Created by

Jason L. Bradbury

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.

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

nxC

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

2 1 1 1 0 1

3 2 23 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

12

212

xc

1

24x c

22 1t dt 4 22 1t t dt 4 22 1t dt t dt dt 4 1 2 1 0 1

2 15 3 1

t t tc

5 31 2

5 3t t t c

3

2

3xdx

x

3

2 2

3xdx

x x

23x x dx 23xdx x dx 1 1 2 1

32 1

x xc

21 3

2x c

x

xe dxxe c

Here are some examples from the homework

7x dx7 1

8

xc

2)

8

8

xc

Here are some examples from the homework

3 2x dx2

13

53

xc

16)

5

33

5

xc

2

3x dx3 23

5

x xc

3 53

5

xc

Here are some examples from the homework

5xe dx

5u due

28)

1

5ue du

5u x

51

5xe c

1

5ue c

5du dx5

dudx