题目
MAT135H5_F25_ALL SECTIONS 4.2 Preparation Check
多重下拉选择题
Suppose you are asked to find the Linear Approximation of 𝑓 ( 𝑥 ) = tan 𝑥 at 𝑥 = 𝜋 4 . Remember that the Linear Approximation is given by: 𝐿 ( 𝑥 ) = 𝑓 ( 𝑎 ) + 𝑓 ′ ( 𝑎 ) ( 𝑥 − 𝑎 ) (a) What is 𝑓 ( 𝜋 4 ) ? [ Select ] 2/sqrt(3) 1 -1 0 1/sqrt(2) (b) What is 𝑓 ′ ( 𝑥 ) ? [ Select ] f'(x)= sec^2x f'(x)= tan x f'(x) = sec x tan x f'(x) = sec x f'(x) = tan^2 x (c) What is 𝑓 ′ ( 𝜋 4 ) ? [ Select ] 1 2 sqrt(2) 0 4 1/sqrt(2)
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标准答案
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思路分析
Let’s break down each part separately and examine the options one by one so the reasoning is clear and reasons for each choice are explicit.
(a) What is f(π/4) for f(x) = tan x?
- Option: 2/sqrt(3) ≈ 1.15. This would correspond to tan(π/3) or a related angle, but at π/4 the tangent value is not this number.
- Option: 1. This is the actual value of tan(π/4), since tan(π/4) = sin(π/4)/cos(π/4) = (√2/2)/(√2/2) = 1.
- Option: -1. This would be the tangent value at an angle in a quadrant where both sine and cosine patterns yield a negative ratio, but π/4 lies in the first quadrant where tan is pos......Login to view full explanation登录即可查看完整答案
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