A Quick Review of Math 200A Quick Review of Math 200

Calculus rests on 3 critical ideas/concepts:Calculus rests on 3 critical ideas/concepts:

Total accumulated change

Instantaneous rates of change

Limit

Rate of Change…Rate of Change…

Relates to slope in the real worldRelates to slope in the real world By using limit the “Rise/Run” becomes the By using limit the “Rise/Run” becomes the

Newton QuotientNewton Quotient And the rest…And the rest…

Max/MinMax/Min Related rates Related rates Differential EquationsDifferential Equations

Perhaps one of the most important graphs in history!

Slope is about 1.4 ppm/year

(1958,315)

Snapshot of Earth Breathing!Snapshot of Earth Breathing!

Example: a drug is released into a Example: a drug is released into a patients blood stream at a rate patients blood stream at a rate given bygiven by

g/minute. How much drug does the patient receive in 4 minutes?

Total Accumulated Change…Total Accumulated Change…

Relates to the idea of area in the real worldRelates to the idea of area in the real world

2( ) 4r t t t

Units are g/min

Working with Integrals…Working with Integrals…

Pay attention to 5.2 and 5.4 in textPay attention to 5.2 and 5.4 in text Substitution: simplest of the methods (Chp Substitution: simplest of the methods (Chp

7.1)7.1) Integration by parts (coming in Chp 7)Integration by parts (coming in Chp 7) etc etc

Which of the following are good Which of the following are good candidates for substitution?candidates for substitution?

x

A x dx

xB dx

x

C xe dx

D x x x dx2 3

) tan

sin)

cos

)

) ( 1)( )

Areas Between CurvesAreas Between Curves

Section 6.1Section 6.1

Some warm ups…Some warm ups…

What is the area between the x-axis and What is the area between the x-axis and the graph ofthe graph of

When is an area “equal to” an integral?When is an area “equal to” an integral?

3( )f x x x

What’s the area between the x-axis, the What’s the area between the x-axis, the lines x = 0 and x = 2 and the graph of the lines x = 0 and x = 2 and the graph of the function function

What’s the area between the y-axis and What’s the area between the y-axis and the above function (x = 0 and x = 2)?the above function (x = 0 and x = 2)?

What’s the area between the curves y = xWhat’s the area between the curves y = x33 and y = x?and y = x?

3( )f x x

Checklist of things to watch for when Checklist of things to watch for when finding areas between curves…finding areas between curves…

Can we develop a strategy?Can we develop a strategy?

Some examples…Some examples…

Pg 331: 68,71,72Pg 331: 68,71,72