State Standard

– 15.0b Students use the fundamental theorem of calculus to interpret integrals as antidervivatives.

– 14.0 Students apply the definition of the integral to model problems of physics, economics, and so forth obtaining results in terms of integrals.

Objective – To be able to use the 2nd derivative test to find concavity and points of inflection.

Theorem: G(x) = F(x) + C

Remember:

5 0d

dx 0

dc

dx

3 3d

xdx

dkx k

dx

4 34dx x

dx

1n ndx nx

dx

2

( )( ) ( )( )d f g f f g

dx g g

( ) ( )( ) ( )( )df g f g g f

dx

Remember:

cos sind

x xdx

sin cosd

x xdx

sec sec tand

x x xdx

2tan secd

x xdx

2cot cscd

x xdx

csc csc cotd

x x xdx

2

( )( ) ( )( )d f g f f g

dx g g

( ) ( )( ) ( )( )df g f g g f

dx

Example 1

Find the general solution fo the differential equation 2y

2y x C

2y x

To begin, you need to find a function whose derivative is 2.

Now we can apply the Theorem

Differentiation Formula Integration Formula

0dx C 0dc

dx

kdx kx C dkx k

dx

1

1

nn xx dx C

n

1n nd

x nxdx

( ) ( ) ( ) ( )f x g x dx f x dx g x dx

Differentiation Formula Integration Formula

cos sinxdx x C

sin cosxdx x C 2sec tanxdx x C

cos sind

x xdx

sin cosd

x xdx

sec sec tand

x x xdx

2tan secd

x xdx

2cot cscd

x xdx

csc csc cotd

x x xdx

sec tan secx xdx x C

csc cot cscx xdx x C 2csc cotxdx x C

2x dx2 1

3

xc

3dx0 1

31

xc

3x c 3

3

xc

2

1dx

x2x dx

2 1

1

xc

1c

x

23 2 2x x dx 23 2 2x dx xdx dx 3 2 13 2 2

3 2 1

x x xc

3 2 2x x x c

3cos xdx 3sin x c 25sec xdx 5 tan x c

4sin csc cotx x x dx4sin csc cotxdx x xdx

4cos cscx x c

2dxx

1

22x dx

1

122

12

xc

1

24x c

22 1t dt 4 22 1t t dt 4 22 1t dt t dt dt

5 3 12

5 3 1

t t tc

5 32

5 3

t tt c

3

2

3xdx

x

3

2 2

3xdx

x x

23x x dx 23xdx x dx 2 13

2 1

x xc

2 3

2

xc

x

Pg. 358 – 359

# 9 – 13 odd, 19, 23, 25,

and

33 – 39 odd