Worked Examples
Browse common examples from different branches of math, each written with clear step-by-step instructions.
Showing 8 worked examples in Calculus.
Differentiate a polynomial from first principles
This worked example shows how the derivative definition creates an algebra problem first and a limit problem second. The simplification step is the point of the exercise.
Open worked example →
Evaluate an integral by integration by parts
A simple example like is useful because it makes the choice of and visible without extra algebra getting in the way.
Open worked example →
Evaluate an integral by substitution
Substitution is the chain rule in reverse. In , the factor matches the derivative of the inner expression , so the structure is visible without extra algebra.
Open worked example →
Solve an optimization problem for the maximum area rectangle under a constraint
With perimeter fixed at , the area becomes the quadratic . That makes the optimization workflow concrete: use the constraint to reduce to one variable, then locate the peak of the area function.
Open worked example →
Compare left Riemann sums, right Riemann sums, and midpoint sums for the same function
For on , left, right, and midpoint sums built from the same partition give noticeably different totals. The point of the page is to compare those behaviors directly rather than treating the formulas as interchangeable.
Open worked example →
Find and classify critical points of a quartic function
This worked example shows the standard calculus workflow: differentiate, solve , then use the second derivative and the graph to decide which critical points are local maxima or minima.
Open worked example →
Test convergence of an improper integral
An improper integral is decided by a limit, not by the antiderivative alone. This example tests and compares it with the divergent harmonic-tail integral.
Open worked example →
Find a Taylor polynomial and estimate the error
This example builds the degree-5 Taylor polynomial of centred at , evaluates it at , and confirms that the actual error stays inside the bound given by Taylor's theorem.
Open worked example →