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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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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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 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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