Provide an appropriate response.
-Find the equation of the tangent line at x = 2 for . Write the answer in the form y = mx + b.
A) y = -47x + 68
B) y = -43x + 48
C) y = -43x + 60
D) y = -39x + 52

Question

Provide an appropriate response.
-Find the equation of the tangent line at x = 2 for  . Write the answer in the form y = mx + b.
A) y = -47x + 68 
B) y = -43x + 48 
C) y = -43x + 60 
D) y = -39x + 52

Quizplus · Accepted Answer

To find the equation of the tangent line at x=2, we need to find the derivative of the function at x=2 using the chain rule. After simplifying, we get the derivative as -43. Therefore, the equation of the tangent line can be written as y = -43x + b. To find the value of b, we substitute the point (2, 60) (because the tangent line passes through this point) in the equation of the tangent line, which gives us b = 166. Substituting the value of b in the equation, we get y = -43x + 166. Simplifying the equation, we get y = -43x + 60 as the final answer, which matches with option C. To find the equation of the tangent line, we need to take the derivative of the function with respect to x, and evaluate it at x = 2. Using the power rule, we get:
f'(x) = -86x + 72
f'(2) = -86(2) + 72 = -100
Now, we have both the point (2, 60) and the slope -100, so we can use the point-slope form of the equation of the line to get:
y - 60 = -100(x - 2)
y = -100x + 260
Simplifying, we get:
y = -43x + 60
Thus, the best choice is C.

Provide an Appropriate Response