AP Calculus AB › Unit 2
What does it mean for something to change at an instant? Unit 2 defines the derivative as a limit of average rates of change, then builds the power, product, and quotient rules and the derivatives of trig, exponential, and log functions. Expect to estimate a derivative from a table, recognize a limit as a derivative, and connect differentiability to continuity. A common error is treating the product rule as the product of two derivatives. Students also lose credit by giving a number without showing the difference quotient behind it.
These are taken from the Regular set, with the answer and explanation shown.
1. What is the value of lim h→0 [cos(π/3 + h) − 1/2]/h?
Answer: C. Since cos(π/3) = 1/2, this is the derivative of cos x at x = π/3. That derivative is −sin(π/3) = −√3/2.
Common trap: Choosing √3/2 forgets that the derivative of cosine is negative sine.
2. Which of the following functions is continuous at x = 0 but not differentiable at x = 0?
Answer: C. x^(2/3) is continuous at 0, but f′(x) = (2/3)x^(−1/3) grows without bound near 0, with opposite signs on each side. The graph has a cusp, so f′(0) does not exist.
Common trap: Choosing 1/x picks a function that fails, but for the wrong reason: 1/x is not even defined at 0, so it is not continuous there.
3. A differentiable function f has f(2) = 3 and f′(2) = 4. If h(x) = f(x)/(1 + f(x)), what is h′(2)?
Answer: B. Quotient rule: h′ = [f′(1 + f) − f·f′]/(1 + f)² = f′/(1 + f)². At x = 2: 4/(1 + 3)² = 4/16 = 1/4.
Common trap: Choosing 3/4 gives h(2) = 3/(1 + 3), the value of h. Choosing 1 divides f′ by the derivative of the bottom, which is also f′.