Which of the following is the equation of the line that is tangent to the curve
𝑦
=
4
𝑥
cos
⁡
(
𝑥
)
at the point
(
3
𝜋
2
,
0
)
?

Question

Which of the following is the equation of the line that is tangent to the curve
𝑦
=
4
𝑥
cos
⁡
(
𝑥
)
at the point 
(
3
𝜋
2
,
0
)
?

BlackTom AI · Accepted Answer

d/dx(4x cos x) = 4 cos x − 4x sin x.

1) Relevant knowledge points and concepts
- Product rule: d(uv)/dx = u'v + uv'. Here u(x)=4x and v(x)=cos x.
- Derivative values of trig functions: d/dx(cos x) = −sin x.
- Values of trig functions at special angles: cos(3π/2)=0, sin(3π/2)=−1.
- Point-slope form of a tangent line: y − y0 = m(x − x0).

2) Complete reasoning process step by step
- Compute y' using the product rule for y = 4x cos x.
- Evaluate y' at x = 3π/2 to find the slope of the tangent at the given point.
- Use the point (3π/2, 0) and the slope in point-slope form, then simplify to slope-intercept form.

3) Calculations with intermediate steps
- u(x)=4x ⇒ u'(x)=4. v(x)=cos x ⇒ v'(x)=−sin x.
- y' = u'v + u v' = 4·cos x + 4x·(−sin x) = 4 cos x − 4x sin x.
- Evaluate at x = 3π/2:
  cos(3π/2) = 0, sin(3π/2) = −1.
  y'(3π/2) = 4·0 − 4·(3π/2)·(−1) = 0 + 4·(3π/2) = (4·3/2)π = 6π.
- Slope m = 6π. Use point-slope: y − 0 = 6π(x − 3π/2).
  Expand: y = 6πx − 6π·(3π/2) = 6πx − (18/2)π^2 = 6πx − 9π^2.

4) Why the correct answer is correct
The derivative evaluation gives the unique slope of the tangent at x = 3π/2 as 6π and using the exact point (3π/2, 0) yields the line y = 6πx − 9π^2. Algebraic simplification confirms the constant term is −9π^2.

5) Why other options would be incorrect
Common mistakes: omitting the factor 4 from the derivative, forgetting the negative sign from d(cos x), mis-evaluating sin or cos at 3π/2, or arithmetic errors when multiplying 6π by 3π/2 (which would produce an incorrect constant). Any line with a different slope or intercept does not satisfy both the derivative value and the point, so such choices are wrong.

6) Conclusion
The tangent line at (3π/2, 0) is y = 6πx − 9π^2, which matches the provided answer.

Which of the following is the equation of the line that is tangent to the curve 𝑦 = 4 𝑥 cos ⁡ ( 𝑥 ) at the point ( 3 𝜋 2 , 0 ) ?单项选择题

类似问题

Which of the following is the equation of the line that is tangent to the curve 𝑦 = 5 𝑥 sin ⁡ ( 𝑥 ) at the point ( − 𝜋 , 0 ) ?

Find an equation of the tangent line to the graph of at the point .

Suppose L is the tangent line of y = 3 tan ⁡ x − 2 csc ⁡ x at x = π 3 . Where does L intersect the x -axis?

Using your answer to the previous question, find an equation of the tangent line of 𝑓 ( 𝑥 ) = 𝑥 − 1 at 𝑥 = 5 . Write your answer in the form 𝑦 = 𝑚 𝑥 + 𝑏 .

If you write out the equation of the tangent line of function f(x) at x=2, what are you writing?

Question text 2Marks What is the value of [math: x] where the tangent line to the curve [math: x2+y2+xy=3]x^2+y^2+xy=3 at the point [math: (1,1)] intersects the the [math: x]-axis (i.e. where [math: y=0])? Answer [math: x=] Answer 1[input].Notes Report question issue Question 27 Notes

When evaluating a new project, firms should include in the projected cash flows all of the following EXCEPT:

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