Math 132
Name:
Quiz on 5.1
1. If f ( x ) is given below, write out the sign chart for f ( x ).
6
4
2
1
1
2
3
4
5
6
Solution: The function is decreasing everywhere, except at x = 3,
where the tangent is horizontal. This yields the following sign chart:
1
0
Math 132
Name:
Quiz on 3.1, ctd.
1. Let f ( x ) =
(a)
(b)
x2 + x 6
. Compute the following.
x+3
lim f ( x )
x 3
lim f ( x )
x 3+
(c) lim f ( x )
x 3
Solution: The answers are all 5, since
( x 2)( x + 3)
x2 + x 6
=
= x2
x+3
x+3
whenever x = 3. Therefore f
Math 132
Name:
Quiz on 3.7
The total prot (in dollars) from the sale of x lawn mowers is
P( x ) = 30x 0.03x2 750
0 x 1, 000.
1. Find the average prot per mower if 50 mowers are produced.
Solution: The average prot is
P(50)
$675
=
= $13.50.
50
50
2. Find t
Math 132
Name:
Quiz on 4.1
A note will pay $20,000 at maturity 10 years from now. How much should
you be willing to pay for the note now if money is worth 5.2% compounded
continuously?
Solution: In the formula A = Pert , we have A = 20000, r = .052, and
t
Math 132
Name:
Quiz on 3.5
Compute the following.
1.
d
dx
x5
.
25
Solution: The derivative is
2.
x4
.
5
dy
if y = 2 + 5t 8t3 .
dt
Solution: The derivative is 5 24t2 .
3. w if w =
7
.
5u2
Solution: We rst rewrite w as 7 u2 . Differentiating, we get
5
which
Math 132
Name:
Quiz on 3.4
Let y = f ( x ) = x2 + x, with graph given below.
10
8
6
4
2
4
2
2
4
2
1. Find the slope of the secant line joining (1, f (1) and (3, f (3).
Solution: The slope is given by
f (3) f (1)
12 2
=
= 5.
31
2
2. Find the slope of the s
Math 132
Name:
Quiz on 3.1
4
2
2
1
1
2
3
4
5
2
Let y = g( x ) be given graphically above. Compute the following. Remember to answer in complete sentences!
1. lim g( x )
x 1
Solution: The limit from the left is 1.
2. lim g( x )
x 1+
Solution: The limit fro
Math 132
Name:
Quiz on 3.3
1. Determine where G ( x ) =
1 x2
is continuous.
x2 + 1
Solution: Rational functions are continuous wherever the denominator is nonzero. We solve x2 + 1 = 0, and nd that there are no
solutions (x2 is never negative, so when we a
Math 132
Name:
Quiz on 4.2
1. If f ( x ) = ln( x8 ), nd f ( x ).
Solution: We have ln( x8 ) = 8 ln x, so f ( x ) = 8
2. If y = log2 x, nd
dy
.
dx
Solution: We have log2 x =
ln x
, so
ln 2
dy
11
1
=
=
.
dx
ln 2 x
x ln 2
1
8
=.
x
x
Math 132
Name:
Quiz on 4
Math 132
Name:
Quiz on 4.3
1. If f ( x ) = x2 e x , what is f ( x )?
Solution: It is 2xe x + x2 e x = xe x (2 + x ).
2. If f ( x ) =
3x + 5
, what is f ( x )?
x2 3
Solution: It is
3x2 9 6x2 10x
3( x 2 3) (3 x + 5)2 x
=
( x 2 3)2
( x 2 3)2
=
3. If f ( x )
Math 132
Name:
Quiz on 5.4
Sketch the graph of y = (3 x )e x . Determine roots, intercepts, intervals of
increase/decrease, concavity, extrema, and inection points.
Solution: The graph passes through the points (0, 3) and (3, 0). It is
positive on (3, ) a
Math 132
Name:
Quiz on 5.6
Find two numbers whose difference is 15 and whose product is a minimum.
Solution: Let x, y be our numbers, and P their product. Then we have
x y = 15 and P = xy. We would like to minimize P, but as there are 2
variables, we cant
Math 132
Name:
Quiz on 5.2
1. Find the x and y coordinates of the inection points of y = x4 + 6x2 .
Solution: Differentiating, we obtain
y = 4x3 + 12x
and so
y = 12x2 + 12.
To nd the inection points, we determine where y is discontinuous
or zero. Since y
Math 132
Name:
Quiz on 5.1
1. Find and classify the local extrema of f ( x ) = 3x4 4x3 + 5.
Solution: The derivative is
f ( x ) = 12x3 12x2 .
It is always continuous, so we nd its roots:
12x3 12x2 = 12x2 ( x 1) = 0
which implies x = 0 or 1. We now constru
Math 132
Name:
Quiz on 4.4
1. If f ( x ) = (9 5x )2 , nd f ( x ).
Solution: The inside function is 9 5x, which has deriviative 5.
Therefore
f ( x ) = 5 2(9 5x ) = 10(9 5x ) = 50x 90.
One could also expand out the original function and then take the
deriva