At what point on the given curve is the tangent line parallel to the line 3x - y = 5?

Question

At what point on the given curve is the tangent line parallel to the line 3x - y = 5?

 

y = 4 + 2e^x − 7x

Answer 1

For the tangent line to be parallel to the line 3x - y = 5, the tangent line has to have the same slope.

 

The slope of 3x - y = 5 is 3, since we can rearrange the terms in the equation to the slope intercept form as y = 3x - 5.

 

Now to determine the line tangent to the curve, we first differentiate the curve

y = 4 + 2e^x - 7x

y' = 2e^x - 7

 

At what value of x will y' equal 3?

3 = 2e^x - 7

10 = 2e^x

5 = e^x

ln 5 = x

 

So at the point (ln5,3) the line tangent to the curve y = 4 + 2e^x - 7x is parallel to 3x - y = 5.