Math 161 - Solutions To Sample Exam 2 Problems Worksheet Page 2
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8Solutions To Sample Exam 2 Problems – Math 161
2
Solution:
Using the Chain Rule, we have u (x) = f (g(x)) · g (x), which means that
u (6) = f (g(6)) · g (6)
= f (3) · g (6)
(since g(6) = 3)
1
3
=
(since f (3) = 1/2 and g (6) = 3/2)
·
2
2
3
=
.
4
Similarly, v (1) = g (f (1)) · f (1) = g (4)f (1), but because g has a sharp point at x = 4, we see that g (4)
does not exist and therefore:
v (1) does not exist.
Finally, we have
1
w (1) = f (f (1)) · f (1) = f (4) · f (1) =
· ( 4) =
2.
2
3. For each of the following functions f (x), use the provided formula for f (x) and f (x) to find: (i) the intervals
on which f (x) is increasing/decreasing, (ii) the intervals on which f (x) is concave up/concave down, (iii)
the local maximum and minimum values of f (x), and (iv) the inflection points of f (x).
2
(x
1)(x
70)
x
70
1
2
(a) f (x) =
x
71x + 70 ln x
Given: f (x) =
and
f (x) =
2
2
x
x
x
2
x
2
x
2
(b) f (x) = e
(31
4x
x
)
Given: f (x) = e
(x
+ 2x
35)
and
f (x) = e
(37
x
)
Solution:
1
2
(a) We first note that f (x) =
x
71x + 70 ln x is undefined if x ≤ 0, so we need only consider values of x
2
that are positive.
First Derivative Sign Chart. Looking at the first derivative, we see that f (1) = f (70) = 0, so 1 and
70 are our two critical points. By using test values, we obtain the sign chart shown below:
1
70
f (x)
+
+
We summarize the resulting information below:
Increasing: f is increasing on (0, 1) and on (70, ∞).
Decreasing: f is decreasing on (1, 70).
Local Maxima: f (1) ≈
70.5 is a local maximum.
Local Minima: f (70) ≈
2222.61 is a local minimum.
Second Derivative Sign Chart. Setting the second derivative equal to zero, we have
√
2
x
70 = 0
=⇒
x = ±
70,
√
and since f is only defined when x > 0, the number x =
70 is the only relevant place where f (x)
equals zero. Again using test values, we obtain the following sign chart:
√
70
f (x)
+
We summarize the resulting information below:
√
Concave Down: f is concave down on (0,
70).
√
Concave Up: f is concave up on (
70, ∞).
√
Inflection Points: f has an inflection point at x =
70, and its coordinates are approximately
(8.37, 410.33).
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