Learn Math with live MCPCalc tools
Explore proofs, worked examples, and clear explanations, then try the math yourself with graphing, CAS, proof builder, spreadsheets, and calculators.
Proofs
Structured arguments in Proof Builder, with theorem statements, line-by-line reasoning, and nearby variations.
Worked Examples
Standard problems where the computation and the explanation stay tied to the live tool.
Explanations
Concept pages that build intuition first and then connect it to formal notation and exact calculations.
Proofs
A catalog of interesting and common proofs to learn, study, and revisit.
Proof that the limit of a convergent sequence is unique
A convergent sequence cannot settle down to two different numbers. The proof is short, but it is a good model for how contradiction and the definition of limit work together.
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Proof that the kernel of a linear map is a subspace
This is a standard linear algebra proof because it packages the subspace test into one reusable pattern: show zero is inside, then check closure under addition and scalar multiplication.
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Epsilon-delta proof that
This proof is common because it shows the standard move in early analysis: factor the expression, then bound the extra factor by forcing into a smaller interval first.
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Proof of the chain rule
The chain rule says the derivative of is . The proof uses a continuous extension of the difference quotient to handle all cases cleanly.
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Worked Examples
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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Explanations
Concept pages that build intuition first and then connect it to formal notation and MCPCalc tools.
What a derivative means geometrically
The derivative is the slope of the tangent line, but that phrase only becomes useful when you compare the curve to nearby secant lines and local linear approximations.
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What an epsilon-delta proof is actually controlling
An epsilon-delta proof is a control problem: keep close enough to a point so the function value stays inside a target band around the limit.
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What eigenvectors and eigenvalues mean geometrically
An eigenvector is a direction that survives a linear transformation without rotating away from its own line. The eigenvalue tells you the scale and possible sign flip along that direction.
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What a -value means and what it does not mean
A -value is a measure of how surprising the observed test statistic would be if the null hypothesis were true. It is not the probability that the null hypothesis itself is true.
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What the definite integral means geometrically
The definite integral measures signed area between a curve and the -axis. This page builds that idea from Riemann sums, connects it to antiderivatives, and shows how to read integral notation.
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