Transcript Calculus Ch.6.1-6.2

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
df
 F dx   dx dx   df  f
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
b
b
df
a F dx  a dx dx  f
b
a
 f (b)  f (a )
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)
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 Df = Dx df/dx.
Start at f0, and increase f step by step:
Calculate Df for each Dx
Add f0+ Df, etc. for each Dx
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
df
f   dx   k dx  kx  const
dx
Recall polynomial rule:
p
df
d
(
x
)
p
If f  x then

 p x p 1
dx
dx
n 1
x
If g  x n then  g dx   x n dx 
 const
n 1
Finding f from df/dx
If f  e
ax
df
then
 _________
dx
Find  e dx  __________
ax
d
1
Use  ln x   to find
dx
x
1
 x dx  ___________
Plot inverse functions
If y=log x, then
x=_________
If y=ln x, then
x=__________
Sketch ex
Trigonometric functions
d
If
sin x  _________
dx
d
If
cos x  _________
dx
then  cos x dx  _______
then  sin x dx  _______
Practice 6.2 odd numbered problems p.271-272