Problem-solving recipes

A step-by-step procedure for every problem type this course teaches — what to do first, what to do next, and how to tell which case you're in.

Limits and continuity

Evaluate a limit algebraically

Use when: you're asked for $\lim_{x\to a}f(x)$ and $f$ is built from polynomials, radicals, or rational expressions

Goal: a single number (or a clear reason the limit doesn't exist), reached without a table or a graph

Algebraic techniques for limits

Prove a limit of a linear function with epsilon-delta

Use when: you must prove $\lim_{x\to a}(mx+b)=L$ from the definition, and $f$ is a straight line

Goal: a complete written proof: a formula for delta in terms of epsilon, plus a forward chain from the assumption to the tolerance

Writing an ε-δ proof · extended later in this course

Prove a limit of a nonlinear function with epsilon-delta

Use when: the output distance $|f(x)-L|$ factors as $|x-a|$ times a leftover factor that still depends on $x$ , as for polynomials, square roots, and fractions

Goal: a complete written proof with $\delta=\min(r,\varepsilon/M)$ , where $r$ keeps $x$ in a safe interval and $M$ bounds the leftover factor there

ε-δ proofs beyond straight lines

The derivative

Build a derivative from the definition

Use when: you're asked for $f'(x)$ (or $f'$ at a specific point) and no shortcut rule is available yet

Goal: a simplified expression for $f'(x)$ , or its exact value at one point, with the limit already taken

The definition of the derivative

Chain rule and further techniques

Differentiate implicitly

Use when: an equation relates $x$ and $y$ but isn't (or can't easily be) solved for $y$ first

Goal: a formula for $dy/dx$ in terms of $x$ and $y$ , valid at any point on the curve

Implicit differentiation

Rates and approximations

Solve a related-rates problem

Use when: two or more quantities change over time, tied together by a shared geometric or physical equation, and you're given one rate to find another

Goal: the requested rate, at the specific instant described

Related rates -- the general method

MVT and the shape of a graph

Sketch a curve using calculus

Use when: you need an accurate sketch of $y=f(x)$ without a graphing tool, or need to justify a shape a tool already drew

Goal: a sketch whose every feature -- where it's defined, where it turns, which way it bends -- is proved, not guessed

Curve sketching -- putting it all together

Evaluate a limit with L'Hopital's Rule

Use when: $\lim_{x\to c}\dfrac{f(x)}{g(x)}$ comes out to a genuine $0/0$ or $\infty/\infty$ form

Goal: the limit's value, reached by differentiating the numerator and denominator separately rather than the original expression

Indeterminate forms and L'Hopital's Rule

Optimization

Solve an optimization problem

Use when: a word problem asks for the largest or smallest possible value of some quantity, subject to a constraint

Goal: the input value that actually maximizes or minimizes the quantity, plus that optimal value itself

Optimization: the general method

Antiderivatives and the definite integral

Solve an initial value problem

Use when: you're given a function's derivative and one known value, and asked for the specific function itself

Goal: one particular antiderivative — a single function, not a whole family

Initial value problems

The Fundamental Theorem

Differentiate an accumulation function

Use when: you're asked for $H'(x)$ where $H(x)=\int_a^{g(x)} f(t)\,dt$ and the upper limit is a function of $x$ , not $x$ itself

Goal: a formula for $H'(x)$ , found without ever computing $H(x)$ in closed form

FTC Part 1: the chain rule twist

Substitution and applications

Integrate by u-substitution

Use when: the integrand looks like $f(g(x))\cdot g'(x)$ -- an outer function wrapped around an inner one, times something that could be that inner function's derivative

Goal: an antiderivative in terms of $x$ , found by temporarily rewriting the whole integral in terms of $u$

Choosing u: recognizing a substitution · extended later in this course

Integrate a definite integral by u-substitution

Use when: a definite integral $\int_a^b f(g(x))g'(x)\,dx$ needs u-substitution, and the bounds $a$ , $b$ are given in terms of $x$

Goal: a single number, reached either by converting the bounds to $u$ directly or by back-substituting to $x$ before evaluating

U-substitution with definite integrals

Find the volume of a solid of revolution

Use when: a flat region is revolved around an axis, and you need the resulting solid's volume

Goal: a single number -- the exact volume, found by integrating cross-sectional area rather than measuring the solid directly

Volumes by disks and washers