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Explore proofs, worked examples, and clear explanations, then try the math yourself with graphing, CAS, proof builder, spreadsheets, and calculators.
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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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Proof that eigenvectors for distinct eigenvalues are linearly independent
Distinct eigenvalues do more than label eigenspaces. They force eigenvectors to be independent, which is exactly why diagonalization becomes possible so often in practice.
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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 that every convergent sequence is bounded
This theorem is a standard follow-up to the definition of convergence because it shows how one tail estimate plus finitely many early terms gives a global bound on the whole sequence.
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Proof that the derivative of a constant function is zero
This proof is short, but it matters because it shows the derivative definition already knows that a function with no change has zero slope. Nothing extra has to be added as a rule.
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Proof that for positive integers
This proof shows where the power rule comes from rather than treating it as a black box. The binomial theorem isolates the only term that survives after dividing by and taking the limit.
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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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Proof that a differentiable function is continuous
This proof shows why differentiability is a stronger condition than continuity. The difference factors into a difference quotient times , and that product is forced to zero.
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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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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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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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Why continuity and differentiability are different concepts
Differentiability is stronger than continuity. A graph can be unbroken at a point and still fail to have a derivative there if the local shape has a corner, cusp, or vertical tangent.
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What convergence means for sequences
A sequence converges when its terms eventually stay as close as you want to a single number. The word eventually matters more than the first few terms.
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What convergence means for infinite series
An infinite series converges when its partial sums settle toward a finite number. The important object is the running total, which is why the convergent geometric series and the divergent harmonic series tell different stories even though both have terms that go to zero.
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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 linear independence means geometrically
Linear independence is the condition that nothing in the list is wasted. Geometrically, each new vector must add a new direction instead of repeating one you already had.
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What a basis does in linear algebra
A basis is what turns a vector space into something you can navigate. It reaches every vector, and it does so without redundancy, so coordinates become possible.
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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 a Taylor polynomial is approximating
A Taylor polynomial is not trying to copy a function everywhere. It matches the function and several of its derivatives at one chosen center, so it is designed to be locally accurate near that point.
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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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