Winter wk 7 – Tues.15.Feb.05
Calc. Ch.6.1: Constructing Antiderivatives
• Antiderivatives = integrals
• Finding antiderivatives graphically
• Finding antiderivatives numerically
6.2: Finding antiderivatives analytically
Next week: 6.3: Introduction to differential equations
Energy Systems, EJZ
Ch.6.1: Antiderivatives graphically
Antiderivatives = integrals
If f=x2 then the derivative of f is
F=df/dx=____
If F=2x, then the antiderivative of F is
f=__
In other words, if F=f’=df/dx, then
dfF dx dx df f
dx
Visualizing Antiderivatives
Given a graph of the slope F=df/dx, sketch f.
Practice with conceptests.
Notice that the intercept is undetermined.
Practice with Ch.6.1 Exercises 1-4, 11-13
(p.265)
Conceptest 1
Conceptest 1 answer
Conceptest 2
Conceptest 2 answer
Conceptest 3
Conceptest 3 answer
Now practice with 6.1 Exercises 1-4
Fundamental theorem of calculus
Antiderivatives = integrals
Given the graph of the slope F=df/dx,
sketch f, showing all critical points (f’=0) and inflection points (where f”=0 or
f” changes sign: concavity of f changes)
( ) ( )b b
b
aa a
dfF dx dx f f b f a
dx
Find function f from slope f’
Practice with Ex. 9-13 p.265
Antiderivatives - numerically
How to find f from values of F=df/dx?
Recall that f = x df/dx.
Start at f0, and increase f step by step:
Calculate f for each x
Add f0+ f, etc. for each x
Tabulate f(x)
Conceptest 4
Conceptest 4 answer
Now practice with 6.1 Ex. 5,6 p.265
6.2 Antiderivatives - analytically
What is an antiderivative of f’(x) = 0?
What function f does not change?
What is an antiderivative of g’(x) = constant?
How does g compare to f above?
Write f and f’ for these
plots of f(x)
Finding f from df/dx
If df/dx=constant (f’=k) then
Recall polynomial rule:
dff dx k dx kx const
dx
1( )pp pdf d x
If f x then p xdx dx
1
1
nn n x
If g x then g dx x dx constn
Finding f from df/dx
_________
__________
1ln
1___________
ax
ax
dfIf f e then
dx
Find e dx
dUse x to find
dx x
dxx
Plot inverse functions
If y=log x, then If y=ln x, then
x=_________ x=__________
Sketch ex
Trigonometric functions
sin _________
cos _______
dIf x
dx
then x dx
cos _________
sin _______
dIf x
dx
then x dx
Practice 6.2 odd numbered problems p.271-272