At what point on the curve y = 7 + 2ex − 4x is the tangent line parallel to the line 4x − y = 4?

Question

At what point on the curve y = 7 + 2e^x − 4x

 is the tangent line parallel to the line 4x − y = 4?

(x,y)=(___, ___,)

Answer 1

For this question, we are trying to find the tangent line of a curve, which means we will need to take a derivative. The slope of this tangent line should be equal to the slope of the line in the second equation in order for them to be parallel. Rearranging the second equation to slope-intercept form, we get y = 4x-4, thus the slope is 4. Now we take the derivative of the first equation and set it equal to 4, then solve:

y = 7 + 2e^x − 4x

4 = y' = 2e^x - 4

(for information on taking the derivative of e^x go to: https://www.wyzant.com/resources/lessons/math/calculus/derivative_proofs/e_to_the_x)

2e^x = 8

e^x = 4

ln (e^x) = ln (4)

x ln (e) = ln (4)

x = ln (4)

To find y, we simply plug into the original equation:

y = 7 + 2e^ln(4) - 4 ln(4)

y = 7 + 8 - 4 ln(4) = 15 - 4 ln (4)

Thus (x,y) = (ln 4, 15 - 4 ln 4) = (2 ln 2, 15 - 8 ln 2)

Answer 2

Simon,

"Parallel" means the lines have the same slope.

So just find where the derivative of the first curve is equal to the derivative of the second equation.