Provide an appropriate response.
-Find the linearization of $f ( x ) = 8 + \int _ { 1 } ^ { x + 1 } \tan \frac { \pi t } { 4 }$ dt at $x = 0$.
A) $\sqrt { 2 } x + 8$
B) $8 x + 1$
C) $x + 8$
D) 8

Question

Quizplus · Accepted Answer

To find the linearization of the function at x = 0, we first need to evaluate the function and its derivative at x = 0:
f(0) = 8 + ∫₁¹ tan(πt/4) dt = 8
f'(x) = tan(π(x+1)/4)
f'(0) = tan(π/4) = 1
The linearization of the function is then given by:
L(x) = f(0) + f'(0)(x-0) = 8 + x
Therefore, the best choice is C, since it matches the form of the linearization.

Provide an Appropriate Response f(x)=8+∫1x+1tan⁡πt4f ( x ) = 8 + \int _ { 1 } ^ { x + 1 } \tan \frac { \pi t } { 4 }f(x)=8+∫1x+1​tan4πt​

Provide an Appropriate Response $f ( x ) = 8 + \int _ { 1 } ^ { x + 1 } \tan \frac { \pi t } { 4 }$