Math 173 04,

the course of Dr. Mihailovs

Final Exam

December 18, 1998

  1. Find $ \underset{x\rightarrow 2}{\lim}\; \frac{x^2-x-2}{x^2-3x+2}$ .

  2. Find $ \underset{x\rightarrow 2}{\lim}\; \frac{x-\sqrt{x+2}}{x-\sqrt{3x-2}}$ .

  3. Find $ \underset{x\rightarrow \infty}{\lim}\; \frac{x^2-x-2}{2x^2-3x+2}$ .

  4. Use the Squeeze Theorem to find $ \underset{x\rightarrow \infty} {\lim}\; \frac{\cos x}{x}$ .

  5. Find $ f'(x)$ for $ f(x)=5x^3+4\sin x+3\cos x+2\tan x+1$ .

  6. Find an equation of the tangent to the curve $ y=5x^3+4\sin x+3\cos x+2\tan x+1$ at the point $ (0, 4)$ .

  7. Find $ f'(x)$ for $ f(x)=\cos(x^3+\sin 2x)$ .

  8. Find $ y'$ if $ x^{2/3}+y^{2/3}=1$ .

  9. Find the 100th derivative of $ f(x)=x^{89}+3\sin x+2\cos x$ .

  10. Find an approximate value of $ \sqrt[3]{997}$ .

  11. Find maximum and minimum values of $ f(x)=\frac{x^2+5}{3x+2}$, $ -4\leq x\leq -1$ .

  12. Sketch the graph of $ y=\frac{x^3}{1-x^2}$ .

  13. Find $ \int \frac{(3x^2+2)\; dx}{\cos^2 (x^3+2x+3)}$ .

  14. Find $ f'(x)$ for $ f(x)=(x-3)^4\cos 2x$ .

  15. Find $ \int_0^3(4(x-3)^3\cos 2x-2(x-3)^4\sin 2x)\; dx$ .

  16. Find the area of the region bounded by the curves $ y=\cos x$, $ y=1-\cos x$ for $ 0\leq x\leq 2\pi$ .

  17. Find the volume of the solid obtained by rotating the region under the graph of $ y=\sqrt[4]{(x+2)/3}$ from $ x=1$ to $ x=10$ about the $ x$-axis.

  18. Find the volume of the solid obtained by rotating the region bounded by $ y=0$ and $ y=15(x-x^3)$, $ x\geq 0$ about the $ y$-axis.

  19. Find the average value of the function $ f(x)=\sin(x^7+2x^3+3x)$ over the interval $ [-2\pi, 2\pi]$.

Copyright © 1998 Aleksandrs Mihailovs