1
Lesson Plan #62
Class: AP Calculus
Date: Monday February 16th, 2011
1
Topic: Anti-derivative of y  .
x
Aim: How do we find the anti-derivative of
1
du
u
Objectives:
1) Students will be able to find the anti-derivative of
1
du
u
HW# 62:
Evaluate
1
x2
dx 4) 
3) 
dx
3x  2
3  x3
Note: We have formulas for the derivative of eu and ln u , the natural exponential function and the natural logarithmic
function, respectively. Let’s just state the formulas for the derivatives of a u and log a u the general exponential
10
1)  dx
x
1
dx
2) 
x5
function and the general logarithmic function, respectively
Theorem: For
1.
d
 a x   a x ln a
dx  
Theorem: For
1.
a  0 and a  1
2
d u
du
 a   a u ln a 
dx
dx
2.
d
1
du

log a u  
dx
u ln a dx
a  0 and a  1
d
1
log a x  
dx
x ln a
Do Now:
Differentiate
1) y  (ln x) 4
2) y  (log x) 4
3) y  8 x
Procedure:
Write the Aim and Do Now
Get students working!
Take attendance
Give back work
Go over the HW
Collect HW
Go over the Do Now
We have seen that
d
ln x  1 . Then going backwards, what must
dx
x
1
 x dx be?
2
1
1
 x dx  ln x  c  u du  ln u  c
Examples: Evaluate
1)
2
 x dx
6)
 u  3du
7)
 6 x 1dx
8)
 sin  d
9)
 x 1dx
2)
1
1
cos 
1
x
10)
 1 4x
11)
 5  2 sin x dx
2
cos x
dx
1
 3  2 xdx
3)
x
 x 2  1 dx
4)
x2  4
 x
5)
x
2
x3
 6x  7
3
12)
e
x
 1  2e
x
dx
On Your Own
Exercises: Evaluate
sin x
1)
 1  3 cos x dx
2)
 1  3u du
2
e x  e x
3)  x
dx
e  ex
x
4)
 1 4x
5)

6)
9 z
7)
 x ln x
2
dx
2x 1
dx
2x
z
dx
2
dz
4
1
8)

9)
 1  cos xdx
10)
1
x 3 1  x 3 


2
dx
sin x

sec x tan x
dx
sec x 1
Sample Test Questions:
2)