MAT237Y1 Lecture Notes - Maxima And Minima, Minimax, The Thirteen Chairs

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4 Jan 2013

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It"s what you think it is: an absolute max is the biggest point on the graph, while the absolute min is the smallest point. While there can be many di erent relative max"s and min"s, there can be only one absolute max and min (think about it: there can only be one highest point (unless another point reaches the same height, but who cares?)) In general, there is no guarantee that there will be an absolute max or min on a given interval. In fact, i think we can now say something about absolute max"s and min"s. So, here"s another theorem of which you must know the name (there are only a few more in this class! Extreme value theorem if f(x) is continuous over a closed interval [a, b], then it has both an absolute max and min on. Why do we need continuity and a closed interval? x, [ 1, 1].

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Sketch using these guidelines

pul aill al thes information together to sketch graphs that reveal the important You might ask: Why don't we just use a graphing calculator or computer to graph a It's true that modern technology is capable of producing very accurate graphs. But K 21+1+2 features of functions curve? Why do we need to use calculus? even the best graphing devices have to be used intelligently. It is easy to arrive at a misleading graph, or to miss important details of a curve, when relying solely on tech- nology. (See "Graphing Calculators and Computers" at www.stewartcalcelus.com, espe 4 discover the most interesting aspects of graphs and in many cases to and minimum points and inflection points exocily instead of approximately cially Examples 1, 3.4, and 5. See also Section 4.6.) The use of calculus enables us to calculate maximum FIGURE 1 For instance, Figure I shows the gr aph of /(x)-è¨-21x, + 18x + 2.At first glance it seems reasonable: It has the same shape as cubic curves like y x and it appears to have no maximum or minimum point. But if you compute the derivative, you will see that there is a maximum when x-0.75 and a minimum when x 1. Indeed if we zoom in to this portion of the graph, we see that behavior exhibited in Pigure 2 Without calculus, we could easily have overlooked it. 6a, FIGURE 2 In the next section we will graph functions by using the interaction between calculus and graphing devices. In this section we draw graphs by first considering the foilowing information. We don't assume that you have a graphing device, but if you do have one you should use it as a check on your work. â Guidelines for Sketching a Curve every item is relevant to every function. (For instance, a to make a sketch that displays the most important aspects of the function. The following checklist is intended as a guide to sketching a curve y- f(x) by hand. Not given curve might not have an asymptote or possess symmetry.) But the guidelines provide all the information you need A. Domain It's often useful to start by determining the domain D of f, that is, the set B. Intercepts The y-intercept is f(o) and this tells us where the curve intersects the of values of x for which f(x) is defined y-axis. To find the x-intercepts, we set y-0and solve for xã(You can omit this step if the equation is difficult to solve.) ry (a) Even fusction: reflectional symmetry all x in D, that is, the equation of the curve is unchanged when x is replacea by -x, then f is an even function and the curve is symmetric about the y-axis. This means that our work is cut in half. If we know what the curve looks like for 0, then we need only reflect about the y-axis to obtain the com- plete curve [see Figure 3(a). Here are some examples: y-ey-ry x). and for all x in D, then f is an odd function and the curve is symmetric about the origin. Again we can obtain the complete curve if we know what it looks like for x0. [Rotate 180Â° about the origin; see Figure 3(b).] Some b) Oud function rotational symmetry FIGURE 3 simple examples of odd functions are y-xy-x'y - and y-sin

This magnetic resonance image of a patient's brain shows a tumor, which is highlighted in red.

Imagine you are a doctor who has just received a magnetic resonance image of your patent’s brain. The brain has a tumor How large is the tumor?

To be precise, what is the area of the red region? The red cross-section of the tumor has an irregular shape, and therefore it is unlikely that you would be able o find a set of equations or inequalities for the region and then be able to calculate its area by conventional means. You could approximate the area by chopping the region into tiny squares (a Riemann sum approach), but this method always gives an answer with some error.

Instead of trying to measure the area of the region directly, we can use a device called a rolling planimeter to calculate the area of the region exactly, simply by measuring its boundary. In this project you investigate how a planimeter works, and you use Green’s theorem to show the device calculates area correctly.

A rolling planimeter is a device that measures the area of a planar region by tracing out the boundary of that region. To measure the area of a region, we simply run the tracer of the planimeter around the boundary of the region. The planimeter measures the number of turns through which the wheel rotates as we (race the boundary; the area of the shape is proportional to this number of wheel ruins. We can derive the precise proportionality equation using Green’s theorem. As the (racer moves around the boundary of the region, the tracer arm rotates and the roller moves back and forth (but does no; rotate).

(a) A rolling planimeter. The pivot allows the tracer arm to rotate. The roller itself does not rotate; it only moves back and forth. (b) An interior view of a rolling planimeter. Notice that the wheel cannot rum if the planimeter is moving back and forth th the tracer arm perpendicular to the roller.

Let C denote the boundary of region D. the area to be calculated. As the tracer traverses curve C, assume the roller moves along the y-axis (since the roller does not rotate, one can assume it moves along a straight line). Use the coordinates (x, y) to represent points on boundary C, and coordinates (0, y) to represent the position of the pivot. As the planimeter traces C the pivot moves along the y-axis while the tracer arm rotates on the pivot. Watch a short animation of a planimeter in action.

Begin the analysis by considering the motion of the tracer as it moves from point (x, y) counterclockwise to point that is close to (x, y). The pivot also moves, from point (0, y) to nearby point (0, y + dy). How much does the wheel turn as a result of this motion? To answer this question, break the motion into two parts. First, roll the pivot along they-axis from without rotating the tracer arm. The tracer arm then ends up a point while maintaining a constant angle with the x-axis. Second, rotate the tracer arm by an angle d θθ without moving the roller. Now the tracer is at point Let 1 be the distance from the pivot to the wheel and let L be the distance from the pivot to the tracer (the length of the tracer arm).

6.49 Mathematical analysis of the motion of the planimeter.

Conclude that the sea of D equals the length of the air am multiplied by the total rolling distance of the wheel. You now know how a planimeter works and you have used Green’s theorem to justify that it works. To calculate the area of planar region D, use a planimeter to trace the boundary of the region .

The area of region is the length of the tracer arm multiplied by the distance wheel rolled

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MAT237Y1 Lecture Notes - Maxima And Minima, Minimax, The Thirteen Chairs

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