Worked Examples
Browse common examples from different branches of math, each written with clear step-by-step instructions.
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.
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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.
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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.
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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.
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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.
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Diagonalize a matrix and compute
Diagonalization is one of the first places students see why eigenvalues matter computationally. Once , powers of become powers of a diagonal matrix.
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Compute eigenvalues and eigenvectors of a matrix
A first serious 3 by 3 eigenproblem should show new structure without burying the point in arithmetic. This one does that: you see the characteristic polynomial, the eigenspaces, and the diagonalization test in one pass.
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Solve a separable differential equation and plot solution families
A first-order separable equation shows both symbolic and visual reasoning: solve for the family first, then inspect how the integration constant changes the curves.
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Approximate a root with Newton's method and compare successive iterates
Newton's method is easier to trust when you can see the iteration table row by row. A spreadsheet makes the recurrence concrete instead of hiding it inside a single answer.
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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.
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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.
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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.
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