If
𝑓
(
9
)
=
6
,
𝑓
′
is continuous, and
∫
3
9
𝑓
′
(
𝑡
)
𝑑
𝑡
=
4
, what is the value of
𝑓
(
3
)
?

Question

If 
𝑓
(
9
)
=
6
, 
𝑓
′
 is continuous, and 
∫
3
9
𝑓
′
(
𝑡
)
𝑑
𝑡
=
4
, what is the value of 
𝑓
(
3
)
?

BlackTom AI · Accepted Answer

By the Fundamental Theorem of Calculus, the integral of a derivative over an interval equals the net change of the original function on that interval.

1. Relevant knowledge points and concepts
- Fundamental Theorem of Calculus (Part 2): If f′ is continuous on [a, b], then ∫_a^b f′(t) dt = f(b) − f(a).
- Continuity of f′ ensures the theorem applies and the integral equals the change in f.

2. Complete reasoning process step by step
- Given ∫_3^9 f′(t) dt = 4 and f′ is continuous, apply the Fundamental Theorem of Calculus:
  f(9) − f(3) = ∫_3^9 f′(t) dt.
- Substitute the known integral value:
  f(9) − f(3) = 4.

3. Calculations with intermediate steps
- We are given f(9) = 6, so substitute:
  6 − f(3) = 4.
- Solve for f(3):
  6 − f(3) = 4
  => f(3) = 6 − 4
  => f(3) = 2.

4. Why the correct answer is correct
- The equality ∫_3^9 f′(t) dt = f(9) − f(3) is exact because f′ is continuous. Using the provided values leads directly to f(3) = 2.

5. Why other options would be incorrect
- A common mistake would be to add instead of subtracting (e.g., computing f(3) = f(9) + 4 = 10), which misapplies the net-change relation. Another error would be ignoring continuity and treating the integral differently; neither is justified here. Any answer other than 2 contradicts the direct algebraic result from the Fundamental Theorem.

6. Conclusion
- Therefore f(3) = 2, which matches the provided correct answer.

If 𝑓 ( 9 ) = 6 , 𝑓 ′ is continuous, and ∫ 3 9 𝑓 ′ ( 𝑡 ) 𝑑 𝑡 = 4 , what is the value of 𝑓 ( 3 ) ?单项选择题

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