Question

The tangent line to y=f(x) at (1,1) passes through the point (−9,−1). Compute f(1)=_________

A) 0
B) 1
C) 2
D) 3

Answer 1

To find f(1), we need to determine the equation of the tangent line to y=f(x) at (1,1) using the given point (-9,-1). The answer is none of the above because we do not have enough information to find the exact value of f(1).

Step-by-step explanation:

To find the value of f(1), we need to determine the equation of the tangent line to y=f(x) at (1,1) using the given point (-9,-1). Since the tangent line passes through (1,1), the slope of the line is equal to the derivative of f(x) evaluated at x=1. Let's call this derivative as f'(x). To find f(1), we need to find the equation of the tangent line and substitute x=1 into that equation to get the corresponding y-value.

Using the point-slope form of a linear equation, we have:

(y - y1) = m(x - x1)

Since the point (1,1) lies on the tangent line, we substitute x1=1 and y1=1 into the equation:

(y - 1) = f'(1)(x - 1)

Substituting the given point (-9,-1) into the equation, we can solve for f'(1):

(-1 - 1) = f'(1)(-9 - 1)

Simplifying the equation, we get:

-2 = -10f'(1)

Solving for f'(1), we find:

f'(1) = 1/5

To find f(1), we can integrate f'(x) to get f(x) and substitute x=1 into the antiderivative of f'(x). However, since we are not given the function f(x), we cannot find the exact value of f(1). Therefore, the answer is none of the above.