Math 174 02,

the course of Dr. Mihailovs

Final Exam

December 18, 1998

  1. Find $ f^{-1}$ for $ f(x)=\frac{x+2}{2x+3}$ .

  2. Find $ g'(a)$ , where $ g$ is the inverse function of $ f(x)=x^7+x^3+1$ , $ a=3$ .

  3. Find $ f'(x)$ for $ f(x)=\frac{e^{2x}\sqrt{3x+1}}{\sqrt[3]{2x^2+2x+1}}$ .

  4. Find $ \int_{-\infty}^0 \frac{5e^{5x}\: dx}{\sqrt{1-e^{10x}}}$ .

  5. Find $ \int_{-\infty}^0 \frac{5e^{5x}\: dx}{\sqrt{1+e^{10x}}}$ .

  6. Find $ \underset{x\rightarrow 0}{\lim}\; \frac{x^3}{x-\sin x}$ .

  7. Find $ \int_0^{\pi/2}x^2 \cos x \; dx$ .

  8. Find $ \int \sin^3 x \cos^{100} x \; dx$ .

  9. Find $ \int_3^4 \frac{x^3-2x^2+x-1}{x^2- 3x+2}\; dx$ .

  10. Find $ \int_2^3 \frac{dx}{1+\sqrt{x-2}}$ .

  11. Use Simpson's Rule with $ n=2$ to find an approximate value of $ \ln 1.5=\int_2^3\frac{dx}{x}$ .

  12. Find the area of the surface obtained by rotating the curve $ y=\cos x$ from $ x=0$ to $ x=\pi/3$ about the $ x$-axis.

  13. Find the Cartesian equation of the curve $ x=3+2\sin t$, $ y=1+3\cos t$ .

  14. Find the equation of the tangent to the curve $ x=t\tan t$, $ y=3t^2+2$ at $ t=0$ .

  15. Find the length of the cardioid $ r=\sin^2 \frac{\theta}{2}$ .

  16. Test the series $ \sum_{n=1}^{\infty}\frac{\ln n}{n(\ln^2 n+1)}$ for convergence or divergence.

  17. Find the radius of convergence of the series $ \sum_{n=0}^{\infty}\frac{3^n x^{2n}}{\sqrt{n}}$ .

  18. Find $ \sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)!}\left( \frac{\pi}{6}\right)^{2n+1}$ .

  19. Expand $ \sqrt{1-4x^2}$ as a power series.

Copyright © 1998 Aleksandrs Mihailovs