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The second midterm will be on Friday, November 19 at the usual class time (12:40pm). Section 1 will take the exam in room 201 of Heavy Enginnering (same as last time) and Section 2 will take in our usual lecture room, room 152 of Light Enginnering.

Place the letter corresponding to the correct answer in the box next to each question.

\fbox{ \rule[-.1cm]{0cm}{.75cm}$\hphantom{xxx}$ }What is the slope of the curve given by x3 + y3 - 9xy =0 at the point (x,y) = (2,4)? (a) 1(b) $\frac {24}{30}$(c) $\frac 34$(d) $ \frac 9{18}$(e) $ \frac 65$ (f) none of these.

\fbox{ \rule[-.1cm]{0cm}{.75cm}$\hphantom{xxx}$ } Suppose f(x) = | x2 -2x|. The set of critical points of f is
(a) $\{ 0\}$ (b) $\{ 1\}$(c) $\{ 0,1,2 \}$(d) $\{2\}$(e) $\{0,2\}$(f) none of these.

  \fbox{ \rule[-.1cm]{0cm}{.75cm}$\hphantom{xxx}$ }Suppose $g'(x) = \sin^{1999}(x)$. The absolute maximum of g on $[0, 2\pi]$ occurs at
(a) 0(b) $\pi/4$(c) $\pi/2$(d) $\pi$(e) $ 2 \pi $(f) none of these.

\fbox{ \rule[-.1cm]{0cm}{.75cm}$\hphantom{xxx}$ }What is $\lim_{x \to +\infty} \frac{x^2 + 2}{x^3 + x^2 +x}$?
(a) 0(b) 1(c) $\frac 23$(d) $ \frac 12$(e) $+ \infty$(f) none of these.

\fbox{ \rule[-.1cm]{0cm}{.75cm}$\hphantom{xxx}$ }Find the linearization of f(x) = x3 -x at x=1.
(a) L(x) =2x(b) L(x)= 2(x+1) (c) L(x) = -2(x-1)+1(d) L(x) = 2x +1(e) L(x) = 2(x-1) (f) none of these.

\fbox{ \rule[-.1cm]{0cm}{.75cm}$\hphantom{xxx}$ }Use differentials to estimate the change in the surface area of a cube S = 6 x2 when the edge length goes from x0 to x0 + dx
(a) 6 dx(b) 6x0 dx(c) 12 x0 dx (d) 12 dx (e) 18 x0 dx(f) none of these.

\fbox{ \rule[-.1cm]{0cm}{.75cm}$\hphantom{xxx}$ }The formula for finding sucessive approximations in Newton's method is
(a) xn+1 = xn + f(xn) / f'(xn)(b) xn+1 = xn - f(xn)/f'(xn)(c) xn+1 = xn + f'(xn)/f(xn)(d) xn+1 = xn - f'(xn)/f(xn)(e) xn+1 =xn - f(xn) f'(xn)(f) none of these.

\fbox{ \rule[-.1cm]{0cm}{.75cm}$\hphantom{xxx}$ }The solution of the inital value problem $\frac{dy}{dx}= x+1$, y(2) =3 is
(a) y = x+1(b) y = x2 - x(c) $ y = \frac 12 x^2 + 2$(d) y = x2 + x + 1(e) $ y = \frac 12 x^2 + x$ (f) none of these.

\fbox{ \rule[-.1cm]{0cm}{.75cm}$\hphantom{xxx}$ } Suppose $f'(x) = x^2\sin^{10}(x)$. Then on the interval $[0, \frac 12 \pi]$ the function f is
(a) increasing and concave down (b) increasing and concave up (c) decreasing and concave down (d) decreasing and concave up (e) constant (f) none of these.

\fbox{ \rule[-.1cm]{0cm}{.75cm}$\hphantom{xxx}$ }The function f(x) = x3 -3x2 +1 has a point of inflection at x= ?
(a) -2 (b) -1 (c) 0 (d) 1 (e) 2 (f) none of these.

Find each of the following indefinite integrals
$\int x^3 -x^2 + 2 dx $, $\hphantom{xxxxxx}$

$\int \sin(3x) dx $, $\hphantom{xxxxxx}$
$\int \cos(3x +2) dx $, $\hphantom{xxxxxx}$
$ \int \sin^4(t) \cos(t) dt $, $\hphantom{xxxxxx}$
$\int t (t^2 +1)^{1/2} dt $, $\hphantom{xxxxxx}$

State the mean value theorem.

(5 pts) Suppose the second hand on a clock has length 20 cm. At what rate is the distance between the tip of second hand and the 12 o'clock mark changing when the second hand points to 3 o'clock?

Suppose it takes 2 hours to replace the drill bit while drilling for oil. A new drill bit digs quickly at first, but slows down with time. Suppose that in t hours it can drill though f(t) feet of rock.
Suppose the drill bit is used for T hours before being replaced. What is the average speed of drilling (including the 2 hours to install the bit)?
Show that to maximize this average speed the bit should should be replaced after T hours of use where T satisfies f'(T) = f(T)/(T+2).
If f(t) = 100t/(t+5) find this time T.

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Chris Bishop