Find the equation of tangent line when f(x)= 1 + lnx^4 at x=e

Question

 This is for calculus and I am confused on what to do

Answer 1

The tangent line has the following properties - 

 

1.) It intersects the curve at the point of tangency

2.) It has the same slope as the curve at the point of tangency 

 

So first, find the point of tangency by evaluating f(e) = 1 + lne^4 = 1 + 4 = 5.  So the point of tangency is (e,5) 

 

Next, take the derivative of f to find the slope of f at x = e

 

f'(x) = (1/x^4) * 4x^3 = 4/x  

 

So the slope at x = e is f'(e) = 4/e 

 

So the tangent line has slope 4/e and goes through the point (e,5) 

 

y = (4/e)x + b

5 = (4/e)e + b

5 = 4 + b

b = 1

 

So the equation of the tangent line is y = (4/e)x + 1