• 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?
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Lab 03 –Labs provide an opportunity to explore, observe, and record a synthesis of your observations. Often the synthesis will include statements from.
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• 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!
Applied Calculus, 3/E by Deborah Hughes-HalletCopyright 2006 by John Wiley & Sons. All rights reserved.
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.
Applied Calculus, 3/E by Deborah Hughes-HalletCopyright 2006 by John Wiley & Sons. All rights reserved.
Example: Instantaneous Velocity
Thus, .
Average velocity (ft/sec) 84 52 Instantaneous velocity: in between...
Applied Calculus, 3/E by Deborah Hughes-HalletCopyright 2006 by John Wiley & Sons. All rights reserved.
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
Applied Calculus, 3/E by Deborah Hughes-HalletCopyright 2006 by John Wiley & Sons. All rights reserved.
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.
Applied Calculus, 3/E by Deborah Hughes-HalletCopyright 2006 by John Wiley & Sons. All rights reserved.
Graph Exercise
What is ?
What is ?
What is the maximum of ? What is the minimum of ? Where is ?
Applied Calculus, 3/E by Deborah Hughes-HalletCopyright 2006 by John Wiley & Sons. All rights reserved.
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
Applied Calculus, 3/E by Deborah Hughes-HalletCopyright 2006 by John Wiley & Sons. All rights reserved.
Symbolic Exercise
Estimate f ′ (2)f x 5x
Applied Calculus, 3/E by Deborah Hughes-HalletCopyright 2006 by John Wiley & Sons. All rights reserved.
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.
Applied Calculus, 3/E by Deborah Hughes-HalletCopyright 2006 by John Wiley & Sons. All rights reserved.