If

𝑓
(
𝑥
)
=
∫
0
𝑥
(
3
𝑡
3
+
7
𝑡
2
+
2
𝑡
)
d
𝑡
,

then which of the following is
𝑓
″
(
𝑥
)
?

Question

If

𝑓
(
𝑥
)
=
∫
0
𝑥
(
3
𝑡
3
+
7
𝑡
2
+
2
𝑡
)
d
𝑡
,

then which of the following is 
𝑓
″
(
𝑥
)
?

BlackTom AI · Accepted Answer

By the Fundamental Theorem of Calculus (Part 2), the derivative of an integral with a variable upper limit equals the integrand evaluated at that upper limit. Then differentiate that result to get the second derivative.

1. Relevant knowledge points and concepts
- Fundamental Theorem of Calculus, Part 2: If F(x) = ∫_a^x g(t) dt then F'(x) = g(x).
- Basic rules of differentiation for polynomials: d/dx[x^n] = n x^(n-1) and linearity of the derivative.

2. Complete reasoning process step by step
- Given f(x) = ∫_0^x (3 t^3 + 7 t^2 + 2 t) dt.
- Apply the Fundamental Theorem: f'(x) = 3 x^3 + 7 x^2 + 2 x.
- Differentiate f'(x) to obtain f''(x).

3. Calculations with intermediate steps
- f'(x) = 3x^3 + 7x^2 + 2x.
- Compute each derivative term:
  - d/dx[3x^3] = 3 * 3 x^2 = 9 x^2.
  - d/dx[7x^2] = 7 * 2 x = 14 x.
  - d/dx[2x] = 2.
- Sum the results: f''(x) = 9x^2 + 14x + 2.

4. Why the correct answer is correct
- The second derivative is simply the derivative of the integrand evaluated at x (first derivative) and then differentiated once more. The polynomial differentiation above yields 9x^2 + 14x + 2 exactly, so that expression is the correct f''(x).

5. Why the other options are incorrect
- 18x + 12 and 18x + 14 are linear expressions; they would result from incorrectly differentiating a quadratic as if it were 9x^2 + 6x or similar—these do not match the true second derivative which is quadratic.
- 9x^2 + 12x + 2 has the wrong coefficient on the x term (12 instead of the correct 14), indicating a miscalculation of d/dx[7x^2].
- Only 9x^2 + 14x + 2 matches the correct term-by-term differentiation.

6. Conclusion
The correct second derivative is f''(x) = 9x^2 + 14x + 2, which matches the provided answer.

If 𝑓 ( 𝑥 ) = ∫ 0 𝑥 ( 3 𝑡 3 + 7 𝑡 2 + 2 𝑡 ) d 𝑡 , then which of the following is 𝑓 ″ ( 𝑥 ) ?单项选择题

类似问题

If 𝑓 ( 𝑥 ) = ∫ 0 𝑥 ( 7 𝑡 3 + 5 𝑡 2 − 8 𝑡 ) d 𝑡 , then which of the following is 𝑓 ″ ( 𝑥 ) ?

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

If f(x)=∫ x 0 (4t3−3t2+3t)dt, then which of the following is f″(x)?

如果 \（f（16） = 14\，\（f'\）连续，且 \（\int _{0}^{16} f' （t） \， dt = 17\，那么 \（f（0）\）的值是多少？

If 𝑓 ( − 1 ) = − 2 , 𝑓 ′ is continuous, and ∫ − 1 13 𝑓 ′ ( 𝑡 ) 𝑑 𝑡 = − 5 , what is the value of 𝑓 ( 13 ) ?

If 𝑓 ( 16 ) = 14 , 𝑓 ′ is continuous, and ∫ 0 16 𝑓 ′ ( 𝑡 ) 𝑑 𝑡 = 17 , what is the value of 𝑓 ( 0 ) ?

Find the derivative.

Let [math: F(x)=∫03xet2 dt]\displaystyle F(x)=\int _0^{3x}{e^{t^2} dt}. Compute [math: F′(0.5)]F’(0.5). (Correct the answer to 2 decimal places.)

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单