Anton calculus early transcendentals 11th edition pdf free download

Find another number 0 for all positive integers n. How is this fact reflected on your graphs above? Using what you know about limits, compute the following quantities: a lim f 3 x x b lim f 4 x 0 x 1 c lim f 15 x x 1 2. Using what you know about limits, compute the following quantities: 1 a lim f n b lim f n 0 99 2 n c lim f n x , where x n 3.

Sketch g x , paying particular attention to g 1 and values of x close to 1. Are the following quantities defined? If so, what are they? If not, why not? When constructing this problem, 1 was used as an arbitrary smallish number.

Could you have done the previous problems if we replaced 1 by 1 10 ,? How about 1? Reread the first sentence on this page.

How do your answers to Problems 1 and 4 show that lim f 2 x x 0? Do your answers to Problems 2, 3, and 4 show that lim x 2 x 65 1 1 0? We will take values of x closer and closer to zero, and see what value x the function approaches. Fill out the following table. If, for example, your digit is 3, then you would compute sin , sin , sin , sin , etc. Now fill out the table with a different digit.

The algebraic computation of limits: manipulating algebraically, examining left- and right-hand limits, using the limit laws to break monstrous functions into pieces, and analyzing the pieces. The evaluation of limits from graphical representations. The computation of limits when the limit laws do not apply, and the use of direct substitution property when they do. Have the students check if lim x x lim 5 x x 5 x 5 x 5 exists, and then compute left- and right-hand limits.

Then check Present some graphical examples, such as lim f x and lim f x in the graph below. Problem 2 is more conceptual than Problem 1, but makes an important point about the sums and products of limits. No, yes, yes, no 2. Does not exist, 13 4. A more careful geometric argument is given in Section 3. The Squeeze Theorem now gives lim 1. How about lim [m x n x ]?

Justify your answers. Compare these with your answers to part a. Is f x defined for x. For x 0? For x 1? For x 2? What is the domain of f? Compute lim x2 x 4. Notice that one limit exists, and one does not. There are two x -values that are not in the domain of f. Geometrically, what is the difference between the two discontinuities? We say that f x has one hole in it. Where do you think that the hole is? The function g x is not defined at x 0.

Sketch this function. Does it have a hole at x 0? Using a graphing calculator, show that if 0 x 1, then x x3 6 sin x x. Give a rough sketch of the three functions over the interval [0 1] on the graph below.

Again using a graphing calculator, show that if 3. Use the inequalities in parts 1 and 2 to help you. Again using parts 1 and 2, can you find a function f x with f x that lim f x x 0 1?

Using parts 3 and 4, compute lim x 0 x. If you have not sin x. Extending the precise definition to one-sided and infinite limits. If the students are familiar with graphing calculators, this definition can be illustrated with setting different viewing windows for a particular graph.

Make sure the students understand that limit proofs, as described in the book, are two-step processes. This fact is stated clearly in the text, but it is a novel enough idea that it should be reinforced. Discuss how close x needs to be to 4, first to ensure that 1 x 1 20 Then argue intuitively that lim 4 2 x 4 x x 4 2 , and then so that. Emphasize that although this result is obvious from the graph, the idea is to see how the definition works using a function that is easy to work with.

Then estimate how close x must be to 0 to ensure that sin x x is within 0 of 1. Describe what you did in terms of the definition of a limit. Return to the interesting function f x the right- and left-hand limits exist at x 1 1 21 x from Group Work 1 in Section 2. They should show why their method works for Problem 2, and fails for Problem 3.

This can also be shown using the Squeeze Theorem and the fact that 0 f x x 2 , and then using the Limit Laws to compute lim 0 and lim x 2. It does not exist. We can conjecture that the limit does not exist by applying the reasoning from Problem 3. Let them discover for themselves how deceptively difficult it is, and then tell them that they should do the best that they can to show what is happening as x goes to zero. If a group finishes early, x 0 pass out the supplementary problems.

The length of the boundary is infinite. There are infinitely many wiggles, each adding at least 2 to the total perimeter length. The area is finite. It is less than the area of the rectangle defined by 0 x 1, 2 y 1. Problem 1 is designed to lead the students to make a false assumption about the third function, h x. Problem 2 should dispel that assumption. Allow the students plenty of time to do the first three questions, which should help them to internalize and understand the formal definition of a limit.

When the students are finishing up, it is crucial to pass out Problem 2. Students may or may not see the wrinkle in h x at this point. Consider this function: f x 0 x2 if x is rational if x is irrational 1. Try to sketch the graph of y f x. Does lim f x exist? If so, what is its value? Make sure to justify your answer x 0 carefully.

Carefully justify your answer. What do you conjecture about lim f x if a x a 0? It appears that this function is not defined at x even have a right-hand limit. Is the length of the boundary of this region finite or infinite? Is the area of this region finite or infinite? Do you think this result is as interesting as we do?

Why or why not? For what values of x near 0 is it true that 2. The precise definition of lim f x x a 1 x2 1,,? Give reasons for your answer. We now have some reason to believe that the above statements are true.

Now, what do you believe about these limits? Do you wish to change your answer to Problem 3 from 83 2. The graphical and mathematical definitions of continuity, and the basic principles. Examples of discontinuity. The Intermediate Value Theorem: mathematical statement, graphical examples, and applied examples. This would seem to contradict Theorem 7. Does it? Does there have to be a value of x , between 1 and 3, such that f x 0? Only if the function is continuous does the IVT indicate that there must be such a value.

The two quantities are equal. To show that all three statements are important to continuity, have the students come up with examples where the first holds and the second does not, the second holds and the first does not, and where the first two hold and the third does not. Examples are sketched below. Examine x2 if x 1 the function f x 1 and x e. This would be a good ln x 1 if 1 x e at the points x x if x e time to point out that the function x is continuous everywhere, including at x 0 Start by stating the basic idea of the Intermediate Value Theorem IVT in broad terms.

Given a function on an interval, the function hits every y-value between the starting and ending y-values. Show that this process reveals some flaws in our original statement that have to be corrected the interval must be closed; the function must be continuous. To many students the IVT says something trivial to the point of uselessness. It is important to show examples where the IVT is used to do non-trivial things.

Example: A graphing calculator uses the IVT when it graphs a function. A pixel represents a starting and ending y-value, and it is assumed that all the intermediate values are there. This is why graphing calculators are notoriously bad at graphing discontinuous functions. Example: Assume a circular wire is heated. Use the IVT to show that there exist two diametrically opposite points with the same temperature.

If f x 0, then f x 0, so by the IVT there must exist a point at which f 0. Example: Show that there exists a number whose cube is one more than the number itself. This is Exercise So by the IVT, 0 x irrational p Have the students look at the function f x 1 x q , where p and q are integers, q is q positive, and the fraction is in lowest terms This function, discovered by Riemann, has the property that it is continuous where x is irrational, and not continuous where x is rational.

Then discuss the continuity of g x e csc x , and why all the discontinuities of g are removable. Many will not believe that it is. Now look at it using the definition of continuity. They should agree that f 0 0. In the activity it was shown that lim f x existed and was equal to 0. So, this function is continuous at x 0. A sketch such as the one x 0 found in the answer to that group work may be helpful.

Present the following scenario: two ice fishermen are fishing in the middle of a lake. One of them gets up at P. The second one leaves at P. Show that there was a time where they were equidistant from camp. Revisit Exercise 5 in Section 2. The first problem is appropriate for all classes. If they have not, skip it and go directly to Problem 3. After this activity, discuss the continuity of [[x ]] at integer and at non-integer values.

Problem 4 is intended for classes with a more theoretical bent. Then, by the Limit Laws, h a. It will also ease the transition to area functions in Chapter 5. It is important that this activity be well set up. Do Problem 1 with the students, making sure to compute a few values of A r and to sketch it.

The students should try to answer Problems 2 and 3 using their intuition and the definition of continuity. It may be desirable to have the students restrict themselves to r 0.

Students may disagree on the answer to Problem 3. If you are fortunate enough to have groups that have reached opposite conclusions, break up one or more of them, and have representatives go to other groups to try to convince them of the error of their ways. The limits can be shown to exist by looking at the left- and right-hand limits. Let them argue for a while.

Ideally, they will come up with the idea of using the Intermediate Value Theorem to prove that Dr. Stewart was correct. They will have to figure out a way to find a single function that they can use. T t is continuous being a difference of continuous functions , T 0 0 Dr. Stewart is warmer at first , and T f 0 where f represents the end of the vacation; Shasta is warmer at the end.

Notice that most students who try to argue that the conclusion is false using things such as stasis chambers and exceeding the speed of light are really trying to construct a scenario where the continuity of the temperature function is violated. As in the Twin Problem, a first hint might be to use the IVT, and a second could be to find a single continuous function of x.

It is probably best to do this activity after the students have seen the solution to the ice fisherman problem above, or the Twin Problem.

Find c and m, or explain why they do not exist. Recall the function f x 0 x2 if x is rational if x is irrational a Do you believe that f x is continuous at x 0? Consider the function h x a Sketch the graph of the function for 1 x 2. Explain your answers. We know that the function g x x 2 is continuous everywhere. If it is true, prove it. If it is false, give a counterexample.

Let A r be the area enclosed by the x -axis, the y-axis, the graph of the function f , and the line x Would you conjecture that A r is continuous at every point in the domain of f? Let B r be the area enclosed by the x -axis, the y-axis, the graph of the function g, and the line x Would you conjecture that B r is continuous at every point in the domain of g?

Let C r be the area enclosed by the x -axis, the y-axis, the graph of the function h, and the line x Would you conjecture that C r is continuous at every point in the domain of h? James Stewart, that few people know. He has an evil twin sister named Shasta. Although he loves his sister dearly, she dislikes him and tries to be different from him in all things.

Last winter, they both went on vacation. Stewart went to Hawaii. Shasta had planned on going to Aruba, but she decided against it. She hates her brother so much that she was afraid there would be a chance that they might be experiencing the same temperature at the same time, and that prospect was distasteful to her. So she decided to vacation in northern Alaska.

After a few days, Dr. I am cold and uncomfortable here. I think we should switch places for the last half of our trip. So they each traveled again. They each traveled their own different routes, perhaps stopping at different places along the way. Eventually, they had reversed locations. Stewart was shivering in Alaska; Shasta was in Hawaii, warm and happy. She received a call from her brother. Guess what?

At some time during our travels, we were experiencing exactly the same temperature at the same time. So HA! Stewart right? Has Good triumphed over Evil? He would try to write out a proof of his statement, but his hands are too frozen to grasp his pen.

Help him out. Either prove him right, or prove him wrong, using mathematics. The geometric and limit definitions of horizontal asymptotes, particularly as they pertain to rational functions. The computation of infinite limits. The technique and the dangers of using calculators to check limits both numerically and graphically. Why must we do such a thing?

Now we can take the limit of the numerator, and easily divide it by the limit of the denominator. Note that a function can cross its horizontal asymptote. Perhaps include a description of slant asymptotes. Ask students if a function can be bounded but not have a horizontal asymptote. Does sin x have a horizontal sin x sin x asymptote? What about? How is different? Then show that this limit is, in fact, infinity. NOTE 5 is ln ln , which is what a student would come up with by plugging very large numbers into a calculator.

For large values of x , 3x 2x x3 x2 x ln x ln ln x , even though x they all approach infinity. An advanced class can discuss the even larger x. Note that for values of x near zero, x x2 x 3 , although all approach zero. Point out that as x approaches 0, a x approaches 1 and loga x approaches. Graph the function. Also perhaps review how to find the limits as x 2x 3 x3 Graph y 16 , after calculating limits as x 27 3 and as x. Show the students how to find a domain for x such that e x 0 for all x in that domain.

Examine lim x [[x ]] [[x ]] and lim 2. While they should justify their answers, it is important that they also get some feel for how limits as x behave. Pick and choose problems. It is more important to have good introduction and closure on each part than to have all of them worked out. Problem 4 is an extension of Exercise Possible answers: a f x g x x g x 4. This can be done from the graph, or using the definition. This can be seen by a similar bounding argument to the one above.

Describe the horizontal asymptotes, if any, of the following functions. Find lim x 4. Draw an even function which has the lines y 1, x 4, and x 2.

Describe all vertical and horizontal asymptotes of f x 3x 2 1 among its asymptotes. Let P x am x m n, respectively.

Show that lim x 0 [[x ]] x 0 2. Compute lim x 2 x 0 4. Show that lim x x 5. Compute lim x 1 1. The slope of the tangent line as the limit of the slopes of secant lines visually, numerically, algebraically. Physical examples of instantaneous rates of change velocity, reaction rate, marginal cost, and so on and their units.

The derivative notations f a lim x a x a h 0 h 4. Using f to write an equation of the tangent line to a curve at a given point. Using f as an approximate rate of change when working with discrete data. Examples include the following: By definition, the slope of the tangent line is the limit of the slopes of secant lines. The idea is to get them thinking about this question. Estimate the slope of the line tangent to y x 3 x at 1 2 by looking at the slopes of the lines between x 0 9 and x 1 1, x 0 99 and x 1 01, and so forth.

Then examine what happens if you look at the limits of the secant lines. Have students estimate the slope of the tangent line to y sin x at various points. Foreshadow the concept of concavity by asking them some open-ended questions such as the following: What happens to the function when the slope of the tangent is increasing? Slowly changing? Discuss how physical situations can be translated into statements about derivatives. For example, the budget deficit can be viewed as the derivative of the national debt.

Describe the units of derivatives in real world situations. The budget deficit, for example, is measured in billions of dollars per year. Another example: if s d represents the sales figures for a magazine given d dollars of advertising, where s is the number of magazines sold, then s d is in magazines per dollar spent.

Describe enough examples to make the pattern evident. Note that the text shows that if f x x 2 8x 9, then f a 2a 8. Thus, f 55 and f Demonstrate that these quantities cannot be easily estimated from a graph of the function.

Foreshadow the treatment of a as a variable in Section 2. If a function models discrete data and the quantities involved are orders of magnitude larger than 1, we can use the approximation f x f x 1 f x. That is, we can use h 1 in the limit definition of the derivative. For example, let f t be the total population of the world, where t is measured in years since Then f is the world population in , f is the total population in , and f is approximately the change in population from to In business, if f n is the total cost of producing n objects, f n approximates the cost of producing the n 1 th object.

Is it larger than 1? About 1? Between 0 and 1? About 0? Between 1 and 0? Smaller than 1? Draw a function like the following, and first estimate slopes of secant lines between x a and x b, and between x b and x c. Then order the five quantities f a , f b , f c , m P Q , and m Q R in decreasing order. How far has the car travelled after 1, 2, and 3 hours? What is the average velocity over the intervals [0 1], [1 2], and [2 3]?

Ask the students to determine when the car was stopped. Ask the students when the car was accelerating that is, when the velocity was increasing. When was the car decelerating? Ask the students to describe what is happening at times A, C , and D in terms of both velocity and acceleration. What is happening at time B? Estimate the slope of the tangent line to y sin x at x x sin x 0 05 09 0 99 0 1 1 0 0 0 0 0 0 0 1 by looking at the following table of values.

Ask them how far the car has traveled after 1, 2, and 3 hours, and then show them how to compute the average velocity for [0 1], [1 2], and [2 3]. It appears to stop at t 2. This is where the car stops.

If the students do this numerically, they should be able to get some pretty good estimates of ln 3 1 If they use graphs, they should be able to get 1 1 as an estimate. As a increases, the slope of the curve at x 0 is increasing, as can be seen below. The slope is less than 1 at a 2 and greater than 1 at a 3. Now apply the Intermediate Value Theorem. The students are estimating e and should get 2 72 at a minimum level of accuracy. At this point in the course, many students will have the impression that all reasonable estimates are equally valid, so it is crucial that students discuss Problem 4.

If there is student interest, this table can generate a rich discussion. Can A ever be negative? What would that mean in real terms? What would A mean in real terms in this instance?

Percent per dollar 4. This is a better estimate because the same figures now give a two-sided approximation of the limit of the difference quotient. The domain is from 0 cm to the maximum total rainfall. The range is from midnight to the end of the storm. Does the car ever stop? What is the average velocity over [1 3]?

Estimate the instantaneous velocity at t 2. Give a physical interpretation of your answer. Use your calculator to graph y method of your choosing. Estimate the slope of the line tangent to this curve at x 0 using a 2. Estimate the slope of the line tangent to this curve at x 0 using a 3. It is a fact that, as a increases, the slope of the line tangent to y continuous way. Geometrically, why should this be the case? Prove that there is a special value of a for which the slope of the line tangent to y 0 also increases in a a x at x 5.

By trial and error, find an estimate of this special value of a, accurate to two decimal places. Is this likely to be an overestimate or an underestimate? Interpret your answer to Problem 1 in real terms. What are the units of A p? Describe the inverse function f 1 in words. What are the domain and range of f 2. Interpret the following in practical terms. Include units in your answers. Most people who study history never see calculus, and vice versa. We recommend assigning this section as extra credit to any motivated class, and possibly as a required group project, especially for a class consisting of students who are not science or math majors.

The students will need clear instructions detailing what their final result should look like. The concept of a differentiable function interpreted visually, algebraically, and descriptively. Obtaining the derivative function f by first considering the derivative at a point x , and then treating x as a variable. How a function can fail to be differentiable.

Sketching the derivative function given a graph of the original function. This section discusses the derivative f x for some function f. What is the difference, and why is it significant enough to merit separate sections?

So f a is a number the slope of the tangent line and f x is a function. Is this function defined at x 0? Continuous at x 0? Many students incorrectly add velocity to this list. Then the left and right limits exist but are unequal.

Exercise Set 1. False; e. True, by Theorem 1. If on the contrary lim g x did exist then by Theorem 1. By Theorem 1. Calculus: Early Transcendentals, 11th Edition Download You are currently using the site but have requested a page in the site. Would you like to change to the site? Howard Anton , Irl C. Bivens , Stephen Davis. View Instructor Companion Site.

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