Exponential Function - The
Number ‘e’
so
e1 = 2.718
and is irrational
so far so boring
What is e?
the letter after d.
Complete the table, draw the graph y = ex
x
-3
-2
-1
0
1
2
3
y
0.049
0.135
0.368
1
2.718
7.389
20.09
  3.141592654.....
e  2.718281828.....
The exciting bit of e
y = ex dy/dx = ex
So that is why ex is exciting
Examples
y = x3 + 3ex
dy/ = 3x2 + 3ex
dx
y = e5x – 4e3x
dy/ = 5e5x – 12e3x
dx
y = e2x – 3
e3x
y = e2x – 3e-3x
dy/ = 2e2x +9e-3x
dx
Differentiate ex
Formula
y = eax
dy/ = aeax
dx
Differentiate the following
y=
e5x
e -x
3e2x
x3 – e3x
2x5 – 2e-3x
3/ + ex
x
dy/
dx
=
5e5x
-e-x
6e2x
3x2-3e3x
10x4+6e-3x
-3x-2+ex
y=
ex+4
e7x - 5
e8-3x
7(ex+2)
(ex+2)(ex+4)
(e3x-1)(e4x+3)
dy/
dx
=
ex+4
7e7x-5
-3e8-3x
7ex
2e2x + 6ex
7e7x +9e3x-4e4x
Integrating ex
e
x
e c
x
Examples:
Integrate ex
1 3x
e  3e c
3x
x
2
 4e
2 x
1 3
2 x
 x  2e  c
3
Formula
1 ax
e  a e c
ax
Integrate the following
ʃdx
e5x
e -x
3e2x
x3 – e3x
2x5 – 2e-3x
3/ + ex
x2
y=
1/5e5x
-e-x
3/2 e2x
1/
4x
4-1/ e3x
3
1/ x6+2/ e-3x
3
3
-3x-1+ex
ʃdx
y=
ex+4
ex+4
e7x - 5
1/7e7x-5
e8-3x
-1/3e8-3x
7ex + 14x
7(ex+2)
1/ e2x + 6ex + 8x
(ex+2)(ex+4)
2
(e3x-1)(e4x+3) 1/7e7x +e3x-1/4e4x-3x
So there's this function party going on and all of the hippest functions are there
having a good time. Cos(x) is getting down and dirty with ln(x), 10x is making the
moves on the (x - 2) 2 twins and even x! dragged himself out for a good time.
Unfortunately, one little function just wasn't having her night. ex was sitting in a
chair in the corner, tears dripping down exponentially. Sin(x) catches her out of
the corner of his eye and decides to cheer her up. "Come on ex! There's no
reason to sit all on your lonesome crying. Get out into the crowd and have some
fun. You know, INTEGRATE a little bit."
ex looks at Sin(x) with the saddest eyes and exclaims, "But it won't make any
difference!"