Find the equation of the line tangent to the graph of f at the indicated value of x.
-f(x) = 8 + ln x; x = 1
A) y = x + 9
B) y = x - 7
C) y = x + 7
D) y = x - 9

Question

Quizplus · Accepted Answer

To find the equation of the tangent line, we need to first find the slope of the tangent line at x = 1.

The slope of the tangent line is equal to the derivative of the function at x = 1:

f'(x) = 1/x

f'(1) = 1/1 = 1

So, the slope of the tangent line at x = 1 is 1.

Now, we can use the point-slope form of a line to find the equation of the tangent line:

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

y - (8 + ln 1) = 1(x - 1)

y - 8 = x - 1

y = x + 7

Therefore, the equation of the tangent line at x = 1 is y = x + 7, so the correct answer is C. To find the equation of the tangent line, we need to find the slope of the curve at x=1, which is given by the derivative of f(x):

f'(x) = 1/x

f'(1) = 1/1 = 1

So the slope of the tangent line is 1. We also know that the line passes through the point (1, f(1)) = (1, 8+ln(1)) = (1,8). So we can use the point-slope form of the equation of a line:

y - 8 = 1(x - 1)

Simplifying, we get:

y - 8 = x - 1

y = x + 7

Therefore, the equation of the tangent line is y = x + 7, which corresponds to choice C.

Find the Equation of the Line Tangent to the Graph