Note: These problems mirror actual AP exam questions in style and difficulty.

📚 Table of Contents

1. Limits & Continuity
2. Derivatives
3. Integration
4. Series
5. Parametric/Polar
6. Differential Equations

Limits & Continuity 📊

Problem 1

Evaluate $\lim_{x \to 0} \frac{\sin(3x)}{x}$

Solution:

1. $= 3\lim_{x \to 0} \frac{\sin(3x)}{3x}$
2. = 3 ⋅ 1 = 3

Problem 2

Find values of k where f(x) is continuous: $f(x) = \begin{cases} x^2 + k & x < 2 \\ kx + 1 & x \geq 2 \end{cases}$

Derivatives 📈

Problem 3

Find $\frac{d}{dx}[\ln(\sin x)]$

Solution:

1. Chain Rule: $\frac{1}{\sin x} \cdot \cos x$
2. Simplify: cot x

Problem 4

If f′(x) = x2ex and f(0) = 1, find f(x).