MATH 150 FINAL EXAMINATION FALL 1996

(12) 1. Find the following limits. You have to justify your answers.

(a) $\displaystyle\lim_{x\to 0}\frac{\tan 5x}{x}$

(b) $\displaystyle\lim_{x\to \infty}(x-\sqrt{x^2-1})$

(c) $\displaystyle\lim_{x\to 2}\frac{x^2-2x}{x^2-4}$

(10) 2. Use the definition of the derivative to find f'(1), where f(x)=x2+2.

(16) 3. Find the derivative $\frac{dy}{dx}$ of the function.

(a) $y=x\tan^2x$

(b) $y=\frac{x}{(1-x)^3}$

(c) $xy=\cos2x$

(d) $y=\displaystyle\int_1^{x^2}\sin\sqrt{t}\,dt$

(10) 4. Use logarithmic differentiation to find y' if y=xx.

(10) 5. Find an equation of the tangent line to the graph of $y=\sin\, 2x$ at $x=\frac{\pi}{3}$.

(10) 6. Find the absolute extrema of $f(x)=\ln x+\frac{1}x$ on $[\frac{1}2,2]$.

(8) 7. The diameter of a circle is measured to be 12.5 cm with a possible error of 0.5 cm. Use differentials to approximate the relative error in computing the area of the circle.

(10) 8. Find the average value of f(x)=-16x2+144 on the interval [-3,3].

(8) 9. Apply Newton's Method to determine a positive zero of f(x)=x3+2x2-8x-1. Choose an appropriate initial guess and continue the process until two successive approximations differ by less than 0.0001.

(a) Write down your initial guess.

(b) Write down all the approximations used to five decimal digits.

(10) 10. An airplane is flying at an altitude of 5 miles and passes directly over a radar antenna (see figure). When the plane is 13 miles away (s=13), the radar detects that the distance s is changing at a rate of 240 miles per hour. What is the speed of the airplane at this moment?

(10) 11. Find f(x) if $f'(x)=\sec x\, \tan x$ and f(0) = -1.

(15) 12. Evaluate the following indefinite integrals.

(a) $\displaystyle\int \frac{1+\sqrt x}x \, dx$

(b) $\displaystyle\int \cos^2 2x \sin2x \, dx$

(c) $\displaystyle\int 2x \sqrt{x^2+1} \, dx$

(15) 13. Use the Fundamental Theorem of Calculus to compute the following definite integrals.

(a) $\displaystyle\int_0^1\frac{e^x}{1+e^x}\,dx$

(b) $\displaystyle\int_0^{\pi/4}(1+\sec\,^2x)\,dx$

(c) $\displaystyle\int_0^1x\sqrt{1-x}\,dx$

(10) 14. Consider the following integral

\begin{displaymath} \int_0^1xe^x\,dx.\end{displaymath}

(a) Use the Trapezoidal Rule, with n=4, to approximate the value of this integral. Write your answer using five decimal digits.

(b) The error E in approximation of an integral $\int_a^bf(x)\,dx$ with the Trapezoidal Rule is known to satisfy,

\begin{displaymath} \vert E\vert \leq \frac{(b-a)^3}{12n^2} \mbox{Max}\vert f''(x)\vert,\, \mbox{over}\, a\leq x \leq b.\end{displaymath}

Use this result to estimate the maximum error of an approximation you found in part (a).

(10) 15. Find the horizontal asymptotes to the graph of

\begin{displaymath} y=\frac{6-e^{-x}}{3+2e^{-x}}.\end{displaymath}

(16) 16. Let $f(x)=3x-\tan\,x,\ \ \frac{-\pi}2<x<\frac{\pi}2$.

(a) With the aid of the calculator sketch the graph of f(x) in the space below. Choose a scale that allows all relative extrema and points of inflection to be identified on the graph.

(b) Find the relative extrema of f(x) and points where they occur. Write your answer using three decimal digits.

(c) Find the intervals on which f(x) is concave up and concave down.

(d) Find the points of inflection for f(x).

(8) 17. Find the area of the region that is bounded by the curves $y=1+x^2,\ \ y=0,\ \,x=0,\ \ \mbox{and}\ \ x=1.$

(12) 18. Find the volume of the solid formed by rotating the region bounded by the graphs of $y=\frac{\ln x}{\sqrt{x}},\ \ y=0,\ \ x=1, \ \ \mbox {and} \ \ x=e\ \ \mbox {about the x-axis}.$