ap calculus ab top 58 questions

1415- 12S AP AB Math Top 58 Questions (26/01/2015) Page 1 of 35

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12S AP AB Math Top 58 Questions

GRID

Grade 12 S, NS Book ,Chapter 4

1. Given: x and y are functions of time t that are related by the equation 2 3y xy .

a) When t = 10, y= 3 and 8dy

dt . Find

dx

dt

Given also that curve C: 2 3 0y xy

b) Find the gradient of the curve.

c) Find the point(s), if they exist where the curve has a horizontal tangent.

d) Find the set of points which belong to C where the line tangent to the curve is

parallel to 1

3y x .


2. Given: 1

23

dyx y

dx .

a) Sketch a slope field for the given equation at the plotted nine points in the

coordinate plane.

1

2

1 1 O

x

y


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b) In terms of x and y find 2

2

d y

dx.

c) Find the inequality that describes the region in the xy-plane for which all 2

2

d y

dx>0.

d) y f x is a particular solution to the differential equation with the initial

condition 0 2f . Does f(x) has a turning point at x = 0? What is its nature?

e) Determine the values of the constants a and b, for which y ax b is a solution

to the differential equation.


3. Given 23 .........x xy e C .

a) Find the area limited by the graph of (C) and y = 2.

b) Find the area limited by the graph of (C) and y = 1 and y = 2.

c) Find an expression for the volume generated when the region limited by the graph

of (C) and y = 2 is rotated 2 about y = 1.

Grade 12 S, NS Book ,Chapter 6 GRID

4. Given y f x b g where 1 1 2dy y xdx

.

a) Find d y

dx

2

2 at (3, 2).

b) Solve 1 1 2dy

y xdx

with the initial condition 3 2f .


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Grade 12 S, NS Book Chapter 6

5. [A graphic calculator is needed to solve this question.]

Let S be the region in the first quadrant enclosed by the graph of

22 , 2 2cosxy e y x and the y-axis.

a) Find the area of the region S

b) When region S is revolved about the x-axis a solid is generated, find the volume

of this solid.

c) The region S is the base of a solid. Consider each cross section, of this solid,

perpendicular to the horizontal axis as a rectangle whose height is three times the

width, where the width is bounded by the graphs of the given functions. Find the

volume of the solid.

Grade 12 S, NS Book ,Chapter 3 & 5

6. The function m x is given by 3 3 when 0 3

3 3 when 3 0

x xm x

x x

If also given that 0

3

x

g x m t dt

a) Find g g g 1 1 1b g b g b g, , and . b) Determine the values of x where g(x) is increasing.

c) Determine the values of x where g(x) is concave down. Give reasons.

d) Sketch the graph of g(x) for 1 1x .


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7.

Distance x (mm) 0 50 100 150 200 250 300

Diameter L(x) (mm) 20 25 20 35 28 27 35

Suppose that a water pipe is 300 millimeters (mm) long with circular cross sections of

varying diameter. The table given above gives the measurements of the diameter of

the water pipe at selected points along the length of the water pipe, where x represents

the distance from one end of the water pipe and L(x) is a twice-differentiable function

that represents the diameter at that point.

a) Give an integral expression in terms of L(x) that represents the average radius,

in mm, of the water pipe between x = 0 and x = 300.

b) Give an approximate value of your answer from part (a) using the data from

the given table and a midpoint Riemann sum with three subintervals of equal

length. Show all your working.

c) Explain the meaning of the expression

2250

902

L xdx

in terms of the water

pipe, using proper units.

d) Why there must be at least one value x, for 0 < x < 300, such that 0L x .

Grade 12 S, NS Book ,Chapter 3 & 5 GRID

8. Given: 3g x x g x for all real numbers x, where g(2) = 15.

a) Find at 2d

g x xdx

b) Solve 3dy

x ydx

with the initial condition g(2) = 15.


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9. f x is differentiable over

The table gives the values of f and its derivative f . f xb g 0 for 2.5,1x . x -2.5 -2.0 -1.5 1 0 0.5 1

f xb g -2 -6 -8 -3 -1 -5 -3 f xb g -6 -7 -2 1 10 3 4

a) Showing your work, find the value of 1

1.5

3 5f x dx

.

b) i) Find the equation of the line tangent to the graph of f at x = 0.5.

ii) Approximate the value of 0.4f , using the equation of the tangent line.

iii) Compare the approximate value of 0.4f , to the actual value of 0.4f .

c) Find a positive real number r having the property that there must exist a value c

with 1 0c and f c rb g . Give a reason for your answer.


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10.

The graph of g x is given above. g x has a tangent of slope = 0 at x = 7,

x = 2 and x = 5 and a tangent of an undefined slope at x = 4.

In each of the following, you should justify your answer by showing all working.

Over the interval 9,9 ,

a) Find the values of x for which g(x) has a relative minimum.

b) Find all the values of x at which g attains a relative maximum.

c) Find all values of x at which 0g x .

d) Find the value of x at which g attains its absolute maximum?


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11. Given the function 3 22g x x mx nx r , where m, n and r are constants. The

following information is given about g(x):

i) g x has a local minimum at 1x .

ii) the graph of g has a point of inflection at 3x .

a) Find the values of m and n.

b) Assume that 1

0

45g x dx , find the value of r.

Grade 12 S, NS Book ,Chapter 3 & 5 GRID

12. Given the graphs of the two curves 2 3 and 1y x y x . The region A in the first

quadrant is bounded by 2 3 and 1y x y x and the y-axis and the region B in the

first quadrant is bounded by the two curves and the x-axis.

a) Find the area of A.

b) Find the area of B.

c) Find the volume of the solid generated when A is revolved about the x-axis.


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13.

The graph of / 2 3sinxg x e x is shown above, g is the first derivative of the function g(x) defined for 0x with g(0) = 0.

a) Using the graph of g , what type of concavity if any, does g x represent when

1.3 1.6x . Explain.

b) Find x, 0 2x at which g x , has an absolute maximum. Explain.

c) Write the equation of the line tangent to the graph of g x at x = 1.

x

y

-0.5 0 0.5 1 1.5 2

-0.5

0

0.5

1

1.5

2

2.5


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Grade 12 S, NS Book ,Chapter 3, 5 GRID

14.

(5,10)

Time (seconds)

O 5 10 12 15 20

5

10

15

(12,10)

t

v(t)

Vel

oci

ty

(met

ers

per

sec

ond

)

(20,0)

A motorcycle is traveling on a straight road for 0 20t seconds, the motorcycles

velocity v(t) in m/s is represented by the graph above.

a) Find 20

0

v t dt , give its unit and explain its meaning.

b) Find 5 and 16v v if any of these values does not exist state so give the unit of

measure of 16v . c) Express a(t), the acceleration of the motorcycle as a piecewise defined function.

d) Find the average rate of change of v when8 16t , knowing that 16 5v .

e) Does the Mean Value Theorem guarantee a value of K, 8,16K such that

v K equals the average rate of change? Explain.


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15. Given: 43 , 0kxg x kxe k

a) Find

i) lim ( )x

g x

ii) lim ( )x

g x

b) Find the absolute maximum value of g x explaining why your answer is an absolute maximum.

c) Determine the range of g x .

Grade 12 S, NS Book ,Chapter 4 & 6

16. At any time 0t , in days, the rate of growth of bacteria population is given by y ky ,

where k is a constant and y is the number of bacteria present. The initial population is

1,500 and the population doubled during the first 6 days.

a) Write an expression for y

b) At what time t, will the population have increased by a factor of 8?

c) By what factor will the population have increased in the first 12 days?


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Grade 12 S, NS Book, Chapter 4 & 6 GRID

17.

(3, 2)

(5,2)

(4,0)

O

(3,2) (5,3)

Graph of f

a) Given that 5

x

g x f t dt

and 2

13

f

Find 1 , 1g g and 1g . If the value does not exist, state so.

b) Find the abscissa(s) of the point of inflection of g when 5 5x . Justify your

answer.

c) If 5

x

h x f t dt where 5 5x .

i) Find x such that 0h x .

ii) Find all intervals on which h x is decreasing.


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Grade 12 S, NS Book, Chapter 7

18. Consider the graph of the function 1

1f x

x

is graphed when 0x . If A is the

region bounded by the graph of f, the x-axis and y-axis and the line x = m; and B is the

region in the first quadrant bounded by the line

x = m, the graph of f x and 3,x 0 3m GRID

a) Find the area of A in terms of m.

b) Find the volume generated when A is revolved about the x-axis.

c) Find m such that the volume when B is revolved about the x-axis is half that of A

when revolved about x-axis.


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19. The graph of f is shown in the diagram below.

a) For 6 6x , find all the values of x at which f has a relative maximum, give

reasons.

b) For 6 6x , find all the values of x at which f has an inflection point.

c) Find the intervals at which the graph of f is concave down.

d) If given that 6 10f , find the absolute maximum value of

, 6 6f x x .


20. Solve the differential equation y x y , given that when x = 3, y = 25.

Write an expression for in the form y g x .


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21. Consider the two function: 1

1 ; ln 12

th t e k t t .

a) Find the area of the region formed by the graphs of f and g between 1

3t

and t = 0.5.

b) Find the volume of the solid generated when the region formed by the graphs of f

and g between 1

3t and t = 0.5 is revolved about the line y = 3.

c) For the function n t h t k t , find the absolute minimum and the absolute

maximum on 1 1

,3 2

.


22. Given a cone with vertex pointed downwards, whose radius is 10 cm and height double

its radius. Water is pumped into the cone so that the depth of the water h is changing at

the constant rate of 1

4 cm/hr, and the radius of the water surface is r.

a) Find the volume V of water in the container when h = 8 cm.

b) Find the rate of change of the volume of water in the container, with respect to

time, when h = 8 cm.

c) Find the relation between the rate of change of the volume of water and the

surface area of the water. (Hint: directly proportional).


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Grade 12 S, NS Book, Chapter 4

23.

The velocity v(t) of a train in m/s for 0 40t is given in the table which gives values

of v(t) at 4 second intervals of time t.

a) Use the given table to draw a graph of velocity, v(t).

b) During what intervals of time is the acceleration of the train positive?

c) Find an approximation for the acceleration of the train in m/sec2, at t = 28.

d) Approximate 40

0

v t dt with a Riemann sum, using the midpoints of five

subintervals of equal length and give a meaning for your answer.


24. Given 3 4h x x x , and m is a tangent line to the graph of h with equation

2y x . Let S be the region bounded by the graph of h, the line m:

a) Sketch the graph of the given function h and the tangent line m.

b) Prove that line m, is tangent to the graph of y h x at the point x = 1 .

c) Find the area of region S.

d) Find the volume of the solid generated when the region in the second quadrant

bounded by the graph of h x and the x-axis is rotated about the x-axis.

t

(seconds)

v(t)

(meter per seconds)

0

4

8

12

16

20

24

28

32

36

40

0

10

18

25

50

60

73

82

70

50

73


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Grade 12 S ,NS Book ,Chapter 3 GRID

25.

The figure shows the graph of y h x , for 1,6x . The given graph has a

horizontal tangent lines at x = 2 and x = 4. The function h is twice differentiable and

3 2h .

a) Find the abscissa of each of the points of inflection of the graph of h. Justify.

b) For what value of x does h reach its absolute minimum value over 1,6 ?

c) For what value of x does h reach its absolute maximum value over 1,6 ?

d) Given a function g defined by g x x h x . Find an equation of the line

tangent to the graph of g at x = 3.


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Grade 12 S ,NS Book ,Chapter 5

26. Given 4 1 dy

x ydx

.

a) Sketch a slope field for the given differential equation at the twelve points

indicated.

1

2

1 1 O

3

x

y

b) The slope field is defined at every point in the xy-plane. Describe all points in the

xy-plane for which the slopes are negative.

c) Find the particular solution y g x to the given differential equation with the

initial condition 0 0g


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Grade 12 S ,NS Book ,Chapter 4 & 6

27.

Expansion

in cm

(x)

Temperature

M(x)

10

7

5

1

0

100

80

75

63

60

The table gives some values of the x, in cm, and the corresponding values of the

temperature M x in degrees Celsius (C). The function M x is increasing and

twice differentiable.

a) Estimate 6M . Show your working and give units of measure.

b) Give an integral expression in terms of M(x) for the average temperature.

c) Estimate the average temperature using a trapezoidal sum with the four

subintervals using the data given in the table.

i. Find 10

0

M x dx , and give units of measure.

ii. Explain the meaning of 10

0

M x dx


28. Let 1 cosf x x

a) What is the domain of f x ?

b) What is the domain of f x ?

c) Write the equation of the line normal to the curve at 2

x

d) If Rolles theorem applicable for f x in the interval 3

,2 2


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29. Given the function by 3 22 3 2 2cosg x x x x x , Let R be the shaded region in

the quadrant II bounded by the graph of g, the x-axis and y-axis. S is the region

bounded by the graph of g and line t, which is tangent to the graph of g at 1.5x .

a) Find the area of region of R.

b) Find the volume of the solid generated when R is rotated about y = 1.

c) Without actually evaluating, write an integral expression for the area of region of

S.


30. Let R be the region in the first quadrant bounded by 2

2

3 2

xf x

x

, 1 6x

a) Find the exact value for the area R, show all your work.

b) The line x k divides the area R into two equal parts, find the value of k.

c) Calculate the average value of f x in the given interval.


31. Given that the region M is bounded by graphs of the curve 3

22

xy

x

, the line

y = x + 4 and the y-axis.

a) Sketch the given functions.

b) Find the area of M.

c) Find the volume of the solid generated when M is revolved about the line

y = 0.

d) Find the volume of the solid whose base is the region M if the cross sections taken

perpendicular to the x-axis are equilateral triangles.


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32.

t

(min)

v(t)

(km/min)

0 8.0

5 10.3

10 18.2

15 8.0

20 14.5

25 12.6

30 13.6

35 14.3

40 19.7

A particle moves in a straight line with positive velocity v(t), in km per min at time t

minutes, where v is a differentiable function of t. Values of v(t) 0 40t are shown in

the given table.

GRID

a) Use a left point Riemann sum with four subintervals of equal length and values

from the table to approximate 40

0

v t dt . Show your working. Using correct

units, explain the meaning of 40

0

v t dt in terms of the particle motion.

b) Using the values in the table, find the smallest number of instances at which the

acceleration of the particle could equal to zero on the (0, 40)? Justify your

answer.

c) 2sinv t t t t is used to model the velocity of a particle in km/m,

0 40t .

i) Calculate the distance covered in the given interval.

ii) Calculate the average velocity in the given interval.


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33. Consider the difference equation given by 2( 1)dy

x ydx

.

a) Complete the following table

Point

dy

dx

(-1, 0)

(0, 0)

(1, 0)

(2, 1)

(1, 1)

(0, 1)

(-1, 1)

(-2, 1)

b) Sketch the slope field at the 8 points given in the table.

c) Use the slope field of the given differential equation to explain why a solution

could not have the graph shown below.

d) Find the particular solution ( )y f x to the given differential equation with the

initial condition (0) 1f .

0

1

-1

-1 1 2 -2

y

x


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34. Let f be the function given by 2 1

3

xf x

x

a) Find c such that 0f c

b) Find 2f c) Draw the graph of f

d) If 2 2 ,y xy at 4t , 2y and 6dy

dt , find

dx

dt.

35. A pool has 1550 gallons of water at 0t . During the time 0 15t hours, water is

flowing into the pool at a rate of 2

1600 gallons/hr

1 sinI t

x

and during the same time,

water is removed from the pool at a rate of 212ln 2O t x gallons per hour.

a) How many gallons of water are flowing into the pool during the interval

0 12t ?

b) Is the water in the pool increasing or decreasing at 5t hours?

c) How many gallons of water are there at 6t ?


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Grade 12 S, NS Book ,Slope Field

36. Consider the differential equation 2

1 1 sin2

dy xy

dx

.

a) Sketch the slope field for the given differential equation at the nine points

indicated.

1

1

1

1

O

x

y

b) y = k, where k is a constant, satisfies the given differential equation, find the value

of k.

c) Determine the particular solution y h x to the differential equation given that

1 0h .

Grade 12 S, NS Book , Chapters 2 & 3

37. Given the velocity at time 0t of a particle moving along x-axis

is 16 5tan tv t e . At t = 0, the particle is at x = 1.

a) Find the acceleration of the particle at time t = 3.

b) Show that the speed of the particle is increasing at time t = 3.

c) Find all values of t at which the particle is at maximum position. Explain.

d) What is the position of the particle at time t = 3?

e) What is the direction of the particle with respect to the origin at t = 3 is it towards

or away from it?


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38.

t

(sec) 0 10 20 30 35 45 50

v(t)

(m/sec) 10 20 10 12 15 0 15

a(t)

(m/sec2)

1 6 4 1 4 5 3

v t and a t are continuous functions of t, 0 50t , represent the velocity and

acceleration of a particle.

a) Find an estimate for 35

10

v t dt using a trapezoidal method with four subintervals

from the given table and explain the meaning of your answer.

b) Calculate the exact value of 50

0

a t dt and explain the meaning of your answer.

c) For 0 50t , will there be a time t when 8v t ? What theorem guarantees

this?

d) For 0 50t , will there be a time t when 0a t ? What theorem justifies this?

Grade 12 S, NS Book, Slope Field

39. The velocity of a particle at time t is given by 22

2 sin3

tv t t

. The particle

moves along the x-axis and at t = 0 its position is x = 1.

a) Find the acceleration of the particle at t = 3. Does it have an increasing or

decreasing speed at t = 3? Justify.

b) At what time t for 0 < t < 4 does the particle change direction? Justify.

c) Find the total distance traveled for 0 4t .


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Level NS Calculus Chapters 2,7

40. Given a function f defined over the set of real numbers, such that (0) 5f

(0) 2, (0) 1f and f . f is a twice differentiable function.

a) Find (0)g , (0)g if ( ) 3 3 ( )kxg x f x .Give your answer in terms of the constant

k.

b) Given 4

( ) 1,5

h x f x mx m

i) Find ( )h x .

ii) Write the equation of the line normal to the graph of h when x = 0.


41. The position of a particle moving in the xy-plane is given by

sin 2 , 3x t t y t t where t

a) Find the range of x t and that of y t .

b) Find the smallest positive t such that xcoordinate of the particle is a local

maximum.

c) Find the speed of the particle at the time found in b.

d) Find the distance traveled by the particle from t to t . Compare the

answer with 5 .

e) Sketch the graph of the path of the particle indicating clearly the direction of

motion along it.


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Level NS Calculus Chapters 3

42.

The height f of the ramp in a toy is a function of x. Given that:

(i) (0) 0 (0) 0f f

(ii) (4) 1 (4) 1f f

(iii) If 0 < x < 4 then f is increasing

a) Find, if possible, the value of the constant 0k ; so that 2( )f x k x satisfies the

condition in (ii).

b) Find if possible the value of the constant 0m so that 2

3( )16

xf x m x satisfies

the condition in (ii).

c) Use the answer of part b to show that ( )f x does not satisfy the condition (iii)

d) Given ( ) 0, 0mx

f x m nn

. Find n so that ( )f x satisfies condition (ii) then

show that ( )f x satisfies (i) and (iii).

0

Height (meters)

x

f(x)


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(A graphic calculator is needed)

43. a) Find the area of the region R enclosed by the graph of 3 1y x , the vertical

line x = 3, and the x-axis.

b) When R is revolved about the horizontal line y = 3, find the volume of the solid

generated.

c) When R is revolved about the vertical line x = 7, find the volume of the solid

generated.


44. A 17 foot ladder is sliding down a wall at a rate of 5 feet / sec. When the top of the ladder is 8 feet from the ground

a) How fast is the foot of the ladder sliding away from the wall?

b) Let be the angle the ladder form with the horizontal floor. Find d

dt


45. A tank holds 2500 gallons of chemical solution at the time t = 0. During the time

interval, 0 6t hours, water is pumped into the tank at a rate 15

1 3

tP t

t

and is

removed from the tank at a rate 4

2 5sin25

tR t

gallons per hour.

a) How many gallons will be pumped out of the tank during the 6 hours period?

b) W(t) is the volume of the solution in the tank at the time t. Write an expression

for W(t).

c) At what rate is the volume of the solution changing when t = 4.

d) When is the volume of the solution minimum? Find this minimum.


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Level NS Calculus Chapters 3,7

46. Given 3 2 5.dy

x ydx

a) Find the values of a, b and c if c xy a x b e is a solution of 3 2 5.dy

x ydx

b) Find 2

2

d y

dx.

c) ( )y f x is a particular solution to the differential equation with the initial

condition (0) 2f .Approximate (1)f using Eulers method with 0 0x and

step size 1

2 .

d) Let g(x) be another solution of the differential equation with g(0) = m, where m is

a constant.

Find m if Eulers method gives the approximation g(1) = 0 starting at 0 0x with

step size = 1 .


47. A container has the shape of a cylinder with radius r = 5 cm. the height of water in the

container is h at a given time t (seconds).

The volume of water is decreasing at a rate 3 h cm3/ s.

a) Find k ifdh

kdt

.

b) Given that the height of water initially is 10 cm. solvedh

kdt

.

c) When the container becomes empty?


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48. v(t) = 2ln ( 5 7)t t is the velocity of a particle moving along the xaxis (0 5)t :

when t = 0 ; x = 5.

a) i) Find t; 0 5t at which the particle changes direction.

ii) Find the time intervals when the particle moves to the left.

b) Find the acceleration of the particle at t = 2.

c) Find the average speed of the particle over the interval 0 2t .

d) Find the position of the particle when t = 2.


49. Given f(x) a function defined by

1 2 5

( )7 5 8

x xf x

x x

a) Show that f is continuous at 5x .

b) Evaluate the average of ( )f x on 2,8

c) Given g(x) a function defined by1 2 5

( )7 5 8

a x for xg x

bx for x

.

Find the values of the constants a and b if g is differentiable at x = 5.


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50. Water flows out of a cylindrical pipe at a rate F t given in the adjacent table.

a) Use a right endpoint of Riemann sum with 5

subintervals of equal width to find

40

0

F t dt .

b) Give the meaning of the answer of part a in

terms of water flow, specify units.

c) Will there be a time t, 0,40t such that

0F t ? Why?

d) Suppose that the rate of flow

F t can be estimated by

21

536 2482

F t t t .

Use F t to estimate the average rate of

water flow during the period of 40 hours.

51. The rate at which cars enter a workshop on a given day is given by the function

2

123

25 180f t

t t

.

The rate at which cars leave the same workshop on the same day is given by the

function 2

97

36 360g t

t t

.

Both f t and g t are measured in cars per hour and time t is measured in hours

after midnight, these functions are valid for 8 22t , the hours the workshop is open.

At time 8t , there are no cars in the workshop.

a) How many cars, to the nearest whole number, have entered the workshop by

6.00 P.M.?

t

(hours) F t

(litres/hours )

0

4

8

12

16

20

24

28

32

36

40

7.8

12.5

12.7

15.2

18.2

18.5

12.2

12.8

7.3

10.9

7.8


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b) The cost of repairing the car is $20 until 6.00 P.M. After 6.00 P.M., the cost of

repairing the car is $25. Find, to the nearest whole number, the total amount of

money collected on a given day.

c) Let 8

t

K t f x g x dx for 8 22t . The value of 18K to the

nearest whole number is 24. Find the value of 18K and explain the

meaning of 18K and 18K in the context of the workshop.

52. Consider the differential equation 2

2dy y

dx x

where 0x .

a) Sketch the slope field for the given differential equation at the nine points

indicated.

b) Find the particular solution y f x to the differential equation with the

initial condition 3 1f

c) For the particular solution y f x described in part (b), find limx

f x

.

1

2

2 1 3

x

y


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53. The height, in feet, of a cylinder with constant radius is a twice differentiable function h

of time t where t is measured in minutes. For 0 14t , the graph of h is concave

down. The table gives selected values of the rate of change, h t , of the height of the

cylinder over time interval 0 14t

t (minutes) 0 3 5 9 11 14

h t 6.1 3.6 3.2 2.5 2.0 1.7

a) Use a right Riemann sum with five subintervals indicated by the data in the

table to approximate 14

0

h t dt . Using correct units, explain the meaning of

14

0

h t dt in terms of the height of the cylinder.

b) Is your approximation in part (a) greater or less than 14

0

h t dt ? Explain.

54. For a function f, 1

cosf x xx

, 2 5x

a) Find the values of x for which the graph has a stationary point

b) Find the values of x on which the graph is increasing

c) Find the values of x on which the graph is concaved up

d) Does the tangent at 3x lie above or below the graph of f x ?

55. A big rectangular reservoir of length 120 m, width 100 m and depth 50 m, contains

1,500 cubic meters of water at 0t .

Water is leaking from the reservoir at a rate r t cubic meters per hour,

t h 0 3 6 9 12

r t 0 120 150 180 200

During the same time interval, water is poured in at a rate of 0.337 tR t e m3/hr.


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a) Use the midpoint Riemann sum to approximate the amount of water that is

leaked during the 12 hrs.

b) Calculate the amount of water pumped in during the 12 hours.

c) What is the rate of increase of the water in the reservoir at 6t hrs and how

fast is the level of water increasing?

56.

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2

120

200

300

400

500

600

700

Time (hours)

r(t)

There are 50 people waiting to go into an open air concert. The doors are open for 2

hours only and people starts going in at a rate of 400 per hour. The graph shows the rate

at which people arrive to go in during the two hours.

a) How many people arrive between 0t and 0.75t

b) Is the number of people waiting to go in increasing between 0.75t and

1t ?

c) Write an equation involving an integral expression whose solution gives the

earliest time at which there is no more people waiting to go in.


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57. The function f is continuous and differentiable for 3 4x , and 0 3f . The graph

of f is given below.

1 2 3

1

2

1 2 3 4

1

2

(3, 2)

(4, 2)

x

y

a) Find 3f and 3f

b) Find the absolute maximum of f.

c) Find the critical points for the function f and classify each.

d) Find the average value of f x on 3 4x .

58.

x 3 3 1x 2 2 2x 2 2 3x 3

f x 11 Positive 9 Positive 3 Positive 8

f x 6 Negative 0 Negative 0 Positive 1

2

g x 2 Negative 0 Positive 4 Positive 2

g x 3 Positive 5

2 positive 0 negative 3

The twice-differentiable functions f and g are defined for all real numbers x. Values of f,

f , g and g for various values of x are given in the table above.

a) Find the x-coordinate of each relative minimum of f on the interval 3,3 . Justify your answers.

b) Explain why there must be a value c, for 2 2c , such that 0f c .


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c) The function h is defined by lnh x f x . Find 3h . Show the computations that lead to your answer.

d) Evaluate 3

3

f g x g x dx

ap calculus ab top 58 questions

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