Suppose the line tangent to the graph of f at x=3 is y=6x+4, and suppose y = 4x-5 is the line tangent to the graph of g at x = 3.

Question

Find the line tangent to the following curves at x = 3.

y = f(x)g(x)

Answer 1

Hi Steven,

 

This is a basic application of the product rule: (fg)' = fg' + f'g, which will give us the slope of the tangent line of fg at 3:

 

though we don't know f(x), we can find f(3) from the tangent line because they intersect at x = 3:

 

f(3) = 6*3 + 4 = 22

f'(3) = 6 (the slope of the tangent line)

g(3) = 4*3 - 5 = 7

g'(3) = 4

 

This gives (fg)' = 22*4 + 6*7 = 130

fg(3) = f(3)*g(3)

 

And that's all you need to get the formula for the line tangent to fg at 3 (use point slope formula for a line).