Browse Classroom Capsules and Notes

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A General Form of the Arithmetic-Geometric Mean Inequality via the Mean Value Theorem

A weighted arithmetic/geometric mean inequality is proved.

Behold! Two Extremum Problems (and the Arithmetic-Geometic Mean Inequality)

A proof without words approach to the geometric-arithmetic mean inequality

The Fresnel Integrals Revisited

Without using double integrals, a new method is presented to evaluate the two Fresnel integrals about sine and cosine that has a wider applications than previous methods.

Differentiating the Arctangent Directly

The derivative of arctangent is derived directly from the definition of derivative by using some clever inequalities.

Finding Matrices that Satisfy Functional Equations

The author shows how to solve a class of analytic functions using an approach demonstrating a surprising connection between multivariable calculus and linear algebra.

A Counterexample to Integration by Parts

The authors exhibit two differentiable functions for which the integration by parts formula does not apply.

Monotonicity of Sequences Approximating \(e^x\)

How do sequences of the form \((1+x/n)^{n+\alpha}\) with \(x >0\) approach their limits?

On the Remainder in the Taylor Theorem

An inductive proof is presented for the bounds of the remainder of Taylor expansion. This result, with Darboux's theorem, implies the classical formula for the remainder.

Computing Definite Integrals Using the Definition

The author finds Riemann sums that equal exactly the definite integrals for polynomials and negative-integer power functions.

Waiting to Turn Left?

The authors model a real traffic problem by using the fundamental theorem of calculus.