1. Homework Homework Assignment #1 Review Section 2.1/Read Section 2.2
Page 66, Exercises: 1 – 9 (Odd), 13 – 29 (EOO)
Rogawski Calculus
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2. Homework, Page 66
1. A ball is dropped from a state of rest at t = 0. The distance traveled after t sec is s(t) = 16t2. (a) How far does the ball travel during the time interval [2, 2.5]? (b) Compute the average velocity over [2, 2.5].
Rogawski Calculus
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3. Homework, Page 66
1. (c) Compute the average velocity over the time intervals [2, 2.01], [2, 2.005], [2, ]. Use this to estimate the object’s instantaneous velocity at t = 2.
Rogawski Calculus
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4. Homework, Page 66
3. Let Estimate the instantaneous ROC of v with respect to T when T = 300.
Rogawski Calculus
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5. Homework, Page 66
A stone is tossed in the air from ground level with an initial velocity of 15 m/s. Its height at time t is h(t) = 15t – 4.9t2 m. 5. Compute the stones average velocity over the time interval [0.5, 2.5] and indicate corresponding the corresponding secant line on a sketch of the graph of h(t).
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

6. Homework, Page 66
7. With an initial deposit of $100, the balance in a bank account after t years is f(t) = 100(1.08)t dollars. (a) What are the units of the ROC of f(t)? The units of the ROC of f (t) are dollars per year. (b) Find the average ROC over [0, 0.5] and [0, 1].
Rogawski Calculus
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7. Homework, Page 66
(c) Estimate the instantaneous ROC at t = 0.5 by computing the average ROC over intervals to the right and left of t = 0.5.
Rogawski Calculus
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8. Homework, Page 66
Estimate the instantaneous ROC at the indicated point.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

9. Homework, Page 66
Estimate the instantaneous ROC at the indicated point.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

10. Homework, Page 66
17. The atmospheric temperature T (in ºF) above a certain point is T = 59 – h, where h is the altitude in feet (h ≤ 37,000 ft). What are the average and instantaneous ROC of T with respect to h? Why are they the same? Sketch the graph of T. The secant line and the graph of T are the same line, so the average and instantaneous ROC are both equal to –
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

11. Homework, Page 66
21. Assume that the period T (in sec) of a pendulum is where L is the length (in m). (a) What are the units for ROC of the T with respect to L? Explain what this rate measures. The units of the ROC of T are sec/m, giving the rate at which the period changes when length changes. (b) What quantities are represented by the slopes of lines A and B in Figure 10? Lines A and B represent instantaneous and average ROC.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

12. Homework, Page 66
25. The fraction of a city’s population infected by a flu virus is plotted as a function of time in weeks in Figure 14.
(a) Which quantities are represented by the slopes of lines A and B? Estimate these slopes.
The slope of line A represents the average ROC over weeks 4 through 6. The slope of line B represents the instantaneous ROC in week 6. The slopes appear to be about , respectively.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

13. Homework, Page 66
25. (b) Is the flu spreading most rapidly at t = 1, 2, or 3?
The flu is spreading most rapidly at t = 3 , as the slope is greatest at that point.
(c) Is the flu spreading most rapidly t = 4, 5, or 6?
The flu is spreading most rapidly at t = 4, as the slope is greatest at that point.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

14. Homework, Page 66
29. Sketch the graph of f (x) = x (1 – x) over [0, 1]. Refer to the graph and find: (a) The average ROC over [0, 1]. average ROC = 0 (b) The instantaneous ROC at x = 0.5. instantaneous ROC = 0 (c) The values of x for which ROC is positive. ROC is positive for x < 0.5.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

15. Section 2.2: Limits: A Numerical and Graphical Approach
Jon Rogawski Calculus, ET First Edition
Chapter 2: Limits
Section 2.2: Limits: A Numerical and Graphical Approach
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

16. Section 2.2 Notes and Examples
In previous courses, we have considered limits when looking at the end behavior of a function. For instance, the function y = 2 + e–x asymptotically approaches y = 2 as x approaches infinity.
Mathematically, we could write this as:
Similarly:
In this section we will look at the behavior of the values of f (x) as x approaches some number c, whether or not f (c) is defined.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

17. Section 2.2 Notes and Examples
Consider the function as x approaches 0.
The function is not defined at x = 0, but it is defined for
values of x very close to 0. By examining these values
in the table below, we see that f (x) approaches 1 as x
approaches 0 or
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

18. Example, Page 76 Fill in the table and guess the value of the limit. θ
±0.002
±0.0001 ± ±
f (+θ)
f (–θ)
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

19. Section 2.2 Notes and Examples
Recalling that the distance between points a and b is |b – a|,
If the values of f (x) do not converge to any finite value as x
approaches c, we say that
Rogawski Calculus
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20. Example, Page 76 Verify each limit using the limit definition.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

21. Section 2.2 Notes and Examples
Theorem 1 states two simple, yet important concepts,
which are illustrated in the graph below.
The value of constant k is independent of the value of x,
as is the value of c.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

22. Graphical Investigation
Use the graphing calculator to graph the function in the vicinity of the value of x in question.
If the graph appears to approach the same point from either side of the value in question, then we may say that the limit exists, and the limit may frequently be estimated directly from the graph.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

23. Example, Page 76 Verify each limit using the limit definition.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

24. Numerical Investigation
Section 2.2 Notes and Examples
Numerical Investigation
Construct a table of values of f (x) for x close to, but less than c.
Construct a second table of values of f (x) for x close to, but greater than c.
If both tables indicate convergence to the same number L, we take L to be an estimate for the limit.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

25. Section 2.2 Notes and Examples
The graph and table below illustrate graphical and
numerical investigation of:
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

26. Section 2.2 Notes and Examples
The graph and table below illustrate graphical and
numerical investigation of:
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

27. Section 2.2 Notes and Examples
The graph and table below illustrate graphical and
numerical investigation of:
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

28. Section 2.2 Notes and Examples
The graph below doesn’t really answer the question of the
existence of a limit, but the table shows that it does not
exist as the values of sin (1/x) have opposite signs when
the same distance from zero in the positive and negative
directions is considered. Thus:
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

29. Section 2.2 Notes and Examples
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

30. Section 2.2 Notes and Examples
Figure 7 shows the graph of a
piecewise function. The limits
as x approaches 0 and 2 do not
exist, but the limit as x
approaches 4 appears to exist,
as the graph on both sides of
x = 4 seems to converge on the
same value of y. The following one-sided limits also
appear to exist:
but DNE as the values of f (x) continue to
oscillate between ±1 as x gets ever closer to 0.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

31. Definition
The limit of a function at a given point exists if, and only if, the one-sided limits at the point are equal. Mathematically, we state:
Rogawski Calculus
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32. Example, Page 76
38. Determine the one-sided limits at c = 1, 2, 4 of the function g(t) shown in the figure and state whether the limit exists at these points.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

33. Section 2.2 Notes and Examples
Graphs (A) and (B) in Figure 8 illustrate functions with
infinite discontinuities. Graph (C) illustrates a function
with the interesting property:
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

34. Example, Page 76 Draw the graph of a function with the given limits.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

35. Homework Homework Assignment #2 Read Section 2.3
Page 76, Exercises: 1 – 53 (EOO)
Quiz next time
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company