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Browse Classroom Capsules and Notes

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Displaying 131 - 140 of 225

A Property Possessed by Every Differentiable Function

If \(f\)  is differentiable at \(x\), an interesting relationship exists between the tangent and two secant lines.

Logarithmic Differentiation: Two Wrongs Make a Right

Differentiate \(f(x)^{g(x)}\) first as if \(g\) was a constant, then as if \(f\) was a constant. Presto!

Placing Natural Logarithm and Exponential Functions on an Equal Footing

Introduce both the natural log and the exponential as solutions to differential equations

An Apothem Apparently Appears

Varieties of cutting a wire and forming two geometric shapes have unifying properties.

A Quotient Rule Integration by Parts Formula

The rule for differentiation of a quotient leads to an integration by parts formula.

Estimating Definite Integrals

The integral of a function can be bounded by integrating a function that majorizes it.

Approaching \(\ln x\)

 

The integral \(\int_{1}^{x} t^{-1+1/n} dt\) is shown to converge to \(\ln x\).

Snapshots of a Rotating Water Stream

Trace the trajectory of a stream of water as the nozzle rotates upward in a vertical plane.

The Computation of Derivatives of Trigonometric Functions

Find the derivative of the arcsine, then the derivative of the sine by using the fundamental theorem of calculus.

Symmetry at Infinity

The centroid of the region bounded by \(y=x^n\) and \(y=x^{1/n}\) is not the same for all \(n\).

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