Differentiation: Word problem - rate of change

A water trough is 10 m long and a cross-section has the shape of an isosceles trapezoid that is 30cm wide at the bottom, 80 cm wide at the top, and has a height of 50 cm. If the trough is being filled with water at a rate of 0.2m3/min, how fast is the water level rising when the water is 30cm deep?

Minimization

A box with a square base and open top must have a volume of 32,000cm3. Find the dimensions of the box that minimize the amount of amount of material used.

Functions: Limits

Lim [1/(x-1) - 2/(x-1)] x→1

Integral test: Convergence or divergence of a series.

Use the integral test to determine the convergence or divergence of the series: En=1 2 / (3n + 5)

theorem 8.11

Use theorem 8.11 to determine the convergence or divergence of the p-series En=1 3 / (n5/3)

Limit comparison test

Use the limit comparison test to determine the convergence or divergence of the series En=1 2 / (3^n - 5)

Taylor Polynomial

Find the nth Taylor polynomial centered at c f(x)= (x)^1/3 n = 3 c = 8

Power series centered at 0

Use the power series 1 / 1+x = En=0 (-1)^n x^n to determine the power series, centered at 0, for the function h(x) = x / x^2-1 = 1 / 2(1+x) - 1 / 2(1-x)

Interval of convergence

Find the interval of convergence of (a) f(x), (b) f'(x), (c) f''(x), (d) {f(x)dx En=1 [(-1)^n+1 (x-2)^n ] / 2

Basic Derivatives

1.) Find the derivative of the function: a.) f(x) = x + 1/x^2 b.) f(x) = (2/3rd root of x) + 3 cos x 2.) Find equation of tangent line to the graph of f at the indicated point: a.) y = (x^2 + 2x)(x + 1) ; (1,6)