lab 03 –labs provide an opportunity to explore, observe, and record a synthesis of your...

• Lab 03– Labs provide an opportunity to explore, observe, and record a synthesis of your

observations. Often the synthesis will include statements from the text or lecture expressed in your own words. The synthesis should always express a deep understanding of the concepts being explored. The Abstract should include this synthesis.

– For Lab03, did you discover that the cross section graphs are “side views” and contour graphs are “top down views” of the 3D plot?

– Did you discover that the transformation x->x+a shifts the 3D plot along the x-axis, the transformation y->y+a shifts the 3D plot along the y-axis, the transformation f->f+a shifts the 3D plot along the z-axis, the transformation x->ax stretches/compresses the 3D plot along the x-axis, the transformation y->ay stretches/compresses the 3D plot along the y-axis, and/or the transformation f->af stretches/compresses the 3D plot along the z-axis? Did you discover that the y-axis cross sections of f(x,y) are stretches/compressions of each other for different values of x?

Applied Calculus, 3/E by Deborah Hughes-HalletCopyright 2006 by John Wiley & Sons. All rights reserved.

Section 2.1

Instantaneous Rate of Change

Today…

We begin Calculus!


Average Rate of Change

Average rate of change is a difference quotient.

If y = f (x)

Average rate of change of y

between x a and x b

y

x

f b f a b a

Instantaneous Rate of Change

Compute average rates of change (as difference quotients) over smaller and smaller intervals.


Example: Instantaneous Velocity

Thus, .

Average velocity (ft/sec) 84 52 Instantaneous velocity: in between...


Tabular Exercise

Data: Percent of households in the US with cable television as a function of years since Jan. 1, 1990.

Estimate f ′ (6)between 1.65 and 0.35

Interpret f ′ (6)annual increase in percent of households with cable on Jan. 1, 1996

Estimate and .between 1.25 and 0.95; between 0.2 and 0.55

Interpret the above informationthe percent of households with cable has been increasingbut at a decreasing rate


Box on page 14 and problem 11

Average Rate of Change Graphically

Average rate of change of y

between x a and x b

y

x

f b f a b a

Instantaneous Rate of Change Graphically

Compute average rates of change over smaller and smaller intervals.


Graph Exercise

What is ?

What is ?

What is the maximum of ? What is the minimum of ? Where is ?


Average Rate of Change Graphically

How would you write an expression that represents the slope of a line drawn between the two points marked in Figure 2.5?

𝑓 (𝑥+∆ 𝑥 )− 𝑓 (𝑥)∆ 𝑥


Symbolic Exercise

Estimate f ′ (2)f x 5x


Verbal Exercise

• Suppose is my car odometer reading in miles at time hours after the start of a trip.

• What does mean?– Two hours after the start of my trip, my odometer reads 10,435 miles.

• What does mean?– Two hours after the start of my trip, my speedometer reads 35 miles per

hour.

• Suppose is the number of liters of water in a container days after installation.

• What does mean?– Exactly 12 days after installation, the container held 3 liters of water.

• What does mean?– Exactly 12 days after installation, the container is leaking water at a rate

of 0.3 liters per day.


Handout

Start now.

Finish for homework.

lab 03 –labs provide an opportunity to explore, observe, and record a synthesis of your...

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