AP Calculus Unit 5 Day 10

Practice: Given f(x) = x2 – 4

1) Use LRam with 4 equal partitions to estimate the integral from [-4, 4]

2) Use Trapezoidal Rule for same.3) Find using FTC4) Find total area between the curve and the x-

axis.

Part 1: More Practice with yesterday

Practice

5223. sec

xd tdtdx

53

0

2. 3 1xd t t dt

dx

5

4. lnxed tdt

dx

Practice1. co coss

xd tdtdx

x

5

2 2 5 42

3 sec. ec 10s 2xd tdt

dxx x

3 35

0

2. 3 (5 ) 3(5 ) 1 51xd t t x xdt

dx

5

4 ln. lnx

x xe

xd edtd

xtx

e e

2x

a

F x t dt

Is F(x) concave up or down at x = 3?

Applying Concepts . . .

FTC (Antiderivative Part) States:

( )x

a

d dF x f t dt f xdx dx

The interpretation is that the instantaneous rate of change of this accumulating area is in fact the y-value of the curve at the given instant.

From a Real-World Application (KNOW this!!!)—Test NEXT Week

1. An oil rig is spilling oil into the water at the RATE given by r tbarrels/ hour.

0

2. R .x

x r t dtThe total oil spilled in x hours is given by

Similar to the velocity of an object (rate of change of position)

Similar to “accumulated” displacement

The point …..

( )b

a

r t dt

Given a rate of change function r(t)

Then can be interpreted as an accumulated amount

This is important to understand!!!!

Study the next two slides.

Real Life Applications--Business

Rate of Change Function

Real Life MeaningInterpret

C’(t) Business Application:“Marginal Cost” to

produce t units of an item Total cost of producing “x” units

S’(t) Business Application:Rate at which sales increase where t is measured in days

“Accumulated” (total) sales for “x” days

rate of change functionb

a

dt

0

C'(t)x

dt

“Marginal Cost” change in total cost that arises when the quantity produced changes by one unit

0

S'(t)x

dt

Accumulation ExampleRate of Change

FunctionReal Life Meaning

Interpret

N’(t)Industry

Rate at which pollutants enter a lake, measured in

pounds per monthTotal number of

pounds of pollutants that enter the lake

over a period of “x” months

rate of change functionb

a

dt

0

N'(t)x

dt

The rate at which pollutants enter a lake from a factory is where N is the total number of pounds of pollutants in the lake at time t. How much pollutant enters the lake in 16 months? What is the average rate pollutants enter the lake over the 16 month time period? Include units.

3/2'( ) 280N t t

Calc. Active:

Accumulation Example ANSWERS

Rate of Change Function

Real Life MeaningInterpret

N’(t)Industry

Rate at which pollutants enter a lake, measured in pounds per month

Total number of pounds of pollutants

that enter the lake over a period of “x”

months

rate of change functionb

a

dt

0

N'(t)x

dt

The rate at which pollutants enter a lake from a factory is where N is the total number of pounds of pollutants in the lake at time t. How much pollutant enters the lake in 16 months?

What is the average rate pollutants enter the lake over the 16 month time period?

3/2'( ) 280N t t

163/2

0

280 114,688t dt pounds

163/2

0

1 280 7,16816 0

t dt pounds per month

Part 2: Average Value Theorem

Finding the Average y-Value of a Function

Let T=f(t) represent temperature at time t, in hours.The temperature is recorded for a 24-hr time period.Below is a graph of the collected temperatures:

The “average temperature” could be found for a given 24-hr time period by using the readings at 4-hr intervals.

(0) (4) (8) (12) (16) (20)6average

f f f f f fT

If it is a hot summer day and at 2:00 in the afternoon (hour 14) there is a short thunderstorm that cools the air for an hour between readings. This average temperature would not reflect this dip in temperature.

Taking more readings at shorter time intervals would result in a better average value for T.

Average Value Theorem: If a function is integrable on [a,b], then the function’s average y-value is:

averagechange in temperatureT

change in time

Find the average value