AP Calculus AB › Unit 4
Here derivatives describe the real world. You will interpret a derivative's meaning and units in context, analyze position, velocity, and acceleration, solve related rates problems, approximate values with a tangent line, and use L'Hospital's rule. Free-response questions often ask what a value means, in a sentence with units. Students frequently misread the question and find an amount when it asks for a rate. Another common slip is using L'Hospital's rule without first showing an indeterminate form like 0/0, or differentiating the whole quotient instead of the top and bottom separately.
These are taken from the Regular set, with the answer and explanation shown.
1. A particle moves along a line with position s(t) = t³ − 6t² + 9t for t ≥ 0. On which interval is the particle moving to the left?
Answer: A. v(t) = 3t² − 12t + 9 = 3(t − 1)(t − 3). This is negative between its zeros, so the particle moves left on 1 < t < 3.
Common trap: Choosing t > 2 uses a(t) = 6t − 12 > 0. That is where velocity increases, not where the particle moves left.
2. A conical tank, vertex down, is 27 meters tall with a top radius of 9 meters. When the water is h meters deep, its surface has radius r. Which expression gives the volume of water in terms of h alone? (V = (1/3)πr²h)
Answer: B. Similar triangles give r/h = 9/27, so r = h/3. Then V = (1/3)π(h/3)²h = πh³/27.
Common trap: Choosing V = 3πh³ uses r = 3h, flipping the ratio. The radius is smaller than the height here, so r = h/3.
3. Use the line tangent to f(x) = ln x at x = e to approximate ln 3.
Answer: C. f(e) = 1 and f′(e) = 1/e. So L(3) = 1 + (1/e)(3 − e) = 1 + 3/e − 1 = 3/e.
Common trap: Choosing (3 − e)/e finds the change in output but leaves out the starting value f(e) = 1.