“THE MOST MISSED”1. Average rate of change does not mean average value

– Average rate of change deals with derivatives• It is the slope between the two endpoints of an

interval– Average value deals with integrals

b

a

dxxfab

valueavg )(1 .

“THE MOST MISSED”2. FTC (Part 1):

– The derivative of an integral with a lower bound that is a constant and an upper bound that is a function of a variable is the integrand of the upper bound times the derivative of the upper bound.

24

3

2 8)34ln()3ln(x

Cxxdttdxd

= 8x ln(4x2 + 3) + C

“THE MOST MISSED”3. A particle is at rest parametrically when both dx/dt and dy/dt

equal 0. (Think of it as neither coordinate x or y can be moving.)

23 3ttx ttty 1232 23 The position of a particle moving in the xy-plane is given by the parametric equations and For what values of t is the particle at rest?

“THE MOST MISSED”4. Approximating an integral if you cannot find the

antiderivative.

If the function f is defined by and g is an antiderivative of f such that g(3) = 5, then g(1) =

2)( 3 xxf

Use Euler’s formula (program) in your calculator but make the “step size” small for accuracy.

Section 7.1 INTEGRAL AS NET CHANGE

ftmin

minutes

A honey bee makes several trips from the hive to a flower garden. The velocity graph is shown below.

What is the total distance traveled by the bee?

200ft

200ft

200ft

100ft

200 200 200 100 700 700 feet

ftmin

minutes

What is the displacement of the bee?

200ft

-200ft

200ft

-100ft

200 200 200 100 100

100 feet towards the hive

To find the displacement (position shift) from the velocity function, we just integrate the function. The negative areas below the x-axis subtract from the total displacement.

Displacementb

aV t dt

Distance Traveledb

aV t dt

To find distance traveled we have to use absolute value.

Find the roots of the velocity equation and integrate in pieces, just like when we found the area between a curve and the x-axis. (Take the absolute value of each integral.)

Or you can use your calculator to integrate the absolute value of the velocity function.

velocity graph

position graph

1

2

12

12

Displacement:

1 11 2 12 2

Distance Traveled:

1 11 2 42 2

Every AP exam I have seen has had at least one problem requiring students to interpret velocity and position graphs.

Example 5: National Potato Consumption

The rate of potato consumption for a particular country was:

2.2 1.1tC t

where t is the number of years since 1970 and C is in millions of bushels per year.

For a small , the rate of consumption is constant.t

The amount consumed during that short time is . C t t

Example 5: National Potato Consumption

2.2 1.1tC t

The amount consumed during that short time is . C t t

We add up all these small amounts to get the total consumption:

total consumption C t dt

4

22.2 1.1tdt

4

2

12.2 1.1ln1.1

tt

From the beginning of 1972 to the end of 1973:

7.066 million bushels

Work:

work force distance

Calculating the work is easy when the force and distance are constant.

When the amount of force varies, we get to use calculus!

Hooke’s law for springs: F kx

x = distance that the spring is extended beyond its natural length

k = spring constant

Hooke’s law for springs: F kx

Example 7:

It takes 10 Newtons to stretch a spring 2 meters beyond its natural length.

F=10 N

x=2 M

10 2k

5 k 5F x

How much work is done stretching the spring to 4 meters beyond its natural length?

F(x)

x=4 M

How much work is done stretching the spring to 4 meters beyond its natural length?

For a very small change in x, the force is constant.

dw F x dx

5 dw x dx

5 dw x dx 4

05 W x dx

42

0

52

W x

40W newton-meters

40W joules

5F x x