Problem Solving Strategy Guide

Calculus 1 · Axiom Academy

SUMMARY Problem Solving Strategy Guide A field guide to Calculus 1: how to recognize each problem type and reach for the right technique — limits, derivatives, optimization, and integration. Identify the problem type first. Limit, derivative, optimization, or integral — naming it tells you which toolkit to open. Start with the simplest move. Direct substitution for limits, a known antiderivative for integrals — only escalate to harder tools when the easy one fails. Draw a picture. A labeled diagram turns word problems (optimization, related rates) into equations you can actually solve. Always check your answer. Differentiate an antiderivative, plug in test values, confirm units — verification catches most sign and bookkeeping errors. Work from easiest to hardest. Most limits yield to substitution; reach for algebra or L'Hopital only when you hit an indeterminate form. Try direct substitution first. If you get , factor and cancel or rationalize — or apply L'Hopital's rule. For , compare growth rates (divide by the highest power). Check one-sided limits to confirm the two-sided limit exists. Strategy Choosing a Derivative Rule Differentiation is mechanical once you read the structure of the function and pick the matching rule. Identify the function type(s) — product, quotient, composite, or a sum of these. Choose the rule: power, product, quotient, or chain. Apply it systematically — don't skip steps, especially the inner derivative in the chain rule.

This is the written version of the interactive lesson above. See the full Calculus 1 course.