Part II: The Answers
354
Answers
501–600
546.
1 4
+ 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).
To find the derivative of the function f x
t dt
x
( ) =
+
∫ 1 4
0
, simply substitute the upper
limit of integration, x, into the integrand:
′
= +
f x
x
( )
1 4
547.
(
)
2
6 4
+ x
To find the derivative of the function f x
t
dt
x
( )
(
)
=
+
∫ 2
6 4
3
, simply substitute the upper
limit of integration, x, into the integrand:
′
= +
f x
x
( ) (
)
2
6 4
548.
x
3
cos x
To find the derivative of the function f x
t
t dt
x
( )
cos( )
= ∫
3
0
, simply substitute the upper
limit of integration, x, into the integrand:
′
=
f x
x
x
( )
cos
3
549.
−e
x
2
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
f x
e dt
t
x
( ) = ∫
2
4
, first flip the limits of integration and change the sign:
f x
e dt
e dt
t
x
t
x
( ) =
= −
∫
∫
2
2
4
4
Then substitute the upper limit of integration into the integrand to find the derivative:
′
= −
= −
f x
e
e
x
x
( )
(
)
2
2