If
𝑓
(
4
)
=
8
,
𝑓
′
(
𝑥
)
is continuous, and
∫
4
10
𝑓
′
(
𝑥
)
𝑑
𝑥
=
−
1
, what is
𝑓
(
10
)
?

Question

If 
𝑓
(
4
)
=
8
, 
𝑓
′
(
𝑥
)
 is continuous, and 
∫
4
10
𝑓
′
(
𝑥
)
𝑑
𝑥
=
−
1
, what is 
𝑓
(
10
)
?

BlackTom AI · Accepted Answer

To tackle this problem, we start by connecting the integral of a derivative to the original function using the Fundamental Theorem of Calculus.
Since f' is continuous, the integral from 4 to 10 of f'(x) dx equals f(10) − f(4).
We are given that ∫ from 4 to 10 f'(x) dx = −1, so f(10) − f(4) = −1.
Next, we use the provided value f(4) = 8 to substitute into the equation: f(10) − 8 = −1.
Solving for f(10) involves adding 8 to both sides, yielding f(10) = 8 + (−1) = 7.
Thus, the relationship between the function values at the endpoints and the definite integral has been used to determine f(10) from the given data.
It is important to note that the continuity of f' ensures the Fundamental Theorem applies, linking the accumulated change in f over [4, 10] directly to the integral of f'.

MATH-112-301-001 Unproctored Midcourse Exam 4 Practice Exam 1

If 𝑓 ( 4 ) = 8 , 𝑓 ′ ( 𝑥 ) is continuous, and ∫ 4 10 𝑓 ′ ( 𝑥 ) 𝑑 𝑥 = − 1 , what is 𝑓 ( 10 ) ?

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