Math 116 - Lab 6 - Fall 2011.
Second derivatives; concavity; curve sketching; optimization problems (Sections 4.3, 4.4,
4.6 and 4.7 of L/G).
You are to provide full solutions to the following problems. You are allowed to collaborate with your
classmates, use your notes and textbook and ask the TA for guidance. Direct copying of solutions
is not encouraged, nor is it allowed or ethical.
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Math 116 - Lab 6 - Fall 2011.
Student number:
Second derivatives; concavity; curve sketching; optimization problems (Sections 4.3, 4.4, 4.6
and 4.7 of L/G).
1. In each of the following, calculate
f
00
(
x
0
) and determine whether the graph of
f
(
x
) is
concave up or concave down at
x
0
:
a)
f
(
x
) = sin(2
x
);
x
0
=
π/
4 ,
b)
f
(
x
) =
e
-
5
x
;
x
0
= 1 ,
c)
f
(
x
) = ln(5
x
);
x
0
= 12 .
2. In each of the following, find the second order critical points of
f
(
x
) and determine
which ones (if any) are inflection points:
a)
f
(
x
) =
x
ln(
x
) ,
b)
f
(
x
) = cosh(
x
) ,
c)
f
(
x
) = sinh(
x
) ,
d)
f
(
x
) =
1
1+
e
-
x
.
3. For what values of
a
and
b
is the graph of
f
(
x
) =
ax
+
e
-
bx
always concave up?
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- Spring '11
- RobertAndre
- Calculus, Critical Point, Derivative, order critical points
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