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SAMPLE MIDTERM 2 MAT 141
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.
 1.
 Place the letter corresponding to the correct answer in the box
next to each question.
 (a)

What is the slope of the curve given by
x^{3} + y^{3}  9xy =0 at the
point
(x,y) = (2,4)?
(a) 1(b)
(c) (d)
(e)
(f) none of these.
 (b)

Suppose
f(x) =  x^{2} 2x. The set of critical points
of f is
(a)
(b) (c)
(d) (e) (f) none of these.
 (c)

Suppose
.
The absolute maximum
of g on
occurs at
(a) 0(b) (c) (d) (e) (f) none of these.
 (d)

What is
?
(a) 0(b) 1(c) (d) (e) (f) none of these.
 (e)

Find the linearization of
f(x) = x^{3} x at x=1.
(a) L(x) =2x(b)
L(x)= 2(x+1)
(c)
L(x) = 2(x1)+1(d)
L(x) = 2x +1(e)
L(x) = 2(x1)
(f) none of these.
 (f)

Use differentials to estimate the change in the surface area of
a cube S = 6 x^{2} when the edge length goes from x_{0} to
x_{0} + dx
(a) 6 dx(b) 6x_{0} dx(c) 12 x_{0} dx
(d) 12 dx
(e) 18 x_{0} dx(f) none of these.
 (g)

The formula for finding sucessive approximations in Newton's
method is
(a)
x_{n+1} = x_{n} + f(x_{n}) / f'(x_{n})(b)
x_{n+1} = x_{n}  f(x_{n})/f'(x_{n})(c)
x_{n+1} = x_{n} + f'(x_{n})/f(x_{n})(d)
x_{n+1} = x_{n}  f'(x_{n})/f(x_{n})(e)
x_{n+1} =x_{n}  f(x_{n}) f'(x_{n})(f) none of these.
 (h)

The solution of the inital value problem
,
y(2) =3 is
(a) y = x+1(b)
y = x^{2}  x(c)
(d)
y = x^{2} + x + 1(e)
(f) none of these.
 (i)

Suppose
.
Then on the interval
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.
 (j)

The function
f(x) = x^{3} 3x^{2} +1 has a point of inflection at
x= ?
(a) 2
(b) 1
(c) 0
(d) 1
(e) 2
(f) none of these.
 2.
 Find each of the following indefinite integrals
 (a)

,
 (b)

,
 (c)

,
 (d)

,
 (e)

,
 3.
 State the mean value theorem.
 4.
 (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?
 5.
 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.
 (a)
 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)?
 (b)
 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).
 (c)
 If
f(t) = 100t/(t+5) find this time T.
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Chris Bishop
19991108