Calculus of One Variable (Springer Undergraduate Mathematics Series)
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The components of the expression which determine the behaviour when x is large are the greatest powers of x in the numerator and denominator. We demonstrate this algebraically by dividing numerator and denominator by x2 , the largest power of x in the denominator. The denominator therefore tends to 1.
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As in example 2. So we have the numerator tending to zero and the denominator tending to 1. The quotient therefore tends to zero. Limits of Functions Example 2. The denominator tends to zero. It is clear from Figure 1. To correct the mistake we have to take careful account of the signs of the various parts of the expression. We can do this either by plotting the graphs of numerator and denominator, or by means of a tabular approach, as follows. At each of these points one of the factors changes sign, and so the table indicates the sign of each factor in the intervals between these numbers.
In this example highest power of x present is the same in numerator and denominator, namely x2. Solving the relevant quadratic equation shows that the denominator has no real zeros, and so there are no vertical asymptotes. Secondly, the denominator 2. This time we divide numerator and denominator by ex. This example illustrates the comment preceding Example 2. Limits of Functions 67 2. Here we can use it in limit problems. We shall give a proof in Section 6. At this stage we give the statement of the rule, including the conditions under which the rule is applicable, which must be understood.
We shall look at some examples. These are the conditions for the rule to be applicable and must always be checked. Algebraic intuition does not help much here. On the other hand we might argue that for x near to zero but not equal to zero we have 0 70 Calculus of One Variable Example 2. So we have 2 sec x. Limits of Functions 71 2.
It indicates the need for care, and these kinds of problems are often studied in Real Analysis textbooks. The subject is discussed in Howie, Chapter 7. There is a single isolated point at 0, 0. The function is discontinuous at 0. What it will not do of course is to tell us how we could prove the result, or provide us with an understanding of why the result is true.
The result of Example 2. This topic would be dealt with systematically in a course on Multivariate Calculus, and here we simply illustrate some possibilities by means of an example. When we generalise these considerations to functions of two variables f x, y , then x, y 2. Limits of Functions 73 belongs to a two dimensional plane.
By analogy with one-sided limits we can then consider whether f x, y has a limiting value if x, y in restricted in some way as it approaches 0, 0. This may be along some particular curve in the x, y -plane. The following example illustrates this idea. Readers who are interested to see what a graphical representation looks like for the function we have considered can do so using the following MAPLE command. Plot the graph of the function given by the following formula, using a graphical calculator or MAPLE. You may need to use more than one plot to observe the various features of the graph, by varying the domain.
Plot the graphs of the functions given by the following formulae, using a graphical calculator or MAPLE. Use the appropriate symbolic notation in each case. In some cases you will need more than one plot, as in the previous exercise. In cases involving roots you may need to use implicit plotting, as described in Section 1. Limits of Functions 75 2. Describe the limiting behaviour of the functions, given by the following formulae, on each side of the value of x indicated.
Find the following limits, using the appropriate algebraic rules. Prove by squeezing that the value of each of the following limits is zero. The function f x is bounded, i. Use the method of Example 2. Evaluate the following limits, using algebraic manipulation. Use the method of change of variable to evaluate the following limits.
Limits of Functions 77 2. What is the domain of f x? Sketch the graph of f x showing the discontinuities. The other is motion speed, velocity, acceleration and other rates of change. The tangent line at the point A can be considered as the limiting position of the chord AB as B tends towards A. It is important to note that this is not just a one-sided limit, and so a diagram where B is to the left of A, where h is negative, is equally valid. The gradient of the tangent will therefore be the limit as x tends to a of the gradient of the chord. The coordinates of the points labelled in Figure 3.
In fact a variety of terminology is encountered in this topic. In an introductory account such distinctions are not so important, whereas they are in more advanced areas of calculus. Readers will have encoundx dx2 tered these in school calculus. Sometimes the Leibniz notation is more helpful than the dash notation, and vice-versa, and we shall use them according to this criterion. We also prove a basic property of derivatives useful in graph sketching. Example 3.
Here we have used the limit obtained in Example 2. Figure 3. Let x denote an arbitrary number in the domain of f. If h 84 Calculus of One Variable this case both numerator and denominator are negative or zero. Note that if a function is strictly decreasing this does not imply that its derivative is strictly positive everywhere.
We shall prove a converse of this result under appropriate conditions in Section 6. We assume that the derivatives in the table below are known from school mathematics. Its purpose is to enable us to analyse separately the changes in f and the changes in g as h tends to zero. This is apparent in the third line, where we can see the chord-slope quotients for f and for g, whose limits are the respective derivatives as h tends to zero. The rule is stated as follows.
The rule can be stated using the Leibniz notation as follows. This helps to separate the behaviour of f and that of g. A properly rigorous proof is given in Howie Chapter 4. What has been calculated here is the derivative with respect to a, not the derivative with respect to x. In this example we have repeated composition, and we extend the chain rule using two intermediate variables. Firstly we need the product rule since the function is x2 multiplied by a cosine term. Secondly the cosine terms itself is composite, and so we need the chain rule.
However, the squeezing argument used in Example 2. The details of the method are discussed in Howie Chapter 1. The inductive proof is as follows, for readers who are familiar this style of proof. While it is clear that there is a pattern here, it is not easy to formulate an expression for the n-th derivative. The role of the x at the end of the command it important. An error message will result if it is omitted. This procedure could be used to calculate the sixth derivative quoted in Example 3.
Finally we note that one need not have a particular formula, so that for example MAPLE will help us if we forget the product rule. Prove these results by the method of mathematical induction. The derivative of an even function is an odd function. The derivative of an odd function is an even function. Do you think the converses are true, namely that every odd even function is the derivative of an even odd function?
If you think so, give a proof. If you do not think so, give a counter-example. If you think the converses are true only for some kinds of function, describe such a set of functions and prove the converses for this set. We shall also discuss Leibniz Theorem, a result which enables higher derivatives of products to be calculated. The following example illustrates what is meant. Example 4. Figure 4. One of the three intersections of this line with the graph is of course the given point 1, 2.
This value is consistent with Figure 4. When we get used to this procedure we do not normally write y x in full, but just use y throughout, as in the next example. Note that substituting arbitrary values of x and y in this equation is meaningless. The point x, y would have to satisfy the original equation in order dy that could be interpreted as the gradient of the curve.
The following sequence of commands can be used to solve the problem. It is a technique which is useful when we have expressions involving the variable in an exponent. It can also be applied to complicated products. In fact we have already encountered this function, in Example 3. This example combines a product with an exponent. We can approach this in two ways. We reason as follows. Its graph is shown in Figure With the parametric form, a given value of t determines both x and y, and hence a unique point on the curve.
That value of t will determine the gradient at that point. The graphs of the function and its inverse are shown in Figure 4. In Section 1. We have to consider the same approach here. Unlike the previous example, where sinh is over its whole domain, and where the choice of square root was straightforward, in this case we have to consider how the domain is restricted 4. In this section we derive a general formula for this procedure. Theorem 4. It is given by the following formula n!
Each term on the right hand side gives rise to two terms, from the product rule. We use the formula for the n-th derivative of f x which we obtained in Example 3. Hence determine the equation of the tangent to the curve at that point. Find expressions for d3 y d2 y and , in terms of t. This is used in many problems in co-ordinate geometry, as in the following examples. Example 5. Find the distance between the points where these tangents cross the x-axis.
The tangents at A and B cross at right-angles. Find the values of t. It is the place where the value of f x is smallest compared with all other values. Theorem 5. The case of a local minimum is proved in exactly the same way. The converse of this theorem is not true, and there are situations where the derivative is zero but we do not have a maximum or a minimum. This point is not a maximum or a minimum, because x3 is negative for all x 0.
Other terms sometimes used are turning point and critical point. We will discuss three possible approaches to this. Once we know where the stationary points are then the graph should be able to tell us whether they are maxima, minima or neither. Naturally we need to choose carefully the values of x and y to be included in the plot, so that we obtain as clear a picture as we can of the behaviour near to the stationary point under investigation.
The second method involves consideration of the derivative near to the stationary point.
Springer Undergraduate Mathematics: Calculus of One Variable by Keith E. Hirst (2005, Paperback)
Near a local maximum, the graph is increasing to the left of the stationary point, and decreasing to the right, so we should expect the derivative to be positive on the left and negative on the right. The reverse occurs with a local minimum. The third method involves using the second derivative. We note that it is normally possible to investigate case iii using Taylor series, discussed in Chapter 6. We shall try to classify these stationary points using the second derivative test. This result is discussed in Howie, Chapter 3. Global maximum and minimum values may occur at the endpoints, as in this case, or at a stationary point if the value of the function there is greater.
We shall consider problems where the optimum solution corresponds to a maximum or a minimum. Problems such as this are often stated verbally, and we can use a common strategy for their solution, with the following steps. Choose variables to represent the quantities described in the problem.
If possible draw a diagram and indicate these quantities on the diagram. Use these formulae to determine a formula for the quantity to be optimised as a function of one variable.
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Determine any restrictions on the variables arising from the conditions of the problem for example an area should be positive. Use the techniques of calculus to determine the stationary points, paying attention to any restrictions on the variables. Determine maxima and minima and choose the one which provides the solution to the problem.
In this section we restrict attention to problems involving one variable, although in practice optimisation problems often involve many variables, and their investigation forms part of Operational Research. The variables here will be the coordinates of an arbitrary point x, y on the curve, and the distance D of that point from 4, 0. We have now obtained a formula for the quantity to be optimised as a function of one variable. The reasoning used here is typical of this kind of problem, being a mixture of calculation and geometrical reasoning derived from the conditions of the problem.
What should be the height of the prism to maximise the volume of the prism? The variables in this problem are the volume V , the height h and half the base b, as shown in Figure 5. We therefore have a maximum. The factory can produce at most doses in a week. We must note that of course x is an integer variable.
In order to apply the methods of calculus we would need to treat it as a real variable. We would have to be careful if the stationary point turned out not to occur at an integer value. Geometrically the method can be understood through Figure 5. The method proceeds by having an initial guess x0 , possibly obtained from a graph, which we hope is close to the actual solution.
We then join the point x0 , 0 , on the x-axis, by means of a line parallel to the y-axis, to the point x0 , f x0 lying on the curve. We then draw the tangent to the curve at this point, and we denote by x1 the value of x for which the tangent meets the x-axis. Apart from considerations such as these we have now determined the value of x1.
Calculating distance divided by time then gives Calculus of One Variable the average speed over that distance, and this is what the records comprise.
However if you drive a car you see a continuous readout on the speedometer, suggesting some idea of speed at an instant of time. The average speed will be measured by distance travelled divided by time taken, i. We therefore need to take the direction of motion into account if we wish to use the derivative to describe how fast something is moving. From the discussion above it seems that a speedometer measures the magnitude of the derivative, and we use the word speed in this sense. When we want to include the direction of motion we use the word velocity. When we begin a car journey our speed increases.
We say that we are accelerating. When we slow down we say that we are decelerating. We describe acceleration mathematically as the rate at which the velocity changes. In this sense, when the car is decelerating its acceleration is negative. Describe the subsequent motion. The formulae for velocity and acceleration 5. Figure 5. The verbal description of the motion given below corresponds to the important features of these graphs.
The velocity is then zero and the acceleration is negative, with magnitude 4, the maximum Calculus of One Variable possible. This means that at that instant the object has stopped moving. The velocity is negative, meaning that the object is moving upwards as fast as it can at that point. The acceleration is zero there. The object has been speeding up and it is just about to start slowing down. A radar station is located 3km away from the port in a direction perpendicular to the direction of the ship.
What is its velocity at that point? Let x t denote the distance of the ship from port at time t and let y t denote its distance from the radar station. To dedt termine a value for b we need to know the velocity at some particular point in time. The following three formulae are therefore the standard equations used to analyse problems about motion in a straight line under constant acceleration.
The experts tell us that the bacteria divide to produce new bacteria. So for example if there are bacteria then after one division there will be twice as many, namely 2. After another division there will again be twice as many, i. Following this pattern, after n divisions there will be 2n. We can see that the rate of growth depends on how many bacteria there are at any given stage of the process.
The number of bacteria in a colony is large, and division happens very quickly. So we model the process by taking the number of bacteria to be X t , a quantity changing continuously with time, t. With this assumption X t is not restricted to integer values, but it means that we can use calculus to model the growth of the colony. It is found that such models correspond closely with such growth processes in nature. This is known as the exponential growth model. It is clear therefore that the model only applies for a restricted interval of time, for otherwise the colony would eventually weigh more than the earth!
What Calculus of One Variable happens of course is that factors come into play which we did not build into the model. Bacteria die, and growth slows as food runs out. What we need to know for a particular growth process is the value of the constants A and k. These can be determined from measurements, as in the following examples.
After 10 seconds it weighs 1. Determine the rate of growth, assuming an exponential model, and hence deduce the weight of the colony after 20 seconds. This tells us that the exponential growth model agrees with our intuitive ideas about such growth processes. The opposite of growth is decay, and one of the best-known decay processes is the decomposition of radioactive material. We model this situation by assuming that the more atoms there are in a lump of material the more decay decompositions there will be, suggesting that the rate of decay should be proportional to the number of atoms there are in a lump of such material.
Of course a lump of such material will not actually halve in size, because the radioactive element decays into something else, like lead for example. Some of the transuranic elements are rather evanescent, having a very short half-life. Every year a nuclear reactor produces a quantity of this element as waste.
Let X t denote the amount of material remaining after time t. Find the coordinates of Q in terms of the coordinates of P. Two circles of radius a are tangent to one another. The two tangents from the centre of one circle touch the other circle at the two points A and B. Find the distance between A and B. Find the points on this hyperbola which are closest to the point 0, a on the y-axis. Many optimisation problems are formulated in terms of manufacturing boxes, as in this exercise.
The resulting rectangles formed at the edges of the card are then folded to make a rectangular box. Find the maximum volume which can be obtained. A right circular cone has a total surface area of 4m2. Find the dimensions of the cone which give the greatest volume. Stainless steel grain silos are to be manufactured, each with a volume of m3. They will rest on a concrete base and consist of a cylindrical wall and a hemispherical top. A stainless steel hemisphere costs three times as much per unit area to manufacture as a cylinder. Find the radius of the cylinder which will minimise the cost of manufacture.
Find the speed at which the train should travel to minimise the cost of this part of its journey. Plot a graph to help to choose suitable initial approximations corresponding to each of the roots, and use a spreadsheet to obtain 10 decimal places of accuracy. It decelerates due to gravity. A tracking station is 10km away from the launch pad.
What is the rate of change of the distance of the rocket from the tracking station when it has reached a height of 10km? Radium has a half-life of years. It decays and changes into Lead through intermediate elements of relatively short half lives. The radiation from radium is toxic, so it has to be shielded from those nearby. If we have one gram of radium in the laboratory, how long would it take before there was only 1mg one thousandth of a gram of radium left in the sample?
A Chemistry student takes some molten metal from a furnace, but forgets to measure its temperature. The circuitry can do addition, and therefore multiplication, which is in essence repeated addition. It can also do subtraction, and division, which can be thought of as repeated subtraction. Clearly such an exercise should include an analysis of the greatest possible errors arising with such approximations if the digits in the display are to be accurate.
We start with a simple case. The curve shown in Figure 6. This approximates to the curve very well when x is near to 1, but as x increases towards 2 the line diverges from the curve by a considerable amount. Secondly we have drawn the line passing through P and Q. This provides quite a good approximation across the interval, but near to P it is not so accurate as the tangent line.
Each of these lines is a possible candidate for a linear approximation. The tangent line and the curve have the same gradient at P , and so y changes 6. In the next section we shall consider the error which this constant approximation involves. It can be found in Real Analysis books such as Howie, Chapter 4. It is important however to have a clear statement of the theorem with all the requisite conditions, and to give a geometrical interpretation.
Theorem 6. The right-hand side is the Calculus of One Variable gradient of the tangent at the point c, f c. Figure 6. Example 6. So f is an increasing function. So f is still an increasing function, although not necessarily strictly increasing. Geometrically this makes sense, because we should expect the error to depend on how far x is from a and on how large the rate of change of f is.
Indeed if we could, this would tell us the exact value of f x and there would be no need to consider approximations at all. The right hand side is called an error bound for the constant approximation. In order to study the error involved in using a linear approximation we need a version of the Mean Value Theorem applied to two functions simultaneously. Now recall that the linear approximation we obtained in Section 6. If this is very large then the graph will quickly diverge from that of its linear approximation. Maclaurin and Taylor Expansions Example 6. Find an error bound and hence discuss the accuracy of the approximation.
See Section 2. To improve the approximation we replace the straight line by a curve whose gradient changes in such a way as to try to follow the graph of f x , the function being approximated. After a linear approximation the next simplest would be a quadratic. In the case of the tangent line, it passed through the point of contact and had the same gradient as the curve at that point.
We shall pursue this in the next section. A similar error analysis is possible to that for constant and linear approximations. We shall however consider this in the more generalised context of polynomial approximations of degree n in the next section. The Taylor series usually converges to the function f x that we start with, but there are exceptions. There are precise conditions for the validity of these results, discussed in Howie, Chapters 2, 4 and 7. The examples we shall consider have been chosen to satisfy those conditions. Certain basic expansions for elementary functions form part of school mathematics and should be memorised.
We use the result of Example 6. This is often much more complicated than using a known expansion and applying various operations. In the next example we compare the two methods. This was discussed in Example 2. Therefore by squeezing Section 2. Proofs can be found in Real Analysis textbooks, for example in Howie Chapter 4.
As we remarked in Section 6. Strictly speaking we should give a proof of our formula, by mathematical induction, but we shall omit that here. Therefore 0 1, telling us that 0. Firstly there is a built-in command which works out Taylor series, in which we have to specify the function, the point about which the expansion is to take place, and how many terms we want.
This is because the constant term in this expansion is zero.
Where we have a regular pattern which tells us one term in terms of the previous ones MAPLE contains a built-in programming language which would enable us to do this. MAPLE does not label its graphs on a plot such as this, so we have to inspect each individual graph to decide which of them corresponds to each approximation, which is a good exercise in itself. This makes it easier to identify each graph.
In fact the color command can be adapted to specify the colours we wish for each of the approximations. Use the Mean Value Theorem to show that if a function has a negative derivative throughout an interval, then the function is decreasing throughout that interval. Use a calculator or a spreadsheet to compare ex with the quadratic approximation for values of x at intervals of 0.
Maclaurin and Taylor Expansions 6. Find the Maclaurin expansions for the following functions. Find the Taylor series for the following functions about the points indicated. Use En x for even values of n. The Greek mathematician Archimedes 3rd Century B. Bringing the two views of integration together was the work of mathematicians in the 19th Century in particular, culminating in the work of Bernhard Riemann — , whose name is associated with the theory of integration which is often studied in Real Analysis courses see Howie Chapter 5 for example.
We approximate this area using rectangles, from below and from above. These ideas are illustrated in Figure 7. For functions that we commonly use, with continuous Calculus of One Variable graphs for example, it turns out that the approximations from below and from above can be made as close as we wish. This is a limiting process, but a more complicated one than we discussed in Chapter 2. We introduce the notation for these as follows. The fundamental idea is that as the lengths of the intervals of the subdivision all tend to zero, both the upper and lower sum tend to a common limit, the area under the curve.
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