Textbook Guide Mathematics: Semi-Major And Semi-Minor Axes, Parametric Equation, Polar Regions Of Earth

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MAT133Y1 Full Course Notes

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Suppose that x and y are both given as functions of a third variable t (called a parameter) by the equations x = f(t) and y = g(t) (called parametric equations). Parametric curve: each value of t determines a point (x,y), which we can plot in a coordinate plane. In general, the curve with parametric equations has initial point (f(a), g(a)) and terminal point (f(a), g(a)). Graphing devices: most graphing calculators and computer graphing programs can be used to graph curves defined by parametric equations. Areas: we can calculate an area formula by using the substitution rule for definite integrals as follows: We choose a point in the plane that is called the pole (or origin) and is labeled o. Then we draw a ray (half-line) starting at o called the polar axis. Then the point is represented by the ordered pair (r, ) and r, are called polar coordinates of p. Area of a sector of a circle:

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Help with 49,(guiding in second pic ) and 29 please

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

Using a technique of integration and the separation-of-variables technique, you'll discover a function whose antiderivative models the shape of an ideal cable suspended from its endpoints. The function depends on a parameter b, which is determined by the properties of the cable, such as the density of the material out of which it is made.

Using principles from physics, it can be shown that when a perfectly flexible, uniformly dense, and inextensible cable is suspended from its endpoints, then it will assume a shape of a curve y = f(x) satisfying the following second-order differential equation:

where Ï is the density of the material of the cable, g is the acceleration due to gravity, and T is the magnitude of the tension in the cable at its lowest point.

We assume our cable is as in the diagram above and y = f(x) is its shape function; thus, not only does y satisfy the "hanging cable differential equation" above, we also have y '(0) = 0. Hence, making the substitution u =dy/dx in our the hanging cable equation, we obtain the first order, initial-value problem satisfied by u:

where b = Ïg/T Solve the preceding initial-value problem, entering your (explicit) solution in the form u = g(x) where g is a function of x in which the constant b is present.

Textbook Guide Mathematics: Semi-Major And Semi-Minor Axes, Parametric Equation, Polar Regions Of Earth

MAT133Y1 Full Course Notes

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