1. Homework, Page 148 Use the Product Rule to find the derivative.
Rogawski Calculus
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2. Homework, Page 148 Use the Product Rule to find the derivative.
Rogawski Calculus
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3. Homework, Page 148 Use the Quotient Rule to find the derivative.
Rogawski Calculus
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4. Homework, Page 148
Calculate the derivative in two ways. First use the Product or Quotient Rule, then rewrite and apply the Power Rule.
Rogawski Calculus
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5. Homework, Page 148
Calculate the derivative using the appropriate rule(s).
Rogawski Calculus
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6. Homework, Page 148
Calculate the derivative using the appropriate rule(s).
Rogawski Calculus
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7. Homework, Page 148
Calculate the derivative using the appropriate rule(s).
Rogawski Calculus
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8. Homework, Page 148
Calculate the derivative using the appropriate rule(s).
Rogawski Calculus
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9. Homework, Page 148
Calculate the derivative using the appropriate rule(s).
Rogawski Calculus
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10. Homework, Page 148
Calculate the derivative using the appropriate rule(s).
Rogawski Calculus
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11. Homework, Page 148 Rogawski Calculus
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12. Homework, Page 148 Rogawski Calculus
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13. Chapter 3: Differentiation Section 3.4: Rates of Change
Jon Rogawski Calculus, ET First Edition
Chapter 3: Differentiation
Section 3.4: Rates of Change
Rogawski Calculus
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14. The average rate of change (ROC) of a function over an interval is:
The instantaneous ROC of a function at a point is:
We must realize that the average ROC of a function is the slope of
the secant line and the instantaneous ROC is the slope of the
tangent line.
Rogawski Calculus
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15. Figure 1 illustrates measuring the average ROC of f (x) over (x0, x1),
while Figure 2 illustrates measuring the instantaneous ROC at x = x0.
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16. Example, Page 158
2. Find the ROC of the volume of a cube with respect to the length of its side s when s = 3 and s = 5.
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17. Example, Page 158
2. Find the ROC of the volume of a cube with respect to the length of its side s when s = 3 and s = 5.
Rogawski Calculus
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18. Table 1 illustrates some of the data used to construct the graph in
Figure 3. The slope of the secant line in Figure 3 gives the average
ROC of the temperature from midnight to 12:28PM on July 6, 1997.
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19. slope of the tangent to the area curve increases.
Notice that as r increases, the area of the blue band increases, and the
slope of the tangent to the area curve increases.
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20. Example, Page 158
8. Calculate the ROC dA/dD, where A is the surface area of a sphere of diameter D.
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21. Example, Page 158
8. Calculate the ROC dA/dD, where A is the surface area of a sphere of diameter D.
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22. Rogawski Calculus
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23. Marginal cost, in economics, gives the cost of delivering one more
unit. Figure 5 shows how the cost of a flight is less per passenger,
the more passengers there are. Thus, the marginal cost of flying
each additional passenger decreases as the number of passengers
increases.
Rogawski Calculus
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24. From the graph, when is the car moving toward its destination?
When is the car standing still?
When is the car going towards is origin?
Rogawski Calculus
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25. From the graph, when is the car moving toward its destination?
When is the car standing still?
When is the car going towards is origin?
Rogawski Calculus
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26. (b) Is the truck speeding up or slowing down?
A truck enters a freeway off-ramp at t = 0. Its position after t seconds
is s(t) = 84t – t3 ft for 0 ≤ t ≤ 5.
(a) How fast is the truck traveling at the moment it enters the off-ramp?
(b) Is the truck speeding up or slowing down?
Rogawski Calculus
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27. (b) Is the truck speeding up or slowing down?
A truck enters a freeway off-ramp at t = 0. Its position after t seconds
is s(t) = 84t – t3 ft for 0 ≤ t ≤ 5.
(a) How fast is the truck traveling at the moment it enters the off-ramp?
(b) Is the truck speeding up or slowing down?
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

28. Galileo used a device such as in Figure 9 to investigate the motion of
objects moving solely under the influence of gravity. From his
experiments, he deduced that, in the absence of air resistance, the
velocity of a falling object is proportional to the time it has been falling.
Rogawski Calculus
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29. The equations below may be used to determine the vertical position
and velocity of an object, absent air resistance, under the influence
of gravity alone. For the equations, s0 is initial height, v0 is initial
velocity, and t is time.
Rogawski Calculus
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30. Example
A baseball is tossed vertically upward with an initial velocity of 85 ft/s from the top of a 65-ft high building. a. What is the height of the ball after 0.25 s? b. Find the velocity of the ball after 1 s. c. When does the ball hit the ground?
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

31. Example
A baseball is tossed vertically upward with an initial velocity of 85 ft/s from the top of a 65-ft high building. a. What is the height of the ball after 0.25 s? b. Find the velocity of the ball after 1 s. c. When does the ball hit the ground?
Rogawski Calculus
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32. Example, Page 158
18. The escape velocity at a distance r meters from the center of the Earth is vesc = (2.82 x 107)r –½ m/s. Assuming the radius of the Earth is 6.77 x 106 m, calculate the rate at which vesc changes with respect to distance at the Earth’s surface.
Rogawski Calculus
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33. Example, Page 158
18. The escape velocity at a distance r meters from the center of the Earth is vesc = (2.82 x 107)r –½ m/s. Assuming the radius of the Earth is 6.77 x 106 m, calculate the rate at which vesc changes with respect to distance at the Earth’s surface.
Rogawski Calculus
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34. WS 3.4.pdf Rogawski Calculus
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