Writing equations for lines...

Question

Write an equation for the line tangent to the curve f(x) = 4e^x - 7 at x = ln 3

 

Could someone please explain how I might go about answering this question? Thank you!

Answer 1

To find the equation of a line tangent to a curve, you need to find the slope of the line, and the point at which it touches the curve.  Once you have those two pieces of information, you can use the usual techniques from algebra to find the equation of the line.

 

First, find the derivative of the function: 

f(x) = 4e^x - 7

f'(x) = 4e^x

 

When x = ln 3, the value of the function is 4e^{ln 3} - 7 = 4*3 - 7 = 5

 

...and the value of the derivative is 4e^{ln 3} = 4*3 = 12.

 

So the tangent line touches the curve at the point (ln 3, 5).  And its slope is 12.

 

So the equation of the line with have the form y = 12x + b.

 

Substitute ln 3 for x and 5 for y, to find b:

5 = 12 ln 3 + b

5 - 12 ln3 = b

 

So the equation is y = 12x + 5 - 12 ln 3