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Homework Homework Assignment #1 Review Section 2.1/Read Section 2.2 Page 66, Exercises: 1 – 9 (Odd), 13 – 29 (EOO) Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page A ball is dropped from a state of rest at t = 0. The distance traveled after t sec is s(t) = 16t 2. (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 Copyright © 2008 W. H. Freeman and Company

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Homework, Page (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 Copyright © 2008 W. H. Freeman and Company

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Homework, Page Let. Estimate the instantaneous ROC of v with respect to T when T = 300. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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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 – 4.9t 2 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

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Homework, Page 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 Copyright © 2008 W. H. Freeman and Company

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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 Copyright © 2008 W. H. Freeman and Company

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

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

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Homework, Page 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

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Homework, Page 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

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Homework, Page 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

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Homework, Page (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

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Homework, Page 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

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

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company 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.

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company 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

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

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company 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

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

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company 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.

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Graphical Investigation Use the graphing calculator to graph the function in the vicinity of the value of x in question. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company 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.

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

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Numerical Investigation 1.Construct a table of values of f (x) for x close to, but less than c. 2.Construct a second table of values of f (x) for x close to, but greater than c. 3.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

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

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

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

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company 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:

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company 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.

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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 Copyright © 2008 W. H. Freeman and Company

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Example, Page 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

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

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

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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

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