Mathematics
Related Rates
Related rates is a mathematical concept that deals with the changing rates of related quantities. It involves finding the rate of change of one quantity with respect to another, often using the chain rule from calculus. This concept is commonly used to solve problems involving changing geometric shapes, fluid dynamics, and other dynamic systems.
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8 Key excerpts on "Related Rates"
eBook - PDFThe Calculus Lifesaver
All the Tools You Need to Excel at Calculus
- Adrian Banner(Author)
- 2009(Publication Date)
- Princeton University Press(Publisher)
(By the way, sometimes we’ll abbreviate and say “rate” instead of “rate of change.”) The above definition of a rate of change was a little sketchy. If you want to know how fast something is changing over time, you simply have to dif-ferentiate with respect to time. So, here’s the real definition: the rate of change of a quantity Q is the derivative of Q with respect to time . That is, if Q is some quantity, then the rate of change of Q is dQ dt . When you see the word “rate,” you should automatically think “ d/dt .” So, how do you go from an equation relating two quantities to an equation relating the rates of change of these quantities? You differentiate, of course! Section 8.2.1: A simple example • 157 If you differentiate both sides implicitly with respect to t , you’ll find that the rates just pop out, giving you a new equation. The same thing works if you are dealing with three or more quantities which are related (for example, the length, width, and area of a rectangle). Just differentiate implicitly with respect to t and you’ll relate the rates of change. So, let’s look at a general overview of how to solve problems involving Related Rates. Then we’ll use it to solve a bunch of examples. 1. Read the question. Identify all the quantities and note which one you need to find the rate of. Draw a picture if you need to! 2. Write down an equation (sometimes you need more than one) that relates all the quantities. To do this, you may need to do some geometry, possibly involving similar triangles. If you have more than one equation, try to solve them simultaneously to eliminate unnecessary variables. 3. Differentiate your remaining equation(s) implicitly with respect to time t . That is, whack both sides of each equation with a d dt . You end up with one or more equations relating the rates of change. 4. Finally, substitute values for everything you know into all the equations you have. Solve the equations simultaneously to find the rate you need.
eBook - PDFMathematics NQF4 SB
TVET FIRST
- M Van Rensburg, I Mapaling M Trollope(Authors)
- 2017(Publication Date)
- Macmillan(Publisher)
Calculus is divided in two main fields: differentiation and integration . Differentiation allows us to find the rate of change of a function at one particular moment ( instantaneous rate of change ). We’ll discuss this in Unit 6.1. When you simplify a situation or problem in real life in such a way that you express it and solve it mathematically, you use ‘mathematical modelling ’. calculus: a powerful mathematical tool with applications in science, technology and economics differentiation: to use limits and find the derivatives of functions in order to tell the rate of change of such functions at one particular moment New words 121 Module 6 In this module we investigate and use instantaneous rate of change of a variable when interpreting models both in mathematical and real-life situations. Unit 6.1: Derivatives of functions from first principles Rate of change is about one thing changing in relation to another thing. It is often calculated as an average. For example, to calculate speed, you will divide the distance travelled by a certain time interval. Therefore, a straight-line graph showing the relation between speed, time and distance will have a constant gradient , because there is a constant rate of change. This gradient is the average speed in the given time interval. However, scientists and mathematicians may need to find the speed at one specific moment in time, rather than an average over a certain time period. This is known as the instantaneous rate of change. 6.1.1 Instantaneous rate of change Think about the reading on the speedometer of a car. This reading changes as soon as the car speeds up or slows down, so it keeps giving the car ′ s speed at a particular moment in time. How do we calculate rate of change at a particular instant if the formula for calculating rate of change is given as change per change in time? • To pinpoint the rate of change, we would need a very detailed table of values or a detailed graph.
eBook - PDF- Sebastian J. Schreiber, Karl J. Smith, Wayne M. Getz(Authors)
- 2014(Publication Date)
- Wiley(Publisher)
By finding the derivative of functions, we calculate the slopes of tangent lines to these functions and examine rates of change defined by these functions. Applications considered in this chapter include enzyme kinetics, biodiversity, foraging of hummingbirds, wolf predation, and population dynamics of the sockeye salmon depicted in Figure 2.1. 2.1 Rates of Change and Tangent Lines One of the fundamental concepts in calculus is that of the limit. Intuitively, “taking a limit” corresponds to investigating the value of a function as you get closer and closer to a specified point without actually reaching that point. To “motivate limits,” we begin by showing how they arise when we consider rates of change and the tangent lines to graphs of functions. 107 108 Chapter 2 Limits and Derivatives Rates of change Rates of change describe how quantities change with respect to a variable such as time or body mass. For instance, in countries where overpopulation is an issue, projections are constantly made about the population growth rate. For a patient receiving drug treatment, physicians perform experiments to estimate the clearance rate of the drug (i.e., the rate at which the drug leaves the blood stream). To calculate a rate of change, we first select an interval of interest and then compute the average rate of change over that interval using the equation in the box below. To calculate the rate of change at a point of interest requires taking limits, as discussed after the next example. Average Rate of Change The average rate of change of f over the interval [a, b] is f (b) − f (a) b − a Example 1 Mexico population growth The (estimated) population size of Mexico (in millions) in the early 1980s is reported in the following table. Year Time lapsed t Population N( t) 1980 0 67.38 1981 1 69.13 1982 2 70.93 1983 3 72.77 1984 4 74.66 1985 5 76.60 From this table, we see that N(t ) denotes the population size in millions t years after 1980.
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.
eBook - PDFCalculus
Single Variable
- Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
- 2016(Publication Date)
- Wiley(Publisher)
59 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 - PDFCalculus
Early Transcendental Single Variable
- 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- Michael A. Calter, Paul A. Calter, Paul Wraight, Sarah White(Authors)
- 2016(Publication Date)
- Wiley(Publisher)
29 ◆◆◆ OBJECTIVES ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ When you have completed this chapter, you should be able to: • Calculate rates of change. • Calculate the motion of a point. • Solve problems using Related Rates. • Find optimal solutions using derivatives. What is this stuff good for? This chapter gives the main answer to that question, at least for the derivative. You saw in Chapter 28 how applications using the derivative were helpful in solving math- ematical and geometric problems. In this chapter, we will spread out more and look at other applications. These applications have to do with rates of change. This is where calculus shines at problem solving. You saw that the derivative represents the slope of a line—the amount y changes with respect to a change in x. If the x axis becomes a time axis, anything that changes over time, such as velocity or acceleration, can become a derivative calculation. Calculus is the best way of describing anything in motion. The inspiration for the development of calculus came from Newton’s study of gravity. Gravity follows an inverse square law and plots not as a straight line but as a curved line. The curved line has a slope that is changing. Algebra is not very good at describing rates of change, so the need to describe rates of change led Newton to develop calculus. For electrical applications, we use the rate of change of charge to find current, the rate of change of current to find the voltage across an inductor, and the rate of change of voltage to find the current in a capacitor. We will then go on to use the fact that the derivative is zero at a maximum or minimum point to find maximum and minimum values of a varying quantity. Applied Applications of the Derivative 29–1 Rate of Change Rate of Change Given by the Derivative We already established in Chapter 27 that the first derivative of a function gives the rate of change of that function.
eBook - PDF- David Pearson(Author)
- 1995(Publication Date)
- Butterworth-Heinemann(Publisher)
If you have access to software which will plot graphs and which has a 'zoom' capability, use this to investigate scaling. 3. Provide further examples of physical laws which may be expressed in terms of linear relationships between measurable quantities. In each case, identify any limitations in the range of applicability of linearity -is linearity valid over the whole range, or just a local approximation? 4.3 Speed and velocity Perhaps our most direct experience of the idea of derivative as rate of change may be while travelling along a motorway at 70 miles per hour, or faster still on a roller-coaster at a fair. We take it for granted that at any instant of time t we are travelling at some definite speed, and that our speed can change as the value of t changes. Let us see how to describe mathematically a moving object such as a car or a roller-coaster. Let s be the distance travelled by the object in time t. Taking t == 0 to be zero time at which the object begins its journey, s will be defined for t ~ 0, and we can regard s as a function of t. The object need not necessarily move along a straight line, but in any case s(t) will measure the total distance travelled by the object between time 0 and time t. Certainly s(t) can only increase or remain constant as t increases, and if s(t) remains constant then the object must be at rest. Rates of Change, Slopes, Tangents 45 The simplest case, apart from that of a body at rest (in which case s is a constant function), is that in which s depends linearly on t, say s == a + ut, where a and u are constants. In this case, between times tl and t2 the object moves a distance (a + ut2) -(a + utI) == U(t2 -tI), and the value of u, the distance travelled per unit time, is the speed of the object, and remains constant. More generally, if s is not necessarily linear in t, we can define the instantaneous speed to be the rate of change of distance with respect to time.
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