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Mathematics
Mathematics
Problem
For
x
≥
0
x \ge 0
x
≥
0
, let
f
(
t
)
=
ln
(
t
2
+
1
)
f(t)=\ln\bigl(t^{2}+1\bigr)
f
(
t
)
=
ln
(
t
2
+
1
)
. Define the function
g
(
x
)
=
∫
0
x
2
f
(
t
)
d
t
g(x)=\int_{0}^{x^{2}} f(t)\,dt
g
(
x
)
=
∫
0
x
2
f
(
t
)
d
t
. (a) Find a simplified expression for
g
′
(
x
)
g'(x)
g
′
(
x
)
. (b) Evaluate
g
′
(
3
)
g'(3)
g
′
(
3
)
. (c) Write the equation of the line tangent to the graph of
y
=
g
(
x
)
y=g(x)
y
=
g
(
x
)
at
x
=
3
x=3
x
=
3
.
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