Unit 8 Overview (College Board AP® Calculus BC): Exam Questions

The density of a bacteria population in a circular petri dish at a distance centimeters from the center of the dish is given by an increasing, differentiable function , where is measured in milligrams per square centimeter. Values of for selected values of are given in the table below.

Use the data in the table to estimate . Using correct units, interpret the meaning of your answer in the context of this problem.

The total mass, in milligrams, of bacteria in the petri dish is given by the integral expression . Approximate the value of using a right Riemann sum with the four subintervals indicated by the data in the table.

Is the approximation found in part (b) an overestimate or underestimate of the total mass of bacteria in the petri dish? Explain your reasoning.

The density of bacteria in the petri dish, for , is modelled by the function defined by . For what value of , , is equal to the average value of on the interval ?

The temperature of coffee in a cup at time minutes is modeled by a decreasing differentiable function , where is measured in degrees Celsius. For , selected values of are given in the table below.

Approximate using the average rate of change of over the interval . Show the work that leads to your answer and include units of measure.

Use a left Riemann sum with the three subintervals indicated by the data in the table to approximate the value of . Interpret the meaning of in the context of the problem.

For , the rate of change of the temperature of the coffee is modeled by , where is measured in degrees Celsius per minute. Find the temperature of the coffee at time . Show the setup for your calculations.

For the model defined in part (c), it can be shown that . For , determine whether the temperature of the coffee is changing at a decreasing rate or at an increasing rate. Give a reason for your answer.

The functions and are defined by and , as shown in the graph.

Let be the region bounded by the graphs of and , from to , as shown in the graph. Write, but do not evaluate, an integral expression that gives the area of region .

Let be the region bounded by the graph of and the -axis, from to , as shown in the graph. Region is the base of a solid. For this solid, at each the cross section perpendicular to the -axis is a rectangle with height equal to half its base in region . Find the volume of the solid. Show the work that leads to your answer.

Write, but do not evaluate, an integral expression that gives the volume of the solid generated when region , as described in part (b), is rotated about the horizontal line .

From 5 A.M. to 10 A.M., the rate at which vehicles arrive at a certain toll plaza is given by , where is the number of hours after 5 A.M. and is measured in vehicles per hour. Traffic is flowing smoothly at 5 A.M. with no vehicles waiting in line.

Write, but do not evaluate, an integral expression that gives the total number of vehicles that arrive at the toll plaza from 6 A.M. () to 10 A.M. ().

Find the average value of the rate, in vehicles per hour, at which vehicles arrive at the toll plaza from 6 A.M. () to 10 A.M. ().

Is the rate at which vehicles arrive at the toll plaza at 6 A.M. () increasing or decreasing? Give a reason for your answer.

A line forms whenever . The number of vehicles in line at time , for , is given by , where is the time when a line first begins to form. To the nearest whole number, find the greatest number of vehicles in line at the toll plaza in the time interval . Justify your answer.

People enter a line for an escalator at a rate modeled by the function given by

where is measured in people per second and is measured in seconds. As people get on the escalator, they exit the line at a constant rate of 0.7 person per second. There are 20 people in line at time .

(Note: Your calculator should be in radian mode.)

How many people enter the line for the escalator during the time interval ?

During the time interval , there are always people in line for the escalator. How many people are in line at time ?

For , what is the first time that there are no people in line for the escalator?

For , at what time is the number of people in line a minimum? To the nearest whole number, find the number of people in line at this time. Justify your answer.

An invasive species of plant appears in a fruit grove at time and begins to spread. The function defined by models the number of acres in the fruit grove affected by the species weeks after the species appears. It can be shown that .

(Note: Your calculator should be in radian mode.)

Find the average number of acres affected by the invasive species from time to time weeks. Show the setup for your calculations.

Find the time when the instantaneous rate of change of equals the average rate of change of over the time interval . Show the setup for your calculations.

Assume that the invasive species continues to spread according to the given model for all times . Write a limit expression that describes the end behavior of the rate of change in the number of acres affected by the species. Evaluate this limit expression.

At time weeks after the invasive species appears in the fruit grove, measures are taken to counter the spread of the species. The function , defined by , models the number of acres affected by the species over the time interval . At what time , for , does attain its maximum value? Justify your answer.

Fish enter a lake at a rate modeled by the function given by . Fish leave the lake at a rate modeled by the function given by . Both and are measured in fish per hour, and is measured in hours since midnight ().

(Note: Your calculator should be in radian mode.)

How many fish enter the lake over the 5-hour period from midnight () to 5 A.M. ()? Give your answer to the nearest whole number.

What is the average number of fish that leave the lake per hour over the 5-hour period from midnight () to 5 A.M. ()?

At what time , for , is the greatest number of fish in the lake? Justify your answer.

Is the rate of change in the number of fish in the lake increasing or decreasing at 5 A.M. ()? Explain your reasoning.

A customer at a gas station is pumping gasoline into a gas tank. The rate of flow of gasoline is modeled by a differentiable function , where is measured in gallons per second and is measured in seconds since pumping began. Selected values of are given in the table.

Using correct units, interpret the meaning of in the context of the problem. Use a right Riemann sum with the three subintervals , , and to approximate the value of .

Must there exist a value of , for , such that ? Justify your answer.

The rate of flow of gasoline, in gallons per second, can also be modeled by for . Using this model, find the average rate of flow of gasoline over the time interval . Show the setup for your calculations.

Using the model defined in part (c), find the value of . Interpret the meaning of your answer in the context of the problem.

Water is pumped into a tank at a rate modeled by liters per hour for , where is measured in hours. Water is removed from the tank at a rate modeled by liters per hour, where is differentiable and decreasing on . Selected values of are shown in the table below. At time , there are 50 000 liters of water in the tank.

Estimate . Show the work that leads to your answer. Indicate units of measure.

Use a left Riemann sum with the four subintervals indicated by the table to estimate the total amount of water removed from the tank during the 8 hours. Is this an overestimate or an underestimate of the total amount of water removed? Give a reason for your answer.

Use your answer from part (b) to find an estimate of the total amount of water in the tank, to the nearest liter, at the end of 8 hours.

For , is there a time when the rate at which water is pumped into the tank is the same as the rate at which water is removed from the tank? Explain why or why not.

The shaded region is bounded by the graphs of the functions and , where and , as shown in the figure.

(Note: Your calculator should be in radian mode.)

Find the area of . Show the setup for your calculations.

Region is the base of a solid. For this solid, at each the cross section perpendicular to the -axis is a rectangle with height and base in region . Find the volume of the solid. Show the setup for your calculations.

Write, but do not evaluate, an integral expression for the volume of the solid generated when the region is rotated about the horizontal line .

It can be shown that . Find the value of , for , at which the line tangent to the graph of is parallel to the line tangent to the graph of .

A tank has a height of 10 feet. The area of the horizontal cross section of the tank at height feet is given by the function , where is measured in square feet. The function is continuous and decreases as increases. Selected values for are given in the table below.

Use a left Riemann sum with the three subintervals indicated by the data in the table to approximate the volume of the tank. Indicate units of measure.

Does the approximation in part (a) overestimate or underestimate the volume of the tank? Explain your reasoning.

The area, in square feet, of the horizontal cross section at height feet is modeled by the function given by .

Based on this model, find the volume of the tank. Indicate units of measure.

Water is pumped into the tank. When the height of the water is 5 feet, the height is increasing at the rate of foot per minute. Using the model from part (c), find the rate at which the volume of water is changing with respect to time when the height of the water is 5 feet. Indicate units of measure.

A particle moves along the -axis so that its velocity at time is given by

There is one time, , in the interval when the particle is at rest (not moving). Find . For , is the particle moving to the right or to the left? Give a reason for your answer.

Find the acceleration of the particle at time . Show the setup for your calculations. Is the speed of the particle increasing or decreasing at time ? Explain your reasoning.

The position of the particle at time is , and its position at time is . Find the position of the particle at time . Show the setup for your calculations.

Find the total distance traveled by the particle over the interval . Show the setup for your calculations.

When a certain grocery store opens, it has 50 pounds of bananas on a display table. Customers remove bananas from the display table at a rate modeled by for , where is measured in pounds per hour and is the number of hours after the store opened. After the store has been open for three hours, store employees add bananas to the display table at a rate modeled by for , where is measured in pounds per hour and is the number of hours after the store opened.

How many pounds of bananas are removed from the display table during the first 2 hours the store is open?

Find . Using correct units, explain the meaning of in the context of the problem.

Is the number of pounds of bananas on the display table increasing or decreasing at time ? Give a reason for your answer.

How many pounds of bananas are on the display table at time ?

Let and be the functions defined by and . The graphs of and , shown in the figure above, intersect at and , where .

Find the area of the region enclosed by the graphs of and .

For , let be the vertical distance between the graphs of and . Is increasing or decreasing at ? Give a reason for your answer.

The region enclosed by the graphs of and is the base of a solid. Cross sections of the solid taken perpendicular to the -axis are squares. Find the volume of the solid.

A vertical line in the -plane travels from left to right along the base of the solid described in part (c). The vertical line is moving at a constant rate of 7 units per second. Find the rate of change of the area of the cross section above the vertical line with respect to time when the vertical line is at position .

For , a particle moves along the -axis. The velocity of the particle at time is given by

The particle is at position at time .

At time , is the particle speeding up or slowing down? Give a reason for your answer.

Find all times in the interval when the particle changes direction. Justify your answer.

Find the position of the particle at time .

Find the total distance the particle travels from time to time .

A particle, , is moving along the -axis. The velocity of particle at time is given by for . At time , particle is at position . A second particle, , also moves along the -axis. The velocity of particle at time is given by for . At time , particle is at position .

Find the positions of particles and at time .

Are particles and moving toward each other or away from each other at time ? Explain your reasoning.

Find the acceleration of particle at time . Is the speed of particle increasing or decreasing at time ? Explain your reasoning.

Find the total distance traveled by particle over the time interval .

A particle moves along the -axis with velocity given by for time . The particle is at position at time .

Find the acceleration of the particle at time .

Find the position of the particle at time .

Evaluate and evaluate . Interpret the meaning of each integral in the context of the problem.

A second particle moves along the -axis with position given by for . At what time are the two particles moving with the same velocity?

Stephen swims back and forth along a straight path in a 50-meter-long pool for 90 seconds. Stephen's velocity is modeled by , where is measured in seconds and is measured in meters per second.

Find all times in the interval at which Stephen changes direction. Give a reason for your answer.

Find Stephen's acceleration at time seconds. Show the setup for your calculations, and indicate units of measure. Is Stephen speeding up or slowing down at time seconds? Give a reason for your answer.

Find the distance between Stephen's position at time seconds and his position at time seconds. Show the setup for your calculations.

Find the total distance Stephen swims over the time interval seconds. Show the setup for your calculations.

The velocity of a particle, P, moving along the -axis is given by the differentiable function , where is measured in meters per hour and is measured in hours. Selected values of are shown in the table below.

Particle P is at the origin at time .

Justify why there must be at least one time , for , at which , the acceleration of particle P, equals 0 meters per hour per hour.

Use a trapezoidal sum with the three subintervals , , and to approximate the value of .

A second particle, Q, also moves along the x-axis so that its velocity for is given by meters per hour. Find the time interval during which the velocity of particle Q is at least 60 meters per hour. Find the distance traveled by particle Q during the interval when the velocity of particle Q is at least 60 meters per hour.

At time , particle Q is at position . Using the result from part (b) and the function from part (c), approximate the distance between particles P and Q at time .

A student starts reading a book at time minutes and continues reading for the next 10 minutes. The rate at which the student reads is modeled by the differentiable function , where is measured in words per minute. Selected values of are given in the table shown.

Approximate using the average rate of change of over the interval . Show the work that leads to your answer. Indicate units of measure.

Must there be a value , for , such that ? Justify your answer.

Use a trapezoidal sum with the three subintervals indicated by the data in the table to approximate the value of . Show the work that leads to your answer.

A teacher also starts reading at time minutes and continues reading for the next 10 minutes. The rate at which the teacher reads is modeled by the function defined by , where is measured in words per minute. Based on the model, how many words has the teacher read by the end of the 10 minutes? Show the work that leads to your answer.

Figures 1 and 2 illustrate regions in the first quadrant associated with the graphs of and , respectively. In Figure 1, let be the region bounded by the graph of , the -axis, and the vertical lines and . In Figure 2, let be the unbounded region between the graph of and the -axis that lies to the right of the vertical line .

Find the area of region .

Region is the base of a solid. For the solid, at each the cross section perpendicular to the -axis is a rectangle with area given by . Find the volume of the solid.

Find the volume of the solid generated when the unbounded region is revolved about the -axis.

The height of a tree at time is given by a twice-differentiable function , where is measured in meters and is measured in years. Selected values of are given in the table below.

Use the data in the table to estimate . Using correct units, interpret the meaning of in the context of the problem.

Explain why there must be at least one time , for , such that .

Use a trapezoidal sum with the four subintervals indicated by the data in the table to approximate the average height of the tree over the time interval .

The height of the tree, in meters, can also be modeled by the function , given by , where is the diameter of the base of the tree, in meters. When the tree is 50 meters tall, the diameter of the base of the tree is increasing at a rate of 0.03 meter per year. According to this model, what is the rate of change of the height of the tree with respect to time, in meters per year, at the time when the tree is 50 meters tall?

The function is twice differentiable for all with . Values of , the derivative of , are given in the table for selected values of .

For , the function is defined by . Find the value of . Show the work that leads to your answer.

What information does provide about the graph of ?

Use Euler's method, starting at with two steps of equal size, to approximate . Show the computations that lead to your answer.

Find . Show the work that leads to your answer.

Particle moves along the -axis such that, for time , its position is given by . Particle moves along the -axis such that, for time , its velocity is given by . At time , the position of particle is .

Find , the velocity of particle at time .

Find , the acceleration of particle at time . Find all times , for , when the speed of particle is decreasing. Justify your answer.

Find , the position of particle at time .

As , which particle will eventually be farther from the origin? Give a reason for your answer.

The graphs of the functions and are shown in the figure for . It is known that for . The twice-differentiable function , which is not explicitly given, satisfies and .

Find the area of the shaded region enclosed by the graphs of and .

Evaluate the improper integral , or show that the integral diverges.

Let be the function defined by . Find the value of .

Two particles move along the -axis. For , the position of particle at time is given by , while the velocity of particle at time is given by . Particle is at position at time .

For , when is particle moving to the left?

For , find all times during which the two particles travel in the same direction.

Find the acceleration of particle at time . Is the speed of particle increasing, decreasing, or neither at time ? Explain your reasoning.

Find the position of particle the first time it changes direction.

A company designs spinning toys using the family of functions , where is a positive constant. The figure above shows the region in the first quadrant bounded by the -axis and the graph of , for some . Each spinning toy is in the shape of the solid generated when such a region is revolved about the -axis. Both and are measured in inches.

Find the area of the region in the first quadrant bounded by the -axis and the graph of for .

It is known that, for , . For a particular spinning toy, the radius of the largest cross-sectional circular slice is 1.2 inches. What is the value of for this spinning toy?

For another spinning toy, the volume is cubic inches. What is the value of for this spinning toy?

The inside of a funnel of height 10 inches has circular cross sections, as shown in the figure above. At height , the radius of the funnel is given by

where . The units of and are inches.

Find the average value of the radius of the funnel.

Find the volume of the funnel.

The funnel contains liquid that is draining from the bottom. At the instant when the height of the liquid is inches, the radius of the surface of the liquid is decreasing at a rate of inch per second. At this instant, what is the rate of change of the height of the liquid with respect to time?

Let be the region enclosed by the graphs of and , the -axis, and the vertical line , as shown in the figure below.

Find the area of .

Region is the base of a solid. For the solid, at each the cross section perpendicular to the -axis has area . Find the volume of the solid.

Write, but do not evaluate, an integral expression that gives the volume of the solid generated when is rotated about the horizontal line .