Calculus II, Spring 2014

Quiz 2

Mar 20, 2014 Show all details.

1. Suppose that f is infinitely many times differentiable on R. State Taylor’s Theorem for f and write down the formula for R_{n}(x) (in any form you choose).

2. Find the Taylor series generated by sin x at x = 0. For what values of x does sin x equal this series? Explain.

3. Find the sum of the series

\frac{1}{2} − \frac{1}{8} + \frac{1}{24} − ⋯ + (−1)^{n−1} \frac{1}{n2^{n}} + ⋯

4. Let f(x) = exp(\frac{−1}{x^{2}}) for x ≠ 0 and f(0) = 0. Is f differentiable everywhere? Explain.

5. Find the area of the surface generated by evolving x = cos t, y = 2 + sin t, 0 ≤ t ≤ 2π about the x-axis.

Calculus II, Spring 2014

Quiz 2

Mar 20, 2014 Show all details.

1. Suppose that f is infinitely many times differentiable on R. State Taylor’s Theorem for f and write down the formula for R_{n}(x) (in any form you choose).

2. Find the Taylor series generated by sin x at x = 0. For what values of x does sin x equal this series? Explain.

3. Find the sum of the series

\frac{1}{2} − \frac{1}{8} + \frac{1}{24} − ⋯ + (−1)^{n−1} \frac{1}{n2^{n}} + ⋯

4. Let f(x) = exp(\frac{−1}{x^{2}}) for x ≠ 0 and f(0) = 0. Is f differentiable everywhere? Explain.

5. Find the area of the surface generated by evolving x = cos t, y = 2 + sin t, 0 ≤ t ≤ 2π about the x-axis.