This comprehensive guide provides a thorough review of key concepts covered in a typical AP Calculus AB curriculum, offering insights into problem-solving strategies and showcasing sample problems similar to those found on the AP exam. This isn't just a collection of problems; it's a structured learning experience designed to enhance your understanding and improve your performance on the actual exam. We'll cover limits, derivatives, integrals, and applications, equipping you with the tools to succeed.
Section 1: Limits and Continuity
Understanding limits and continuity forms the foundation of calculus. A solid grasp of these concepts is crucial for mastering more advanced topics.
1.1 Limits:
Limits describe the behavior of a function as its input approaches a certain value. We explore different techniques for evaluating limits, including direct substitution, factoring, and L'Hôpital's Rule (for indeterminate forms).
Sample Problem 1: Evaluate lim_(x→2) (x² - 4)/(x - 2).
Solution: Factoring the numerator, we get (x - 2)(x + 2)/(x - 2). Canceling the common factor (x - 2), we have lim_(x→2) (x + 2) = 4.
1.2 Continuity:
A function is continuous at a point if the limit exists at that point, the function is defined at that point, and the limit equals the function value. We'll examine different types of discontinuities and how to identify them.
Sample Problem 2: Determine if the function f(x) = { x² if x ≤ 1; 2x -1 if x > 1 } is continuous at x = 1.
Solution: We need to check if lim_(x→1) f(x) = f(1). lim_(x→1⁻) f(x) = 1 and lim_(x→1⁺) f(x) = 1. Since both one-sided limits are equal to 1, and f(1) = 1, the function is continuous at x = 1.
Section 2: Derivatives
Derivatives measure the instantaneous rate of change of a function. We will cover various derivative rules, including the power rule, product rule, quotient rule, and chain rule.
2.1 Basic Differentiation Rules:
Mastering the basic differentiation rules is paramount. Practice is key to developing fluency and speed in calculating derivatives.
Sample Problem 3: Find the derivative of f(x) = 3x⁴ - 2x² + 5x - 7.
Solution: Applying the power rule, f'(x) = 12x³ - 4x + 5.
2.2 Implicit Differentiation:
Implicit differentiation allows us to find the derivative of functions that are not explicitly solved for y.
Sample Problem 4: Find dy/dx for x² + y² = 25.
Solution: Differentiating both sides with respect to x, we get 2x + 2y(dy/dx) = 0. Solving for dy/dx, we get dy/dx = -x/y.
Section 3: Integrals
Integrals represent the accumulation of a quantity over an interval. We will explore both definite and indefinite integrals, fundamental theorem of calculus, and various integration techniques.
3.1 Indefinite Integrals:
Indefinite integrals represent the antiderivative of a function. Remember to include the constant of integration, "+C".
Sample Problem 5: Find the indefinite integral of f(x) = 4x³ + 6x.
Solution: ∫(4x³ + 6x)dx = x⁴ + 3x² + C
3.2 Definite Integrals:
Definite integrals calculate the area under a curve over a specified interval. The fundamental theorem of calculus connects derivatives and integrals.
Sample Problem 6: Evaluate ∫₁³ (2x + 1) dx.
Solution: The antiderivative is x² + x. Evaluating at the limits of integration: (3² + 3) - (1² + 1) = 10.
Section 4: Applications of Derivatives and Integrals
This section focuses on applying derivatives and integrals to solve real-world problems.
4.1 Related Rates:
Related rates problems involve finding the rate of change of one quantity with respect to time given the rate of change of another related quantity.
4.2 Optimization:
Optimization problems involve finding the maximum or minimum value of a function.
4.3 Area and Volume:
Integrals are used to calculate areas and volumes of irregular shapes.
This sample examination provides a glimpse into the topics covered in the AP Calculus AB exam. Thorough preparation, consistent practice, and a deep understanding of the underlying concepts are crucial for success. Remember to consult your textbook and class notes for more detailed explanations and practice problems. Good luck!
