Exponential functions?

Question

Let L be the tangent Line to g(x)=e^x at point (0,1)

a. Find the equation of the line, L

b. Let c be the x intercept of the line L: What is the distance between c and 0????


2. Now let T be the tangent line to g(x)=e^x at the point (a,e^a) where a is a real number:
a. Find the equation of the line T

b. Let c be the x intercept of the line T
What is the distance between c and a?????

I have the first one done I think the equation of L is y=x+1 and then distance is 1. Does that sound right

The second one is getting me.

Answer 1

For number 1,
a) y=x+1
b) 1

gotta go, update later

Answer 2

For the first, x-intercept is -1. Distance would be 1.

2. Derivative is e^x at a, e^a is the slope.

y - e^a = e^a(x - a)
y = e^ax + e^a(1 - a)

Set y = 0
-e^a(1 - a) = e^ax
-(1 - a) = x or x = a - 1

You're 2 points are (a-1, 0) and (-1, 0).
If you put it into the distance formula, you will get:

sqrt(((a-1)-(-1))^2+0) = sqrt(a^2) = a

Answer 3

y=e^x and y´=e^x
At (a,e^a) the tangent is
y-e^a= e^a(x-a) and the intercept is y=0
x=a-1
The distance between (a,e^a) and (a-1,0) is sqrt(1+e^2a)
Inthe case of a=0 this gives sqrt2 for distance