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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………
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