Calculus AB — limits, continuity and the derivative

The first two units, and the ones where the marks are lost to definitions rather than to algebra. Ten questions, no calculator needed for any of them.

Five on limits and continuity. A limit where substitution gives nothing and the factor has to be cancelled first. A limit at infinity settled by comparing degrees. A piecewise function approached from below, where the rule that applies at the point itself is irrelevant and the equals sign in the other piece is the thing that catches people. A function whose limit exists and does not match its value, so that exactly one of the three conditions for continuity fails. And the Intermediate Value Theorem.

That last one is the question worth reading the explanation for even when you get it right. The theorem guarantees at least one value, not exactly one, and the most instructive wrong answer is numerically correct: the average rate of change across the interval really is 2, and a point where the derivative equals it really does exist — but that is the Mean Value Theorem, which needs differentiability as well as continuity. Two theorems, one hypothesis apart.

Five on the derivative. Recognising a difference quotient as a particular derivative at a particular point. The product rule and the chain rule, with the standard omissions offered as answers. A velocity read out of a position function with its units. And the absolute value function at zero, which is continuous and has no derivative, because the slope from the left is minus one and from the right is one.

Every function and every number here is invented, and every piecewise definition is written out in full because there are no graphs in these materials.

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• Evaluate a limit that gives an indeterminate form, by factoring and cancelling before substituting
• Find a limit at infinity of a rational function by comparing the degrees of numerator and denominator
• Compute a one-sided limit of a piecewise function using only the rule that applies on that side
• State the three conditions for continuity at a point and identify which one fails
• State what the Intermediate Value Theorem guarantees, and distinguish it from the Mean Value Theorem by hypothesis
• Recognise a difference quotient as the derivative of a named function at a named point
• Apply the product and chain rules correctly, interpret a derivative with units, and explain why a continuous function can fail to be differentiable

A function $f$ is continuous on the closed interval $[0, 4]$, with $f(0) = -3$ and $f(4) = 5$. What does the Intermediate Value Theorem guarantee?