Chapter 6

AP Calculus BC

6.1 Slope Fields and Euler’s MethodEuler’s Method the chart…

Orig. Pt. dx dy/dx dy New pt.( , )o ox y 0.1 given ( )dydx

dx( , )o ox dx y dy

( , )o ox dx y dy Continue………

Do an example with the chart……..Slope Fields: draw it with the derivative, you see the original(antiderivative) in the graph.

WS Packet….Differential EquationInitial Value Condition

+C

6.1 cont’d. & 6.2 AntiderivativesGeneral Antiderivatives: ( ) ( )f x dx F x C

Integral Formulas:1

1nn xx dx Cn

lndx x Cx

kxkx ee dx Ck

Basic Trig Integrals

2

2

cossin

sincos

sec tan

csc cot

sec tan sec

csc cot csc

kxkxdx Ckkxkxdx Ck

xdx x C

xdx x C

x xdx x C

x xdx x C

Properties p.332

6.2 cont’d.Examples:

5

3

1

cos2

x

x dx

dxxe dxxdx

If given an Initial condition (point on graph)

FIND CIntegration by substitution…..

1

1nn uu dx Cn

Examples:

5

2

( 3)

4 1

cos(7 5)

1cos 2tan

x dx

x dx

x dx

dxx

xdx

2 3

4

( 2 3) ( 1)

sin cos

x x x dx

x xdx

6.2 cont’d.Substitution in Definite Integrals….

( )

( )( ( ( ))( '( )) ( )b g b

a g af g x g x dx f u du

240

tan secx xdx Separable differential equations:

( ) ( )dy g x h ydx

1 ( )( )dy g x dx

h y

222 (1 ) xdy x y edx

Either always change values or always don’t … pick a way and stick with it……

6.3 Integration by PartsIntegration by parts formula: u dv uv v du What to make u ?

LIATE

LogarithmInv. Trig…AlgebraicTrigonometryExponential

Examples:2

cos

cos

x

x

x xdx

x e dx

e xdx

Tabular Integration – use when u = algebraic…….

Examples: 3

4sin

xx e dx

x xdx

6.4 Exponential Growth/DecaySeparable Differential Equations ( ) ( )dy f y g x

dx

Separate the variables, integrate, then solve for y.2( )

(1,1)

dy xydx

Law of Exponential change – If y changes at a rate proportional to the amount present

dy kydt

And if when t = 0 then0y y 0kty y e k is the growth constant

or decay constant.Money Formulas

0

0

(1 )nt

rt

rA A n

A A e

Radioactive Decaydy kydt

Half-life = ln2

k

Newton’s Law of Cooling….

0( ) ktS ST T T T e

6.5 Logistic Growth

Partial Fractions:2

4

2

132 7 13

3 11

x dxx x

x dxx

Logistic Differential Equation (carrying capacity = M)

( )dP kP M Pdt

Fastest rate of growth is at half of carrying capacity.

.0003 (1000 )

(0) 61

dP P Pdt

P

Example: long way 1st

Shortcut is equation on page 367………