1. Homework Homework Assignment #4 Read Section 5.5
Page 335, Exercises: 1 – 49(EOO)
Rogawski Calculus
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2. Homework, Page 335
1. Write the area function of f (x) = 2x + 4 with lower limit a = –2 as an integral and find a formula for it.
Rogawski Calculus
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3. Homework, Page 335 Rogawski Calculus
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4. Homework, Page 335
Find formulas for the functions represented by the integrals.
Rogawski Calculus
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5. Homework, Page 335
Find formulas for the functions represented by the integrals.
Rogawski Calculus
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6. Homework, Page 335
Express the antiderivative F(x) of f (x) satisfying the given initial condition as an integral.
Rogawski Calculus
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7. Homework, Page 335 Calculate the derivative. Rogawski Calculus
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8. Homework, Page 335
25. Make a rough sketch of the graph of the area function of g(x) shown in Figure 12.
Rogawski Calculus
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9. Homework, Page 335 Calculate the derivative. Rogawski Calculus
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10. Homework, Page 335 Calculate the derivative. Rogawski Calculus
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11. Homework, Page 335 37. Find the min and max of A(x) on [0, 6].
Rogawski Calculus
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12. Homework, Page 335
41. Area Functions and Concavity. Explain why the following statements are true. Assume f (x) is differentiable. (a) If c is an inflection point of A(x), then f ′(c) = 0. Since f(x) = A′(x), f ′(x) =A″(x) and A″(x) = 0 at inflections points of A(x). (b) A(x) is concave up if f (x) is increasing. If f(x) is increasing, then f ′(x) > 0, hence A″(x) > 0 and A(x) is concave up. (c) A(x) is concave down if f (x) is decreasing. If f(x) is decreasing, then f ′(x) < 0, hence A″(x) < 0 and A(x) is concave up.
Rogawski Calculus
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13. Homework, Page 335
45. Sketch the graph of an increasing function f (x) such that both f ′(x) and A (x) are decreasing.
Rogawski Calculus
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14. Homework, Page 335
49. Determine the function g (x) and all values of c such that
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

15. Calculus, ET First Edition
Jon Rogawski
Calculus, ET First Edition
Chapter 5: The Integral
Section 5.5: Net or Total Change as the Integral of a Rate
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

16. If water enters an empty bucket at the rate r(t), as shown in Figure 1,
then the amount of water in the bucket at any time on the interval [0, 4]
is equal to the area under the curve up to the vertical line at the time
in question.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

17. the signed areas and give us the net change in s. This is the basis of
If s′(t) is both positive and negative on [t1, t2], the integral will total
the signed areas and give us the net change in s. This is the basis of
Theorem 1.
Rogawski Calculus
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18. Example, Page 341
4. A survey shows that a mayoral candidate is gaining votes at the rate of 2000t votes per day since she announced her candidacy. How many supporters will the candidate have after 60 days, assuming she had no supporters at t = 0.
Rogawski Calculus
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19. The table shows the flow rate of cars passing an observation point in
units of cars per hour. Estimate the number of cars using the highway
in the two hour period by taking the average of the left and right
Riemann sums.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

20. Remembering that displacement is a vector quantity, the integral of
velocity, another vector quantity, over an interval of time gives us the
net displacement. To find distance traveled, we must take the integral
of the absolute value of velocity, as illustrated in Figure 2.
Figure 3 illustrates
the path of the
particle with velocity
v(t) from Figure 2.
Rogawski Calculus
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21. Theorem 2 formalizes what we observed in Figure 2.
Rogawski Calculus
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22. Example, Page 341
8. A projectile is released with initial vertical velocity 100 m/s. Use the formula v(t) = 100 – 9.8t for velocity to determine distance traveled in the first 15 seconds.
Rogawski Calculus
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23. Example, Page 341
18. Suppose that the marginal cost of producing x video recorders is 0.001x2 – 0.6x dollars. What is the cost of producing 300 units? If production is set at 300 units, what is the cost of producing 20 additional units?
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

24. Homework Homework Assignment #5 Read Section 5.6
Page 341, Exercises: 1 – 19(Odd)
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company