Exercises
UF Calculus Set 27
1. For the polynomials below, calculate the intervals of increase/decrease and concavity. Use
these, the intercepts, and end behavior to sketch their graphs.
(a)
f
(
x
) =
x
4
+ 2
x
3
(b)
f
(
x
) =
x
(3

x
)
4
(c)
f
(
x
) = (1

x
2
)
5
Count the number of turning points and inflection points, and consider how this relates to
the multiplicity of the roots to
f
0
and
f
00
for polynomials.
2. Graph the functions below. In the process, consider each of the following details to use in
your graph.
(1) Domain;
x, y
intercepts, when calculable
(2) interval(s) on which the function increases/decreases
(3) interval(s) on which the function is concave up/down
(4) critical numbers, local extrema, inflection points
(5) asymptotes, local behavior at discontinuities/endpoints of domain
(a)
f
(
x
) = 2
x
3
+
6
x
(b)
f
(
x
) =
20
x
4
x
2
+ 1
(c)
f
(
x
) =
x
+ 1
x
2
(d)
f
(
x
) =
ln
x
x
2
(e)
f
(
x
) =
x
√
15

x
(f)
f
(
x
) =
x
2
√
15

x
(g)
f
(
x
) =
x
1
/
3
(6

x
)
(h)
f
(
x
) =
4
x
2
/
3
4 +
x
(i)
f
(
x
) =
x
2
+ 8 ln(
x
)

10
x
(j)
f
(
x
) =
√
x
ln(
x
)
(k)
f
(
x
) =
xe
1

x
2
(l)
f
(
x
) =
e
x
1

x
(m)
f
(
x
) =
e

1
/x
(n)
f
(
x
) = 2
e

x
+
e
x/
4
3. Graph the functions below. In the process, consider the details as in Exercise 2 to use in your
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