Math 243: Calculus II |
1. Evaluate \(\displaystyle \int_0^1 e^x\, dx\) by using the Riemann
sum definition and noticing that it's a geometric sum. You may check your
answer using the Fundamental Theorem, but to receive credit you must do the
Riemann sum.
2. Find the MacLaurin series for \(f(x) = e^{-x^2}\) and use it to
evaluate \(\displaystyle \int_0^1 e^{-x^2}\, dx\) correct to 4 decimal places.
3. Find the Taylor series for \(y = \ln x\) centered at \(x_0 = 1\).
What is its interval of convergence?
4. Find the Taylor series for \(y = 3x^3 - 2x^2 + 9x + 4\) centered
at \(x_0 = 2\).
5. Use series to evaluate the limit, \(\displaystyle \lim_{x\to 0}
\dfrac{x^3 - 3x + 3\arctan(x)}{x^5}\).
6. Use series to evaluate the limit, \(\displaystyle \lim_{x\to 0}
\dfrac{\tan x - x}{x^3}\).
7. Find the radius of convergence of the series, \(\displaystyle
\sum_{n=1}^\infty \dfrac{(2n)!}{(n!)^2} (x - x_0)^n\).
8. Find the MacLaurin series for \(y = \sqrt{1 + x^4}\) and use it
to approximate \(\displaystyle \int_0^1 \sqrt{1 + x^4}\, dx\).
9. You must be able to successfully apply all convergence tests
for series.
10. Consider the parametric curve \(\mathbf{r}(t) = \langle
t\cos t, t\sin t\rangle\). Compute \(\tfrac{dy}{dx}\) and
\(\tfrac{d^2y}{dx^2}\).
11. Find the area enclosed by the curve \(r^2 = 9\cos(5\theta)\).
12. Find the length of the curves.
| a.) | \(\mathbf{r}(t) = \langle 3t^2, 2t^3\rangle\),
\(0 \leq t \leq 1\) |
|
| b.) | \(r = \sin^3(\theta/3)\),
\(0\leq \theta\leq \pi\) |
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