AP Calculus BC
Friday, 05 February 2016
•
OBJECTIVE TSW (1) explore properties of inverse
•
ASSIGNMENTS DUE
trigonometric functions, (2) differentiate inverse trig functions,
and (3) review basic differentiation rules.
– WS Bases Other Than e  to the left of the wire basket
– WS Differential Equations: Growth and Decay
– WS Newton’s Law of Cooling  black tray
 wire basket
•
QUIZ: Other Bases; Growth and Decay will be given
after the lesson.
•
TEST: Transcendental Functions – Differentiation and
Integration will be on Tuesday, 09 February 2016.
•
2ND PERIOD: For those participating in the Economics
Fair next Friday, try to come for the lesson next Friday
during 1st period.
Inverse Trigonometric
Functions – Integration
Inverse Trigonometric Functions
– Integration
Inverse Trigonometric Functions
– Integration
Ex:
Evaluate
Let a = 5, u = x
du = dx

dx
25  x
2

du
a2  u 2
u
 arcsin  C
a
x
 arcsin  C
5
Inverse Trigonometric Functions
– Integration
Ex:
Evaluate
Let a = 2, u = x
du = dx

dx
4x
2

du
a2  u 2
u
 arcsin  C
a
x
 arcsin  C
2
Inverse Trigonometric Functions
– Integration
Inverse Trigonometric Functions
– Integration
Ex:
1
du
dx
  2
Evaluate: 
2
2 a  u2
3  4x
Let a  3 , u = 2x
1 1
u
  arctan  C
2 a
a
du = 2 dx
½ du = dx
1
2x

arctan
C
2 3
3
Inverse Trigonometric Functions
– Integration
Inverse Trigonometric Functions
– Integration
Ex:
Evaluate 
Let a = 4, u = 7x
dx
x 49 x 2  16
1 7  du


2
2
1
7
u
u

a
 

du
u u 2  a2
1/7 u = x
du = 7 dx
1/7 du = dx
u
1
 arcsec  C
a
a
7x
1
 arcsec
C
4
4
Inverse Trigonometric Functions
– Integration
Ex:
Evaluate
Let a = 1, u = e x
du = e x dx
du / e x = dx
du / u = dx

dx
e
2x
1

du
u u 2  a2
u
1
 arcsec  C
a
a
 arcsec e x  C
AP Calculus BC
Monday, 08 February 2016
•
OBJECTIVE TSW (1) finish exploring integration
•
ASSIGNMENTS DUE TOMORROW
of trigonometric functions, and (2) review for
tomorrow’s test covering differentiation and
integration of transcendental functions.
–
–
•
WS Inverse Trigonometric Functions:
Differentiation
WS Inverse Trigonometric Functions: Integration
TEST: Transcendental Functions –
Differentiation and Integration is tomorrow,
Tuesday, 09 February 2016.
Inverse Trigonometric Functions
– Integration
Ex:

x2
4x

Evaluate
2
dx  
Let u = 4 – x 2
du = –2x dx
–1/2 du = x dx
x
4  x2
x2
4x
dx  
2
dx
2
4  x2
dx
1 du
x
 
 2arcsin
2 u
2
1
x
12
   2  u  2arcsin  C
2
2
x
  4  x  2arcsin  C
2
2
Inverse Trigonometric Functions
– Integration
Ex:
3
3
x
2
Evaluate
 x 2  4 dx
The numerator's degree is greater – use long
division to separate.
Inverse Trigonometric Functions
– Integration
12 x  2
3x  2
x 4
x 2  0x  4 3x 3  0x 2  0x  2
3
2
3 x  0 x  12x
12 x  2
Inverse Trigonometric Functions
– Integration
Ex:
3
3
x
2
Evaluate
 x 2  4 dx
12 x  2 

   3x  2
dx

x 4 

12x
2 

   3x  2
 2
dx

x 4 x 4

3 2
2x
1
 x  6 2
dx  2 2
dx
2
x 4
x 4
3 2
x
2
 x  6ln x  4  arctan  C
2
2


Inverse Trigonometric Functions
– Integration
Completing the Square
Ex:
Evaluate
dx
 x 2  4x  7



dx
2
x  4x  4  7  4

dx
 x  2
2
3
1
x 2

arctan
C
3
3
Inverse Trigonometric Functions
– Integration
Completing the Square
dx
Ex: Evaluate  2
2 x  8 x  10
dx

2 x 2  4 x  4  10  8



dx
2  x  2  2
2
dx
1

 arctan  x  2   C
2
2  x  2   1 2


Inverse Trigonometric Functions
– Integration
Completing the Square
Ex: Find the area bounded by the graph of
3
the x-axis, and the lines x  and x 
2
94
dx
A
32
3x  x 2
94
dx

32
 x 2  3x  9 4  9 4



94
dx
32
9 4   x  3 2
2
1
3x  x 2
9
.
4

A
6
94

x  3 2
 arcsin

3
2

3 2