Major Topics Problem Sets – Accumulation



A.P Calculus Review – 8: Derivatives

ALL NO CALCULATOR

1. Let f be a function that is continuous on the interval [0, 4). The function f is twice differentiable except at

x = 2. The function f and its derivatives have the properties indicated in the table below, where DNE indicates that the derivatives of f do not exist at x = 2.

a. For 0 < x < 4, find all values of x at which f has a relative extremum. Determine whether f has a relative maximum or a relative minimum at these values. Justify your answer.

b. On the axes below (left), sketch the graph of a function that has all the characteristics of f.

c. Let g be the function defined by [pic] on the open interval (0, 4). For 0 < x < 4, find all values of x at which g has a relative extremum. Determine whether g has a relative maximum or a relative minimum at each of these values. Justify your answer.

d. For the function g defined in part (c), find all values of x, 0 < x < 4, at which the graph of g has a point of inflection. Justify your answer.

2. The graph of the function f (above, right) consists of three line segments.

a. Let g be the function given by [pic]. For each of[pic], [pic] and [pic], find the value or state that it does not exist.

b. For the function g defined in part (a), find the x-coordinate of each point of inflection of the graph of g on the open interval –4 < x < 3. Explain your reasoning.

c. Let h be the function given by [pic]. Find all the values of x in the closed interval

–4 < x < 3 for which [pic] = 0.

d. For the function h defined in (c), find all intervals on which h is decreasing. Explain your reasoning.

(This assignment continued on the next page.)

3. The graph of the function f shown at right consists of a semicircle and three line segments. Let g be the function given by [pic].

a. Find [pic] and [pic].

b. Find all values of x in the open interval (–5, 4) at which g attains a relative maximum. Justify your answer.

c. Find the absolute minimum value of g on the closed interval [–5, 4]. Justify your answer.

d. Find all values of x in the open interval (–5, 4) at which the

graph of g has a point of inflection.

A.P Calculus Review – 9: Motion Along a Line

NO CALCULATOR

1. A car is traveling on a straight road. For 0 ≤ t ≤ 24 seconds, the car’s velocity v(t), in meters per second, is modeled by the piecewise-linear function defined by the graph below.

a. Find [pic]. Using correct units, explain the meaning of [pic].

b. For each of [pic] and[pic], find the value or explain why it does not exist. Indicate units of measure.

c. Let [pic] be the car’s acceleration at time t, in meters per second per second. For 0 < t < 24, write a piecewise-defined function for [pic].

d. Find the average rate of change of v over the interval 8 ≤ t ≤ 20. Does the Mean Value Theorem guarantee a value of c, for 8 < c < 20, such that v'(c) is equal to this average rate of change. Why or why not?

2. Two particles move along the x-axis. For 0 ( t ( 6, the position of particle P at time t is given by [pic]while the position of particle R at time t is given by r(t) = t3 – 6t2 + 9t + 3.

a. For 0 ( t ( 6, find all times t during which particle R is moving to the right.

b. For 0 ( t ( 6, find all times t during which the two particles travel in opposite directions.

c. Find the acceleration of particle P at time t = 3. Is particle P speeding up, slowing down, or doing neither at time t = 3 ? Explain your reasoning.

d. Write, but do not evaluate, an expression for the average distance between the two particles on the interval 1 ( t ( 3.

WITH CALCULATOR

3. A particle moves along the y-axis so that its velocity v at time t ( 0 is given by [pic].

At time t = 0, the particle is at y = –1.

a. Find the acceleration of the particle at time t = 2.

b. Is the speed of the particle increasing or decreasing at time t = 2? Give a reason for your answer.

c. Find the time t > 0 at which the particle reaches its highest point. Justify your answer.

d. Find the position of the particle at time t = 2. Is the particle moving toward the origin or away from the origin at time t = 2? Justify your answer.

A.P Calculus Review – 10: Area and Volume

NO CALCULATOR

1. Let R be the region in the first quadrant enclosed by the graphs of [pic] and [pic], as shown in the figure above.

a. Write an equation for the line tangent to the graph of f at [pic].

b. Find the area of R.

c. Write, but do not evaluate, an integral expression for the volume of the solid generated when R is rotated about the horizontal line y = (1.

WITH CALCULATOR

2. Let f and g be the functions given by [pic] and [pic]. Let R be the shaded region in the first quadrant enclosed by the graphs of f and g as shown in the figure

a. Find the area of R.

b. Find the volume of the solid generated when R is revolved about the

x-axis.

c. The region R is the base of a solid. For this solid, the cross sections perpendicular to the x-axis are semicircles with diameters extending from [pic] to[pic]. Find the volume of this solid.

3. Let R be the shaded region bounded by the graph of y = ln x

and the line y = x – 2, as shown.

a. Find the area of R.

b. Find the volume generated when R is rotated about the horizontal line y = –3.

c. Write, but do not evaluate, an integral expression that can be used to find the volume generated when R is rotated about the y-axis.

A.P Calculus Review – 11: Accumulation

ALL WITH CALCULATOR

1. A water tank at Camp Newton holds 1200 gallons of water at time t = 0. During the interval 0 ≤ t ≤ 18 hours, water is pumped into the tank at the rate

[pic] gallons per hour.

During the same interval, water is removed from the tank at the rate

[pic] gallons per hour.

a. Is the amount of water in the tank increasing at time t = 15? Why or why not?

b. To the nearest whole number, how many gallons of water are in the tank at time t = 18?

c. At what time t, 0 ≤ t ≤ 18, is the amount of water in the tank at an absolute minimum? Show the work that leads to your conclusion.

d. For t > 18, no water is pumped into the tank, but water continues to be removed at the rate R(t) until the tank becomes empty. Let k be the time at which the tank becomes empty. Write, but do not solve, an equation involving an integral expression that can be used to find the value of k.

2. Traffic flow is defined as the rate at which cars pass through an intersection, measured in cars per minute. The traffic flow at a particular intersection is modeled by the function F defined by

[pic] for 0 ( t ( 30,

where F(t) is measured in cars per minute and t is measured in minutes.

a. To the nearest whole number, how many cars pass through the intersection over the 30-minute period?

b. Is the traffic flow increasing or decreasing at t = 7? Give a reason for your answer.

c. What is the average value of the traffic flow over the time interval 10 ( t ( 15? Indicate units of measure.

3. There is no snow on Janet’s driveway when snow begins to fall at midnight. From midnight to 9 A.M., snow accumulates on the driveway at a rate modeled by [pic] cubic feet per hour, where t is measured in hours since midnight. Janet starts removing snow at 6 A.M. (t = 6). The rate [pic], in cubic feet per hour, at which Janet removes snow from the driveway at time t hours after midnight is modeled by

[pic]

a. How many cubic feet of snow have accumulated on the driveway by 6 A.M.?

b. Find the rate of change of the volume of snow on the driveway at 8 A.M.

c. Let [pic] represent the total amount of snow, in cubic feet, that Janet has removed from the driveway at time t hours after midnight. Express h as a piecewise-defined function with domain 0 ( t ( 9.

d. How many cubic feet of snow are on the driveway at 9 A.M.?

A.P Calculus Review – 12: Differential Equations and Slope Fields

ALL NO CALCULATOR

1. Consider the differential equation [pic].

a. On the axes at right, sketch a slope field for the given differential equation at the twelve points indicated.

b. Let [pic] be the particular solution to the differential equation with the initial condition[pic]. Write an equation for the line tangent to the graph of f at (1, –1) and use it to approximate [pic].

c. Find the particular solution [pic] to the differential equation with the initial condition [pic].

2. Consider the differential equation [pic].

a. On the axes provided, sketch a slope field for the given differential equation at the nine points indicated.

b. Find [pic] in terms of x and y. Describe the region in the xy-plane in which all solution curves to the differential equation are concave up.

c. Let [pic] be a particular solution to the differential equation with the initial condition [pic] = 1. Does f have a relative minimum, relative maximum or neither at x = 0? Justify your answer.

d. Find the values of the constants m and b for which y = mx + b is a solution of the differential equation.

3. At the beginning of 2010, a landfill contained 1400 tons of solid waste. The increasing function W models the total amount of solid waste stored at the landfill. Planners assume that W will satisfy the differential equation [pic] for the next 20 years. W is measured in tons, and t is measured in years from the start of 2010.

a. Use the tangent line to the graph of W at t = 0 to approximate the amount of solid waste that the landfill contains at the end of the first 3 months of 2010 (time [pic]).

b. Find [pic] in terms of W. Use [pic] to determine whether your answer in part (a) is an underestimate or an overestimate of the amount of solid waste that the landfill contains at time [pic].

c. Find the particular solution W = W(t) to the differential equation [pic] with initial condition W(0) = 1400.

Solutions

Derivatives

1a. f has a relative maximum at x = 2 because f ' changes from positive to negative at x = 2.

b.

2a. [pic].

g'(–1) = f(–1) = –2.

g"(–1) does not exist because f is not differentiable at x = –1.

b. x = 1. g' = f changes from increasing to decreasing at x = 1.

c. x = –1, 1, 3

d. h is decreasing on [0, 2]. h' = –f < 0 when f > 0.

3a. [pic]

g'(0) = f(0) = 1

b. g has a relative maximum at x = 3. This is the only x-value where g' = f changes from positive to negative.

c. The only x-value where f changes from negative to positive is x = –4. The other candidates for the location of the absolute minimum value are the endpoints.

g(–5) = 0

[pic]

[pic]

So the absolute minimum value of g is –1.

Motion Along a Line

1a. [pic]

The car travels 360 meters in these 24 seconds.

b. v'(4) does not exist because [pic]

[pic] m/s2

c. [pic] a(t) does not exist at t = 4 and t = 16.

d. The average rate of change of v on [8, 20] is [pic] m/s2.

No, the MVT does not apply to v on [8, 20] because v is not differentiable at t = 16.

2a. [pic] > 0 for 0 ≤ t < 1 and 3 < t ≤ 6.

b. [pic]= 0 for t = 0, t = 4. vp < 0 for 0 < t < 4 and vp > 0 for 4 < t < 6.

The particle travel in opposite directions on 0 < t < 1 and 3 < t < 4.

c. [pic]

Since vp(3) < 0 and ap(3) > 0, the particle is slowing down at t = 3.

d. [pic]

3a. a(2) = v'(2) = –0.132 or –0.133

b. v(2) = –0.436

Speed is increasing since a(2) < 0 and v(2) > 0.

c. v(t) = 0 when tan–1(et) = 1

t = ln(tan(1)) = 0.443 is the only critical value for y.

v(t) > 0 for 0 < t < ln(tan(1))

v(t) < 0 for t > ln(tan(1))

y(t) has an absolute maximum at t = 0.443.

d. [pic]

The particle is moving away from the origin since v(2) < 0 and y(2) < 0.

Area and Volume

1a. [pic]. [pic] so [pic]. Tangent line: [pic]

b. [pic]

c. [pic]

2. The graphs of f and g intersect in the first quadrant at (S, T) = (1.13569, 1.76446)

a. Area = [pic]

b. [pic]

c. Volume = [pic]

3a. ln(x) = x – 2 when x = 0.15859 = a and x = 3.14619 = b

Area = [pic]

b. Volume = [pic]

c. Volume = [pic]

Accumulation

1a. No, the amount of water is not increasing at t = 15 since [pic]= –121.09 < 0.

b. [pic]

1310 gallons.

c. [pic] ( t = 0 ( t = 6.4948 ( t = 12.9748 hours

The Absolute minimum occurs when t = 6.494 or 6.495.

d. [pic]

2a. [pic] cars

b. [pic]= –1.872 or –1.873

Since [pic] < 0, the traffic flow is decreasing at t = 7.

c. [pic] cars/min

d. [pic] cars/min2

3a. [pic] cubic feet

b. [pic] cubic feet per hour

c. [pic]

d. [pic] cubic feet

Differential Equations and Slope Fields

1a.

2a.

d. y = mx + b ( y' = m and the DE becomes: [pic] ( [pic].

This will be true for all x only if [pic] and [pic].

Alternate solution: If y = mx + b, then y" = 0 and [pic] so [pic] and [pic].

12. a. [pic]. The tangent line is [pic] or y = 1400 + 44x.

Then [pic] tons.

b. [pic]. Since [pic]on [pic], W is concave up and the tangent line estimate will underestimate the amount of solid waste at time t = [pic].

c. [pic] ( [pic]( [pic]([pic]([pic]

W(0) = 1400 ( A = 1100 so [pic]

-----------------------

|x |0 |

|0 |1200 |

|6.495 |525 |

|12.975 |1697 |

|18 |1310 |

b. The line tangent to f at (1, –1) is y + 1 = 2(x – 1).

So, [pic].

c. [pic]

ydy = –2xdx

[pic]

[pic]

y2 = –2x2 + 3

Since the solution passes through (1, –1), y must be negative.

So [pic]

y

x

O

–1

1

–1

2

1

–2

2

-1 O 1

x

y

1

b. [pic].

The solutions are concave up when y" > 0 ( [pic]

c. [pic] and [pic] so f has a relative minimum

at x = 0.

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