The tire pressure can be modeled by the following equation: P ( t ) = 35 ( 2 ) − t 20 P\left(t\right)=35\left(2\right)^{-\frac{t}{20}} P ( t ) = 3 5 ( 2 ) − 2 0 t where you can find an estimate of P ′ ( t ) = − 1.5 ( 2 ) − t 20 P'\left(t\right)=-1.5\left(2\right)^{-\frac{t}{20}} P ′ ( t ) = − 1 . 5 ( 2 ) − 2 0 t Write the equation of the local linearization of P(t) at t = 20

y = − 0.75 ( x − 20 ) + 17.5 y=-0.75\left(x-20\right)+17.5 y = − 0 . 7 5 ( x − 2 0 ) + 1 7 . 5

y = − 0.75 ( x − 20 ) − 17.5 y=-0.75\left(x-20\right)-17.5 y = − 0 . 7 5 ( x − 2 0 ) − 1 7 . 5

The population of a town x years after 2000 is modeled by P(x). What is the best interpretation of P'(10) = 50.

In 2010, the population was increasing 50 people per year.

In 2010, the population was 50 people.

It took 10 years for the population to increase by 50 people.

The average change in population over the 10 years was 50 people per year.

For t ≥ 0 hours, D is a differentiable function of t that gives the depth, in meters, of the water at the end of a pier. Which of the following is the best interpretation of f'\left(12\right) D'(12)?

The rate of change in the depth at the end of the 12th hour.

The change in the depth of the water during the first 12 hours.

The change in the depth during the 12th hour.

The average rate of change in the depth during the 12th hour.

AP Calc Unit 4 Review

Authored by Julie Lowman

Mathematics

11th Grade - University

CCSS covered

Used 108+ times

AI Actions

Add similar questions

Adjust reading levels

Convert to real-world scenario

Translate activity

More...

About

This quiz covers related rates problems and applications of derivatives, which are fundamental topics in Advanced Placement Calculus AB and BC. The material is appropriate for grade 11-12 students who have mastered basic differentiation techniques and are ready to apply calculus to real-world scenarios. Students must demonstrate proficiency in implicit differentiation, understanding of rate relationships between variables, and the ability to set up and solve complex word problems involving geometric shapes and physical situations. The core concepts include the chain rule, geometric relationships (Pythagorean theorem, similar triangles, volume and surface area formulas), and the interpretation of derivatives as rates of change. Students need strong algebraic manipulation skills, spatial reasoning to visualize three-dimensional problems, and the ability to identify which variables are changing with respect to time in multi-step problems involving ladders, expanding circles, deflating spheres, and draining containers. Created by Julie Lowman, a Mathematics teacher in US who teaches grade 11-University. This comprehensive review quiz serves multiple instructional purposes, functioning effectively as a unit assessment, homework assignment, or intensive review session before the AP Calculus exam. The problems progress from classic related rates scenarios to more sophisticated applications involving exponential functions and linear approximations, allowing students to build confidence while tackling increasingly complex material. Teachers can use this quiz for formative assessment to identify areas where students need additional support, particularly in setting up differential equations and interpreting derivatives in context. The variety of problem types makes it ideal for differentiated instruction, as students can work through problems at their own pace while reinforcing essential calculus concepts. This assessment aligns with College Board AP Calculus standards, specifically addressing topics in differential calculus applications, rates of change, and mathematical modeling that are central to the AP curriculum framework.

Content View

Student View

18 questions

Show all answers

MULTIPLE CHOICE QUESTION

1 min • 1 pt

Rachel is standing atop a 13 ft ladder. The ladder is leaning against a vertical wall. The ladder starts sliding away from the wall at a rate of 3 ft/sec. How fast is the ladder sliding down the wall when the tip of the ladder is 5 ft high?

3 ft/sec

-7.2 ft/sec

7.2 ft/sec

Tags

CCSS.8.G.B.8

MULTIPLE CHOICE QUESTION

15 mins • 1 pt

Rachel is standing atop a 13 ft ladder. The ladder is leaning against a vertical wall. The ladder starts sliding away from the wall at a rate of 3 ft/sec. How fast is the angle between the tip of the ladder and the house changing when the ladder is 5 ft high? Hint: Use a trig function.

1 deg/sec

-.5 deg/sec

.6 deg/sec

The angle is not changing.

Tags

CCSS.HSG.SRT.C.6

CCSS.HSG.SRT.C.8

MULTIPLE CHOICE QUESTION

15 mins • 1 pt

Rachel is standing atop a 13ft ladder. The ladder is leaning against a vertical wall. The ladder starts sliding away from the wall at a rate of 3ft/sec. How fast is the angle changing between the base of the ladder and the ground when the ladder is 5 ft high?

-0.6 rad/sec

1.5 rad/sec

.5 rad/sec

.65 rad/sec

Tags

CCSS.HSG.SRT.C.6

CCSS.HSG.SRT.C.8

CCSS.HSF.TF.A.1

MULTIPLE CHOICE QUESTION

15 mins • 1 pt

Chris is sitting on the edge of a dock tossing rocks into the water. As each rock hits the water, small circles appear traveling outward from the point of impact. The radius of the circle is changing at a rate of 5 in/sec. How fast is the area of the outer circle changing when the diameter is 8 in?

80pi in/sec

20pi in/sec

60pi in/sec

4opi in/sec

Tags

CCSS.HSG.GMD.A.1

CCSS.HSA.SSE.A.1

MULTIPLE CHOICE QUESTION

15 mins • 1 pt

Brandon is starting to clean up after a birthday party. He begins deflating each spherical balloon by puncturing a hole in each. The air leaves the balloon at a constant rate of 2 cm³/sec. How fast is the diameter changing when the diameter is 8 cm?

-1/(16pi) cm/sec

-1/16 cm/sec

-1/(4pi) cm/sec

1/(4pi) cm/sec

Tags

CCSS.HSG.GMD.A.3

MULTIPLE CHOICE QUESTION

15 mins • 1 pt

A water tank, shaped like an inverted circular cone, has a base radius of 6 ft and a height of 9 ft. The tank is completely full and needs to be drained. The valve is opened and the water begins to decrease at a rate of 2 ft³/sec. How fast is the height of the water changing when the water is 2 ft deep?

-9/(8pi) ft/sec

9/(8pi) ft/sec

-8/(9pi) ft/sec

8/(9pi) f/tsec

Tags

CCSS.HSG.GMD.A.3

CCSS.HSG.GMD.A.1

CCSS.HSG.MG.A.1

MULTIPLE CHOICE QUESTION

15 mins • 1 pt

A water tank, shaped like an inverted circular cone, has a base radius of 6 ft and a height of 9 ft. The tank is completely full and needs to be drained. The valve is opened and the water begins to decrease at a rate of 2 ft³/sec. How fast is the radius changing when the water is 2 ft deep?

4/(3pi) ft/sec

3/(4pi)ft/sec

-3/(4pi) ft/sec

-4/(3pi) ft/sec

Tags

CCSS.HSG.GMD.A.3

CCSS.HSA.CED.A.2

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

The tire pressure can be modeled by the following equation: $P\left(t\right)=35\left(2\right)^{-\frac{t}{20}}$
where you can find an estimate of $P'\left(t\right)=-1.5\left(2\right)^{-\frac{t}{20}}$
Write the equation of the local linearization of P(t) at t = 20

$y=-0.75\left(x-20\right)+17.5$

$y=-0.75\left(x-20\right)-17.5$

Tags

CCSS.HSA.CED.A.2

CCSS.HSF.LE.B.5

CCSS.HSF.BF.A.1

CCSS.HSF.IF.C.8

CCSS.HSF.LE.A.2

Access all questions and much more by creating a free account

Create resources

Host any resource

Get auto-graded reports

Continue with Google

Continue with Email

Continue with Classlink

Continue with Clever

or continue with

Microsoft

Apple

Others

Already have an account?

Similar Resources on Wayground

14 questions

Multiplying and Dividing Fractions

Quiz

•

10th - 12th Grade

20 questions

Central Limit Theorem for Sums

Quiz

•

11th - 12th Grade

17 questions

graphing inequalities

Quiz

•

8th - 12th Grade

19 questions

Budgeting Terms & Vocabulary

Quiz

•

12th Grade

20 questions

Algebra 2

Quiz

•

11th Grade

20 questions

Benchmark 2 Review

Quiz

•

9th - 12th Grade

15 questions

Limit Theorems

Quiz

•

11th Grade

20 questions

Resultant Vector

Quiz

•

University

Popular Resources on Wayground

15 questions

Fractions on a Number Line

Quiz

•

3rd Grade

20 questions

Equivalent Fractions

Quiz

•

3rd Grade

25 questions

Multiplication Facts

Quiz

•

5th Grade

29 questions

Alg. 1 Section 5.1 Coordinate Plane

Quiz

•

9th Grade

$fractions$

22 questions

fractions

Quiz

•

3rd Grade

11 questions

FOREST Effective communication

Lesson

•

20 questions

Main Idea and Details

Quiz

•

5th Grade

20 questions

Context Clues

Quiz

•

6th Grade

Discover more resources for Mathematics

14 questions

Transformations of Quadratic Functions

Quiz

•

KG - University

AP Calc Unit 4 Review

Rachel is standing atop a 13 ft ladder. The ladder is leaning against a vertical wall. The ladder starts sliding away from the wall at a rate of 3 ft/sec. How fast is the ladder sliding down the wall when the tip of the ladder is 5 ft high?

Rachel is standing atop a 13 ft ladder. The ladder is leaning against a vertical wall. The ladder starts sliding away from the wall at a rate of 3 ft/sec. How fast is the angle between the tip of the ladder and the house changing when the ladder is 5 ft high? Hint: Use a trig function.

Rachel is standing atop a 13ft ladder. The ladder is leaning against a vertical wall. The ladder starts sliding away from the wall at a rate of 3ft/sec. How fast is the angle changing between the base of the ladder and the ground when the ladder is 5 ft high?

Chris is sitting on the edge of a dock tossing rocks into the water. As each rock hits the water, small circles appear traveling outward from the point of impact. The radius of the circle is changing at a rate of 5 in/sec. How fast is the area of the outer circle changing when the diameter is 8 in?

Brandon is starting to clean up after a birthday party. He begins deflating each spherical balloon by puncturing a hole in each. The air leaves the balloon at a constant rate of 2 cm³/sec. How fast is the diameter changing when the diameter is 8 cm?

A water tank, shaped like an inverted circular cone, has a base radius of 6 ft and a height of 9 ft. The tank is completely full and needs to be drained. The valve is opened and the water begins to decrease at a rate of 2 ft³/sec. How fast is the radius changing when the water is 2 ft deep?

(3 min) The radius r of a sphere is increasing at the uniform rate of 0.3 inches per second. At the instant when the surface area S becomes 100 π square inches, what is the rate of increase, in cubic inches per second, in the volume V? SA = 4 π r² and V = (4/3) π r³

The tire pressure can be modeled by the following equation: $P\left(t\right)=35\left(2\right)^{-\frac{t}{20}}$
where you can find an estimate of $P'\left(t\right)=-1.5\left(2\right)^{-\frac{t}{20}}$
Write the equation of the local linearization of P(t) at t = 20

Given that f (-3) = 4 and f ' (-3) = 2, which of the following is the tangent line approximation of f (-3.1)?

Access all questions and much more by creating a free account

Similar Resources on Wayground

Popular Resources on Wayground

Discover more resources for Mathematics

AP Calc Unit 4 Review

Rachel is standing atop a 13 ft ladder. The ladder is leaning against a vertical wall. The ladder starts sliding away from the wall at a rate of 3 ft/sec. How fast is the ladder sliding down the wall when the tip of the ladder is 5 ft high?

Rachel is standing atop a 13 ft ladder. The ladder is leaning against a vertical wall. The ladder starts sliding away from the wall at a rate of 3 ft/sec. How fast is the angle between the tip of the ladder and the house changing when the ladder is 5 ft high? Hint: Use a trig function.

Rachel is standing atop a 13ft ladder. The ladder is leaning against a vertical wall. The ladder starts sliding away from the wall at a rate of 3ft/sec. How fast is the angle changing between the base of the ladder and the ground when the ladder is 5 ft high?

Chris is sitting on the edge of a dock tossing rocks into the water. As each rock hits the water, small circles appear traveling outward from the point of impact. The radius of the circle is changing at a rate of 5 in/sec. How fast is the area of the outer circle changing when the diameter is 8 in?

Brandon is starting to clean up after a birthday party. He begins deflating each spherical balloon by puncturing a hole in each. The air leaves the balloon at a constant rate of 2 cm3/sec. How fast is the diameter changing when the diameter is 8 cm?

A water tank, shaped like an inverted circular cone, has a base radius of 6 ft and a height of 9 ft. The tank is completely full and needs to be drained. The valve is opened and the water begins to decrease at a rate of 2 ft3/sec. How fast is the radius changing when the water is 2 ft deep?

(3 min) The radius r of a sphere is increasing at the uniform rate of 0.3 inches per second. At the instant when the surface area S becomes 100 π square inches, what is the rate of increase, in cubic inches per second, in the volume V? SA = 4 π r2 and V = (4/3) π r3

Given that f (-3) = 4 and f ' (-3) = 2, which of the following is the tangent line approximation of f (-3.1)?

Access all questions and much more by creating a free account

Similar Resources on Wayground

Popular Resources on Wayground

Discover more resources for Mathematics

Brandon is starting to clean up after a birthday party. He begins deflating each spherical balloon by puncturing a hole in each. The air leaves the balloon at a constant rate of 2 cm³/sec. How fast is the diameter changing when the diameter is 8 cm?

A water tank, shaped like an inverted circular cone, has a base radius of 6 ft and a height of 9 ft. The tank is completely full and needs to be drained. The valve is opened and the water begins to decrease at a rate of 2 ft³/sec. How fast is the radius changing when the water is 2 ft deep?

(3 min) The radius r of a sphere is increasing at the uniform rate of 0.3 inches per second. At the instant when the surface area S becomes 100 π square inches, what is the rate of increase, in cubic inches per second, in the volume V? SA = 4 π r² and V = (4/3) π r³