AP Calculus BC Summer Assignment

Introduction

Welcome to AP Calculus BC! This summer assignment is designed to help you refresh and strengthen your understanding of key concepts from Pre-Calculus and AP Calculus AB that will be essential for success in the BC course. A solid foundation in these areas will allow us to delve more deeply into the exciting new topics that Calculus BC offers.

Objectives

After completing this summer assignment, you should be able to:

• Evaluate limits algebraically and conceptually
• Understand and apply derivative rules efficiently
• Interpret and apply the Mean Value Theorem
• Use definite integrals to solve problems involving area and accumulation
• Apply integration techniques including substitution and integration by parts
• Analyze properties of functions, including concavity and points of inflection
• Connect differential and integral calculus through the Fundamental Theorem of Calculus

Part I: Functions and Graphs

1. Algebraic Functions

Review the properties and graphs of polynomial, rational, power, and piecewise functions. Be able to:

• Find domain and range
• Identify intercepts and asymptotes
• Sketch graphs based on function behavior
• Perform operations and compositions of functions
• Find inverse functions when they exist

2. Exponential and Logarithmic Functions

Master the properties of exponential and logarithmic functions:

• Properties of exponents
• Properties of logarithms
• The natural exponential function e^x and natural logarithm ln(x)
• Solving exponential and logarithmic equations
• Applications such as growth and decay models

3. Trigonometric Functions

Develop complete understanding of trigonometric functions:

• Unit circle and special angles
• Graphs of sine, cosine, tangent and their transformations
• Reciprocal and inverse trigonometric functions
• Trigonometric identities
• Solving trigonometric equations

Part II: Limits and Continuity

1. Evaluating Limits

Be proficient in evaluating limits using algebraic techniques, including:

• Direct substitution
• Factoring and cancellation
• Rationalization
• Using special limits (e.g., lim(x0) sin(x)/x = 1)
• One-sided limits

2. Continuity

Understand the concept of continuity:

• Definition of continuity at a point
• Types of discontinuities (removable, jump, infinite)
• Intermediate Value Theorem
• Maximum and Minimum Value Theorems

Part III: Differentiation

1. Basic Derivative Rules

Be fluent in applying derivative rules:

• Power Rule: d/dx[xⁿ] = nx^n-1
• Product Rule: d/dx[fg] = f'g + fg'
• Quotient Rule: d/dx[f/g] = (f'g - fg')/g²
• Chain Rule: d/dx[f(g(x))] = f'(g(x))g'(x)

2. Derivatives of Special Functions

Memorize the derivatives of:

• Exponential functions
• Logarithmic functions
• Trigonometric functions
• Inverse trigonometric functions
• Implicit differentiation

3. Applications of Derivatives

Apply derivatives to solve problems involving:

• Rates of change
• Tangent lines and normal lines
• Curve sketching (increasing/decreasing, concavity)
• Optimization (maxima and minima)
• Related rates
• Mean Value Theorem

Part IV: Integration

1. Antiderivatives and Indefinite Integrals

Find antiderivatives using:

• Basic integration rules
• Integration by substitution
• Integration by parts: u dv = uv - v du

2. Definite Integrals

Apply definite integrals to:

• Calculate areas under curves
• Areas between curves
• Volumes of solids with known cross-sections
• Volumes of revolution (disk/washer and shell methods)

3. The Fundamental Theorem of Calculus

Understand and apply both parts of the FTC:

• FTC Part 1: d/dx[_a^x f(t) dt] = f(x)
• FTC Part 2: _a^b f(x) dx = F(b) - F(a)

Part V: Differential Equations

1. Basic Differential Equations

Review solving simple differential equations:

• Separation of variables
• Initial value problems
• Exponential growth and decay models

Practice Problems

Limits and Continuity

1. Evaluate lim(x2) (x² - 4x + 4)/(x - 2)
2. Find lim(x) (3x² + 5x - 2)/(7x² - x + 9)
3. Determine where f(x) = (x³ - 8)/(x - 2) is discontinuous and classify the discontinuities.

Differentiation

4. Find dy/dx for y = e^xln(2x)
5. Find the derivative of f(x) = cos^-1(3x² + 1)
6. Use implicit differentiation to find dy/dx for x²y + y³ = 6
7. Find the equation of the tangent line to y = x(x² + 1) at x = 1

Applications of Derivatives

8. Find all critical numbers of f(x) = x³ - 3x² - 9x + 7
9. Determine the intervals on which f(x) = x/(x² + 1) is increasing and decreasing.
10. Find the absolute maximum and minimum of f(x) = x³ - 12x on the interval [0, 4]
11. A 15-foot ladder is leaning against a wall. If the bottom of the ladder slides away from the wall at a rate of 2 ft/sec, how fast is the top sliding down the wall when the bottom is 9 ft from the wall?

Integration

12. Evaluate x(x+5) dx
13. Compute xln(x) dx
14. Find ₀^/4 sin³(x)cos²(x) dx
15. Set up an integral to find the area between y = x² - 4x + 3 and y = x + 1
16. The region bounded by y = x², the x-axis, and x = 2 is revolved around the y-axis. Find the volume using the appropriate method.

Differential Equations

17. Solve the differential equation given dy/dx = x/y with the initial condition y(0) = 3
18. A population of bacteria grows at a rate proportional to the current population. If there are 100 bacteria initially and 250 bacteria after 2 hours, how many will there be after 4 hours?

Resources for Review

If you need help with any of these topics, consult these resources:

• Your Pre-Calculus and AP Calculus AB textbooks
• Khan Academy (khanacademy.org) - Sections on Limits, Derivatives, and Integrals
• Paul's Online Math Notes (tutorial.math.lamar.edu)
• PatrickJMT (patrickjmt.com) - YouTube videos on calculus topics

Submission Guidelines

Please follow these guidelines when submitting your assignment:

• Show all work clearly and neatly for each problem
• Answer the problems in order
• Box or highlight your final answers
• Include your name and the date on the first page

Course Website

For additional information and updates, please visit our course website: www.yourschool.edu/calculusbcsummer