tangent line and derivatives

Question

The function in the figure below has f (7)= 27 and f '(7)=2.2

Find the coordinates of the points A, B, C.

 

The figure has a curve with a tangent line. A B and C are on the tangent line.

A is at 7 on the x-axis and is the point that touches both the curve and the tangent line.

B is at 7.2

And C is at 6.9

 

How do I go about solving this problem? Can you tell me even though the graph is not available?

 

 

Added info: The hint that I am given is "The derivative at point A can be interpreted as the slope of the tangent to the curve at A. The points B and C are on the tangent line to the curve at A"

Answer 1

The equation of a line is y = mx + b where m is the slope and b is the y-intercept.  The derivative of f at x=A is the slope of the tangent line: f '(A) = 2.2 = m.  So the equation of the tangent line so far is y = 2.2x + b.  To find b, you need to plug in the x and y-values of the point A = (7,27) and solve for b:

 

     y = 2.2x + b

     27 = 2.2*7 + b

     27 = 15.4 + b

     11.6 = b

 

So the equation of the tangent line is y = 2.2x + 11.6.  To find the coordinates of point B, plug the given x-value for B (7.2) into the equation for the tangent line and solve for y:

 

     y = 2.2x + 11.6

     y = 2.2*7.2 + 11.6

     y = 27.44

 

B is located at (7.2, 27.4).  Do the same for point C.