AP Calculus AB › Unit 5
Unit 5 uses derivatives to analyze graphs. You will apply the Mean Value and Extreme Value theorems, find where a function increases or decreases, locate relative and absolute extrema, test concavity, and solve optimization problems. Many exam questions give the graph of f' and ask about f. Justifications must cite a derivative's sign or behavior. A classic error is reading features of f' as if they were features of f. Vague phrases like "it goes up" cost points; name f, f', or f'' instead.
These are taken from the Regular set, with the answer and explanation shown.
1. How many critical points does f(x) = |x² − 4| have?
Answer: A. For −2 < x < 2, f(x) = 4 − x², so f′(x) = −2x = 0 at x = 0. At x = ±2 the graph has sharp corners, so f′ does not exist there. That makes 3 critical points.
Common trap: Choosing 2 counts only the corners at x = ±2. The flipped hump also has a horizontal tangent at x = 0.
2. What is the absolute maximum value of f(x) = x sin x on the interval [0, 3]?
Answer: A. f′(x) = sin x + x cos x. A calculator gives the zero x ≈ 2.029. Then f(2.029) ≈ 1.820, while f(0) = 0 and f(3) ≈ 0.423. The largest value is 1.820.
Common trap: Choosing 2.029 reports where the maximum occurs. The question asks for the value f(2.029) ≈ 1.820.
3. A rectangle has its base on the x-axis and its two upper corners on the parabola y = 12 − x², at (−x, y) and (x, y) with x > 0. Which function gives its area?
Answer: A. The base runs from −x to x, so its length is 2x. The height is the y-value of the corners, 12 − x². Area = base × height = 2x(12 − x²).
Common trap: Choosing x(12 − x²) uses only the right half of the rectangle. The base goes from −x to x, so its width is 2x.