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