Mathematics
Derivatives of Polar Functions
Derivatives of polar functions involve finding the rate of change of a polar curve with respect to the angle. This is done using the polar form of the chain rule and involves expressing the polar coordinates in terms of the Cartesian coordinates. The derivative of a polar function gives the slope of the tangent line to the curve at a specific point.
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eBook - PDFA Course of Mathematical Analysis
International Series of Monographs on Pure and Applied Mathematics
- A. F. Bermant, I. N. Sneddon, S. Ulam, M. Stark(Authors)
- 2016(Publication Date)
- Pergamon(Publisher)
182 COURSE OF MATHEMATICAL ANALYSIS is called the graph of function /. The question arises of the geo-metrical meaning of the derivative άρ/άφ = /' (φ), measuring the rate of change of the radius vector with the polar angle. This is answered by the following theorem. FIG. 66 THEOREM. The rate of change ρ' of the radius vector Q of a curve C* = /(ίΡ) with respect to the polar angle φ is equal to the radius vector multiplied by the cotangent of the angle Θ between the radius vector and the tangent to the curve at the point in question: ρ' = Qcotd. Proof. Let the curve M X M 2 (Fig. 66) be the graph of the function q = f((p). Let the angle φ receive the increment Δφ. Then the point M (ρ, φ) passes to the point M'(ρ -f-Δρ, φ -f Δφ). With the pole P as centre, we draw the arc of a circle MN and drop from M a perpen-dicular ΜΝ Ύ on to the radius vector ΡΜ'. Then (see Fig. 66), Δρ ΝΜ' Δφ (since MN = ρΔφ) or Δρ N^'-N^N MN X _ (Ν λ Μ ! N t N ' Q ~MN^ Δφ NM' = ρ—^ MN Δφ where ΜΝ τ MN MN :) MN X MN ,(t) N X M' ΜΝ^ MN X MN = cot z. N^M'M = cot L PM'M, PN — PN X ρ — ρ cosZl φ ΜΝ λ ρ sinzl φ ρΔφ ρ sinZl^ &ϊηΔφ Δφ = tan Δφ DERIVATIVES AND DIFFERENTIALS 183 Thus, if Δφ -> 0, then N 1 N/MN 1 -> 0, MNJMN -> 1, and since the secant MM' tends to the tangent MT, then L PM'M -> Z. PMT = Θ and N^'jMN^ -> cot0. By virtue of this, on passing to the limit in equation (f) as Δ φ ->· 0, we get: ρ' = ρ cotO (or ρ'/ρ = cotö). This is what we wanted to prove. It is clear from this how, with the aid of the differential calculus, we can solve problems connected with tangents and normals to curves given in polar co-ordinates. Examples. (1) Since the polar equation of a circle with centre at the origin (the pole) has the form p = a = const, we have cot Θ == ρ'/ρ = 0, i.e. Θ = n, i.e. the tangent to a circle is perpendicular to the radius through the point of contact.
eBook - PDFCalculus
Early Transcendentals
- Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
- 2016(Publication Date)
- Wiley(Publisher)
79 2 One of the crowning achievements of calculus is its ability to capture continuous motion mathematically, allowing that motion to be analyzed instant by instant. THE DERIVATIVE Many real-world phenomena involve changing quantities—the speed of a rocket, the inflation of currency, the number of bacteria in a culture, the shock intensity of an earthquake, the voltage of an electrical signal, and so forth. In this chapter we will develop the concept of a “derivative,” which is the mathematical tool for studying the rate at which one quantity changes relative to another. The study of rates of change is closely related to the geometric concept of a tangent line to a curve, so we will also be discussing the general definition of a tangent line and methods for finding its slope and equation. 2.1 TANGENT LINES AND RATES OF CHANGE In this section we will discuss three ideas: tangent lines to curves, the velocity of an object moving along a line, and the rate at which one variable changes relative to another. Our goal is to show how these seemingly unrelated ideas are, in actuality, closely linked. TANGENT LINES In Example 1 of Section 1.1 we used an informal argument to find the equation of a tangent line to a curve. However, at that stage in the text we did not have a precise definition of a tangent line. Now that limits have been defined precisely we can give a mathematical definition of the tangent line to a curve y = f (x) at a point P(x 0 , f (x 0 )) on the curve. As illustrated in Figure 2.1.1, the slope m PQ of the secant line through P and a second point Q(x, f (x)) on the graph of f is m PQ = f (x) − f (x 0 ) x − x 0 If we let x approach x 0 , then the point Q will move along the curve and approach the point P. Suppose the slope m PQ of the secant line through P and Q approaches a limit as x → x 0 . In that case we can take the value of the limit to be the slope m tan of the tangent line at P. Thus, we make the following definition. Figure 2.1.1
eBook - PDF- Daniel Ashlock(Author)
- 2022(Publication Date)
- Springer(Publisher)
2 p 2; 2 p 2/. ˙ One of the major uses for converting between the two coordinate systems is to permit us to plot polar points on normal graph paper when we are graphing a polar function. If you have a good eye, or a protractor, it is possible to plot polar points directly, but typically a person just learning polar coordiantes have far more practice plotting .x; y/-points. One of the nice things about polar coordinates is that they let us deal very easily with circles centered at the origin. Circles centered at the origin are constant functions in polar coordinates. The next example demonstrates this. Example 2.25 Graph the polar function r D 2. Solution: r D 2 (0,3) (0,-3) (3,0) (-3,0) ˙ 64 2. PARAMETRIC, POLAR, AND VECTOR FUNCTIONS We usually write polar functions in the form r D f . / making the angle the independent variable and the r the dependent variable. This makes it very easy to give polar functions as parametric functions. Knowledge Box 2.4 Parametric form of polar curves If r D f . / on 1 2 is a polar curve, then a parametric form for the same curve is: .f .t/ cos.t/; f .t/ sin.t// for t 2 OE 1 ; 2 . So far in this section we have established the connections between polar coordinates and the rest of the systems developed in this text. It is time to display polar curves that have unique characteristics that are most easily seen in the polar system. Example 2.26 Graph the polar function r D cos.3 / on OE 0; /. Solution: (0,1) (0,-1) (1,0) (-1,0) The arrows show the drawing direction. ˙ 2.2. POLAR COORDINATES 65 Definition 2.3 Petal curves are curves with equations of the form: r D cos.n / or r D sin.n /; where n is an integer. If no restriction is placed on n, then the curve is traced out an infinite number of times. For odd n, a domain of 2 OE 0; / traces the curve once; when n is even, 2 OE 0; 2 / is needed to trace the entire curve once. Example 2.27 Compare the curves r D sin.5 / and r D cos.5 / on the range 2 OE 0; /.
eBook - PDF- James Stewart, Daniel K. Clegg, Saleem Watson, , James Stewart, James Stewart, Daniel K. Clegg, Saleem Watson(Authors)
- 2020(Publication Date)
- Cengage Learning EMEA(Publisher)
107 We know that when an object is dropped from a height it falls faster and faster. Galileo discovered that the distance the object has fallen is proportional to the square of the time elapsed. Calculus enables us to calculate the precise speed of the object at any time. In Exercise 2.1.11 you are asked to determine the speed at which a cliff diver plunges into the ocean. Icealex / Shutterstock.com 2 Derivatives IN THIS CHAPTER WE BEGIN our study of differential calculus, which is concerned with how one quantity changes in relation to another quantity. The central concept of differential calculus is the derivative, which is an outgrowth of the velocities and slopes of tangents that we considered in Chapter 1. After learning how to calculate derivatives, we use them to solve problems involving rates of change and the approximation of functions. Copyright 2021 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 108 CHAPTER 2 Derivatives Derivatives and Rates of Change In Chapter 1 we defined limits and learned techniques for computing them. We now revisit the problems of finding tangent lines and velocities from Section 1.4. The special type of limit that occurs in both of these problems is called a derivative and we will see that it can be interpreted as a rate of change in any of the natural or social sciences or engineering.
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