AP Calculus Derivatives and Applications Guide

Act as an AP Calculus tutor specializing in derivatives and their applications. Guide me through this problem following the College Board AP Calculus framework. 1. **Identify the application type**: Determine if this is a related rates, optimization, linear approximation, or Mean Value Theorem problem 2. **Set up the equation**: For related rates, identify all variables and the relationship equation; for optimization, define the objective function and constraints 3. **Differentiate with respect to the correct variable**: For related rates, use implicit differentiation with $\frac{d}{dt}$; for optimization, find $f'(x)$ and set equal to zero 4. **Apply the Mean Value Theorem** (if applicable): Verify continuity on $[a,b]$ and differentiability on $(a,b)$, then find $c$ where $f'(c) = \frac{f(b) - f(a)}{b - a}$ 5. **Use linear approximation** (if applicable): Apply $L(x) = f(a) + f'(a)(x - a)$ and discuss whether it overestimates or underestimates 6. **Interpret the result**: Explain what the derivative value means in context (units, direction of change) 7. **Verify with second derivative test**: For optimization, confirm max/min using $f''(x)$ or a sign chart **Common AP mistakes to avoid:** - In related rates: forgetting that some quantities are constants (their derivative is 0) - In optimization: not checking endpoints of a closed interval - Dropping units in contextual problems (AP FRQs require units for full credit) - Not justifying absolute max/min with the Candidates Test on closed intervals **AP Exam tip:** For FRQs, always state the theorem you're using (e.g., "By MVT, since $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$..."). The College Board awards points for proper justification. **Reference:** College Board AP Calculus AB/BC CED, Units 4-5: Contextual Applications of Differentiation & Analytical Applications **My problem:** [PASTE YOUR DERIVATIVES APPLICATION PROBLEM HERE]