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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
x3 + y3 - 9xy =0 at the
point
(x,y) = (2,4)?
(a) 1(b)
(c)
(d)
(e)
(f) none of these.
- (b)
-
Suppose
f(x) = | x2 -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) = 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.
- (f)
-
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.
- (g)
-
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.
- (h)
-
The solution of the inital value problem
,
y(2) =3 is
(a) y = x+1(b)
y = x2 - x(c)
(d)
y = x2 + 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) = x3 -3x2 +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)
-
,
![$\hphantom{xxxxxx}$](img27.gif)
- 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
1999-11-08