tank after 4 hours?

1600

f(x)={(e⁻ⁿ+2 for x<0)/(ax+b for x≥0)} f(x) = Let f be the function defined above, where a and b are constants. If f is differentiable at x=0, what is the value of a+b?

2

The area of the region in the first quadrant enclosed by the y-axis and the graphs of y=3cosx and y=x is

2.078

Let f be the function given by f(x) = tan x and let g be the function given by g(x) = x². At what value of x in the interval 0≤x≤π do the graphs of f and g have parallel tangent lines?

2.083

The table above gives values of a differentiable function f. What is the appropriate value of f'(4)? (table)

2.340

The volume of the solid formed by revolving the region bounded by the graphs of y=9 and y=(x-3)² about the line y=9 is given by which of the following integrals?

2π∫₀³(9-(x-3)²)²dx

The line y=mx+b with b≥2 is tangent to the graph of f(x)=-2(x-2)²+2 at a point in the first quadrant. What are all possible values of b?

2≤b<12

The function f(x)=tan(3ⁿ) has one zero in the closed interval [0, 1.4]. The derivative at this point is ... ? (note n=x)

3.451

As shown in the figure above, the function f(x) consists of a line segment from (0,4) to (8,4) and one-quarter of a circle with a radius of 4. What is the average (mean)value of this function on the interval [0,12]? (insert graph later)

3.714

For the function whose values are given in the table above, ∫₀⁶f(x) dx is approximated by a Riemann sum using the value at the midpoint of each of three intervals of width 2. The approximation is ... ? (table)

3.76

The graph of a polynomial function, f(x), is shown above. In the interval shown, how many times will the graph of f'(x) cross the x-axis?

4

The table above gives values of a function y and its derivative at selected values of x. If dy/dx is continuous on [1,5], what is the value of ∫₂⁵(dy/dx)dx?

46

The region enclosed by the line x+y=1 and the coordinate axes is rotated about the line y=-1. What is the volume of the solid generated?

4π/3

The derivative of f is given by f'(x)=eⁿ(-x³+3x)-3 for 0<x<5. At what value of x is f(x) an absolute minimum? (note n=x)

5

The area of the region enclosed by the graphs of y = e^x² -2 and y = 4-x² is

5.050

The base of a solid is the region enclosed by the graph of y=3(x-2)² and the coordinate axes. If every cross section perpendicular to the x-axis is a square, then the volume of the solid is

57.6

The region in the first quadrant enclosed by the coordinate axes, the line x=π, and the curve y=cos(cosx) is rotated about the x-axis. What is the volume of the solid generated?

6.040

Let f be the function given by f(t)=∫₀¹ [e^(xcosx)](cosx-xsinx)dx,0≤x≤10. At which of the following values of t does f attain its absolute minimum value?

6.437

The region in the first quadrant enclosed by the graphs of y=x and y=2sinx is revolved about the x-axis. The volume of the solid generated is ... ?

6.678

A rectangle inscribed in a semicircle of radius 8 has one side lying on the diameter of the circle. What is the maximum possible area of the rectangle?

64

A company manufactures x calculators weekly that can be sold for 75 - 0.01x dollars each. The cost of manufacturing x calculators is given by 1850 + 28x - x² +0.001x³. The number of calculators the company should manufacture weekly in order to maximize its weekly profit is

683

Silt is being dredged out of a river bed at the rate given by R(t)=100[(5t²-t-1)/(5t²+t)] gallons per minute, where t is measured in minutes. Approximately how much silt is dredged from the river bed between t=2 to t=10 minutes?

731 gallons

Two cars start at the same place and at the same time. One car travels west at a constant velocity of 50 miles per hour and a second car travels south at a constant velocity of 60 miles per hour. Approximately how fast is the distance between them changing one-half hour later?

78 miles per hour

The figure above shows the graph of y=f(t). Let g(x)=g(0) +∫(from 0 to x)f(t)dt. If g(-3)=2, what is the value g(0)?

8

Let g(x)=∫(from a to x)f(t)dt, where a<x<b. The figure above shows the graph of polynomial function g on [a,b]. Which of the following could be the graph of y=f'(x) on (a,b,)?

B in book (graph)(downwards parabola)

If f is differentiable and increasing on the interval [0,b] and c is the number guaranteed by the Mean Value Theorem on this interval, then which statement must be true?

F '(c) > 0

A population increases according to the equation P(t)=6000-5500e⁻⁰'¹⁵⁹ⁿ for t≥0, t measured in years. This population will approach a limiting value as time goes on. During which year will the population reach half of this limiting value. (note n=t)

Fourth

The figure above shows the graph of a function f(x) which has horizontal asymptotes of y=3 and y=-3. Which of the following statements are true? I. f'(x)<0 for all x>0 II. xf'(x)=0 III. x- f'(x)=3

I and II only

If lim(x→2) f(x)/(x-2)=f'(2)=0, which of the following must be true? I. f(2)=0 II. f(x) is continuous at x=2. III. f(x) has a horizontal tangent line at x=2.

I only

Let g be the function defined by g(x)=∫(from 0 to x)[(5+4t-t²)(2-t)]dt. Which of the following statements about g must be true? I. g is increasing on (3,5). II. g is increasing on (5,7). III g(7)<0

I only

The figure above shows the graph of f''(x), the second derivative of a function f(x). The function f(x) is continuous for all x. Which of the following statements about f are true? I. f is concave up for x<0 and b<x<c. II. f has a relative maximum in the open interval b<x<c. III. f has points of inflection at x=0 and x=b.

I only

f(x) = {2x-1 for x≤2, ax +4 for x >2 Let f be the function given above If f is continuous for all x, which of the following statements are true about f? I. lim f(x) = 3 II. f is differentiable at x=2. III. f has point of inflection at x=2.

I only

If f is a continuous odd function and the lim(x→-∞) f(x)=3, which of the following statements must be true? I. lim(x→∞) f(x)=3 II. There are no vertical asymptotes. III. The lines y=3 and y =-3 are horizontal asymptotes.

I, II, and III

Let f(x) be a differentiable function defined for all real numbers. The table below gives the value of f(x) and its derivative f'(x) for selected values of x. (table) Which of the following statements are true about f(x)? I. At x=2, the function is increasing. II. There is a relative minimum in the interval -1≤x≤1, but not necessarily at x=0 III. There is a relative maximum in the interval -1≤x≤1.

I, II, and III

The function y = sin x + cos x is a solution of which differential equation? I. y+dy/dx=2sin x II. y+dy/dx=2cos x III. (dy/dx)-y=-2sin x

II and III

Let f be a function which is continuous on [2,10] and whose derivative is given by f'(x)=cosx/ln(x+1). Which of the following are true about f(x) on the interval [2,10]? I. f(x) always increases or always decreases II. f(x) has a relative minimum, not at an endpoint III. f(x) has three points of inflection

II and III only

If f is the continuous function shown in the figure above, then the area of the shaded region is

Integral B

On the interval a≤x≤b the function f is positive, increasing and concave upwards. Let A=∫(from a to b) f(x)dx, L= the left Riemann sum approximation of ∫(from a to b) f(x)dx with n subdivisions of equal length, R= the right Riemann sum approximation of ∫(from a to b) f(x)dx with n subdivisions of equal length, and let T= the trapezoidal sum approximation of ∫(from a to b) f(x)dx with n subdivisions of equal length. Which of the following inequalities is true?

L < A < T < R

How many relative extreme values does the function whose derivative is given by y'=sin(lnx) have in the interval 0≤x≤1?

More than four

The expression 4∫₀² ρ(x)dx gives the number of people living on one side of a 4-mile long stretch of highway, where x is the number of miles from the highway. What are the units of ρ (x)?

People per square mile

The graph above shows the distance s(t) from a reference point of a particle moving on a number line, as a function of time. Which of the points marked is closest to the point where the acceleration first becomes negative?

Point C

If f'(x)=(x-a)(x-b)(x-c) and a<b<c, then which of the following could be the graph of f(x)?

Saggy boobs graph (B in book)

The second derivative of a function is given by f''(x)= 0.5+cosx-e⁻ⁿ. How many points of inflections does the function f(x) have on the interval 0≤x≤20? (note n=x)

Six

A particle moves along the x-axis so that its position at any time t>0 is given by x(t)=t³+22t+3-6 cos (πt). For what value of t, if any, is the velocity negative?

The velocity is never negative.

Suppose that f(x), f '(x), and f "(x) are continuous for all real numbers x, and that f has the following properties. I. f is negative on (-∞, 6)and positive (6,∞). II. f is increasing on (-∞,8) and decreasing on (8,∞). III. f is concave down on (-∞,10) and concave up on (10,∞). Of the following, which has the least numerical value?

f "(4)

If f is a function such that lim(x→a) [f(x)-f(a)]/(x-a) = 0, which of the following must be true?

f '(a)=0

The graph of the derivative of a twice-differentiable function f is shown above. If f(1) = -2, which of the following is true?

f(2) < f '(2) < f "(2)

Let f be a function that is everywhere differentiable. The value of f'(x) is given for selected values of x in the table below. (table) If f is always increasing, which statement about f(x) must be true?

f(x) has a relative minimum at x=0

Which of the functions given below has an average (mean) value of zero on the interval -a≤x≤a, a>0?

sin(x)

A particle moves along the x-axis so that its position at any time t>0 is given by x(t) = t⁴-10t³+29t²-36t+2.For which of the following values of t is the speed the greatest?

t=4

For what values of x is the function f(x)=5+15x+6x²-x³ decreasing?

x < -1 or x > 5

For what values of x does the curve y²-x³-15²=4have horizontal tangent lines?

x=-10 and x=0 only

Let f be a differentiable function with f(3) = -6 and f'(3) = 4, and let g be the function defined by g(x) = 4x+[f(x)/x] for all x≠0. Which of the following could be an equation of the line tangent to the graph of g at the point where x=3?

y-10=6(x-3)

Let f and g be functions that are differentiable for all real numbers x with g(x)=f(x)/x. If y=2x-3 is an equation of the line tangent to the graph of f at x=1, what is the equation of the line tangent to the graph of g at x=1?

y=2x+3

The region bounded by the x-axis and the part of the graph of y=cosx between x=0 and x=π/2 is divided into two regions by the line x=c. If the area of the region for 0≤x≤c is equal to the area of the region for c≤x≤π/2, then c must be

π/6

Let R be the region in the first quadrant enclosed by the lines x = 0 and y = 2 and the graph of y=eⁿ. The volume of the solid generated when R is revolved about the x-axis is given by (note n=x)

π∫(from 0 to ln2) (4-e²ⁿ)dx (note n=x)

As shown in the figure above, a square with vertices (0,0), (2,0), (2,2) and (0,2) is divided into two regions by the graph of y=-x²+2x. If a point is picked at random from inside the square, what is the probability that the point lies in the region above the parabola?

⅔

Let f(t)=1/t for t > 0. For what value of t is f '(t) equal to the average rate of change of f on the closed interval [a,b]?

√(ab)

In the interval 0 < x < 5 in the graph of y = cos 2x and y = sin 3x intersect four time. Let a, b, c, and d be the x-coordinates of these points so that 0 < a < b < c < d < 5. Which of the definite integrals below has the greatest value?

∫(from c to d)(cos 2x-sin 3x)dx

The figure above shows the graph of a function f(x) on the interval [0,5]. Which of the following definite integrals has the greatest value?

∫₀² f(x)dx

Let R(t) represent the rate at which water is leaking out of a tank, where t is measured in hours. Which of the following expressions represents the total amount of water in gallons that leaks out in the first three hours?

∫₀³ R(t)dt

The current price of a compact car is $14,500. The price of a compact car is changing at a rate of 120+180√(t) dollars per year. What will be the approximate price of a compact car five years from now?

$16,440

The functions f and g are piecewise linear functions whose graphs are shown above. If h(x)=f(x)/g(x), then h'(3)=

-2/9

If f is the function defined by f(x)=³√(x²+4x) and g is an antiderivative of f such that g(5)=7, then g(1)≈

-3.882

The slope of the function f(x)=∫(from 0 to x)(arcsint)dt when x = 0.4 is

0.412

Let f be cts such that ∫₂³f(2x)dx=8. What is the value of ∫₄⁶f(x)dx?

16

A region R is enclosed by the coordinate axes and the graph of y=k(x-5)², k>0. When this region is revolved around the x-axis, the solid formed has a volume of 2500 cubic units.

2

(Table) The acceleration of a particle from 0 to 8 seconds is given in the table above. If the velocity at t=0 is 4 feet per second, the approximated value of the velocity, in feet per second, at t = 8 seconds, computed using the Riemann sum with four subdivisions of equal length is

28

d/dx ∫(from x to x³) sin(t2)dt = ?

3x²sin (x⁶)-sin (x²)

The average value of a continuous function f(x) on the closed interval [3,7] is 12. What is the value of ∫₃⁷ f(x)dx ?

48

Let f be the function given by f(x)=5+5.8sin(πx/4)-15.7cos(πx/3). For 0≤x≤12 f is increasing most rapidly when x equals

7.566

If f(x)=2x³, then the average rate of change of f on [0,2] is

8

The figure above shows the graph of the derivative of a polynomial function f. How many points of inflection does the graph of f have? (insert graph later)

Four

Let f(x) be a continuous function and let A be the area of the shaded region in the figure above. Which of the following must be true? I. A=∫(from 0 to a)f(x)dx II. A=∫(from 0 to b)f⁻¹(x)dx III. A=∫(from 0 to b)f⁻¹(y)dy

I, II, and III

The position of an object attached to a spring is given by y(t)=1/3 sin (4t)-1/8 cos (4t) where t is time in seconds. How many times does the acceleration of the object change from negative to positive in the first 5 seconds?

Three

Let f be a function whose derivative is given by f'(x)=x/15+sin(e^0.2x). How many relative maximum points does f(x) have in the interval 0≤x≤12?

Two

The local linear approximation of a function f will always be greater than or equal to the function's value if, for all x in an interval containing the point of tangency,

f"<0

If lim(x→a) [g(3)-g(x)]/(3-x) = -0.628, then near the point where x=3, the graph of g(x)

is decreasing

The equation of the horizontal asymptote for the graph of y=[2-e^(1/x)]/[2+e^(1/x)] is

y=⅓

At what point on the curve x³ + 3x² +y² = 4 is the tangent line vertical?

(-2,0) and (1,0)