Part II: The Answers
358
Answers
501–600
′
= −
( )

 

 
−




=
( )
f x
x
x
x
x
( )
sin
sin
2
2
2
2
1
1
1
554.
3
10
x x
+
Part of the fundamental theorem of calculus states that if the function g is continuous
on [a, b], then the function f defined by
f x
g t dt
a x b
a
x
( )
( )
(
)
=
≤ ≤
∫
where
is continuous on [a, b] and is differentiable on (a, b). Furthermore, f '(x) = g(x).
Note also that to find d
dx
g t dt
a
h x
( )
( )
∫
, you can use the substitution u = h(x) and then
apply the chain rule as follows:
d
dx
g t dt
d
dx
g t dt
d
du
g t dt du
dx
g u du
d
a
h x
a
u
a
u
( )
( )
( )
( )
( )
∫
∫
∫
=
=
(
)
= ′
(
) x x
g h x
du
dx
= ′ (
)
(
)
( )
All this tells you to substitute the upper limit of integration into the integrand and multiply by the derivative of the upper limit of integration. Therefore, the derivative of
f x
t t
dt
x
( ) =
+
∫
1
4
1
3
is
′
=
+ ( )
( )
=
+
= +
f x
x
x
x
x
x
x
x x
( )
1
3
3
3
3
3
4
2
2
3
1 2
10
555.
3
3
3
2
3
2
x
x
x
x
+
−
+
sec
tan
Part of the fundamental theorem of calculus states that if the function g is continuous
on [a, b], then the function f defined by
f x
g t dt
a x b
a
x
( )
( )
(
)
=
≤ ≤
∫
where
is continuous on [a, b] and is differentiable on (a, b). Furthermore, f '(x) = g(x).
To use the fundamental theorem of calculus, you need to have the variable in the
upper limit of integration. Therefore, to find the derivative of the function