The tangent line to the graph of f (x) at is shown. On the tangent line, P is the point of tangency and A is another point on the line. Find .
A) 2
B) -2
C) 1
D) -5
E) does not exist

Question

The tangent line to the graph of f (x) at  is shown. On the tangent line, P is the point of tangency and A is another point on the line. Find  . ​  ​
A) 2 
B) -2 
C) 1 
D) -5 
E) does not exist

Quizplus · Accepted Answer

Since the point of tangency P on the tangent line has coordinates (1,3) and the slope of the tangent line is -3, we can find the equation of the tangent line:

y - 3 = (-3)(x - 1)

y = -3x + 6

To find point A on the line, we just need to plug in a value for x. Since we are looking for a point with integer coordinates, we can try x = 0:

y = -3(0) + 6 = 6

Therefore, point A has coordinates (0,6).

Finally, we can find the value of f(0) by plugging in x=0 into the original function.

f(0) = 3(0) + 5 = 5

Thus, the slope of the secant line passing through (0,f(0)) and (1,3) is:

(f(0) - 3)/(0 - 1) = (5 - 3)/(-1) = -2

Since the slope of the tangent line is -3, which is not equal to -2, the limit does not exist.

Therefore, the best choice is A) 2, which gives us the coordinates of point A on the tangent line, but does not attempt to evaluate the limit.

The Tangent Line to the Graph of F (X) at