Rates of Change

10 Key excerpts on "Rates of Change"

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  Mathematics 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.

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  Single Variable Calculus, Metric Edition

  • James Stewart, Daniel K. Clegg, Saleem Watson, , James Stewart, James Stewart, Daniel K. Clegg, Saleem Watson(Authors)
  • 2020(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  The instan- taneous rate of change is the limit of the average Rates of Change, so it is measured in the same units: billions of dollars per year. 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. 116 CHAPTER 2 Derivatives rate of change of production cost with respect to the number of items produced; the rate of change of the debt with respect to time is of interest in economics. Here is a small sample of other Rates of Change: In physics, the rate of change of work with respect to time is called power. Chemists who study a chemical reaction are interested in the rate of change in the concentration of a reactant with respect to time (called the rate of reaction). A biologist is interested in the rate of change of the population of a colony of bacteria with respect to time. In fact, the computation of Rates of Change is important in all of the natural sciences, in engineering, and even in the social sciences. Further examples will be given in Section 2.7. All these Rates of Change are derivatives and can therefore be interpreted as slopes of tangents. This gives added significance to the solution of the tangent problem. Whenever we solve a problem involving tangent lines, we are not just solving a problem in geom- etry. We are also implicitly solving a great variety of problems involving Rates of Change in science and engineering. 2.1 Exercises 1. A curve has equation y - f s xd. (a) Write an expression for the slope of the secant line through the points Ps3, f s3dd and Qs x, f s xdd. (b) Write an expression for the slope of the tangent line at P .

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  Mathematics for Business

  • Gary Bronson, Richard Bronson, Maureen Kieff(Authors)
  • 2021(Publication Date)
  • Mercury Learning and Information

    (Publisher)

  We reserve our study of these applications, however, until Chapter 6. In this chapter, we direct our attention to the derivative as a rate of change and providing specific rules for calculating derivatives of commonly encountered mathematical functions. 5.1 AVERAGE Rates of Change An extremely useful measure in business forecasting is the average rate at which a quantity changes. For example, it is useful in predicting sales for the month of April to know that sales in March totaled 10,000 units. Even more useful, however, is the additional information that sales have been increasing, on average, at a rate of 2,000 units per month. Similarly, a company needs information on wage scales for the past year before it can prepare next year’s budget. In addition, however, information on the average yearly increase in salaries is also important. In both these situations knowledge of the rates that quantities change, either changes in sales per month or changes in wages paid per hour, increases one’s ability to forecast future requirements accurately. One rate of change you are probably familiar with is average speed, which measures the change in distance for a given change in time. For example, if an individual traveled 500 miles in 10 hours, the average change in distance with respect to time is 50 miles per hour. That is, on the average, each hourly increase in driving time resulted in an increase of 50 miles traveled. Formally, this average speed is calculated by dividing the total change in miles driven by the total change in hours traveled using the formula: Example 1: A woman drove from her house to a destination 400 miles away. The following table shows how far she traveled at 1 hour intervals

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  Calculus

  Single Variable

  • Deborah Hughes-Hallett, Andrew M. Gleason, William G. McCallum, Daniel E. Flath, Patti Frazer Lock, David O. Lomen, David Lovelock, Brad G. Osgood, Douglas Quinney, Karen R. Rhea, Jeff Tecosky-Feldman, Thomas W. Tucker, Otto K. Bretscher, Sheldon P. Gordon, Andrew Pasquale, Joseph Thrash(Authors)
  • 2014(Publication Date)
  • Wiley

    (Publisher)

  2.2 THE DERIVATIVE AT A POINT Average Rate of Change In Section 2.1, we looked at the change in height divided by the change in time; this ratio is called the difference quotient. Now we define the rate of change of a function f that depends on a variable other than time. We say: Average rate of change of f over the interval from a to a + h = f (a + h) − f (a) h . The numerator, f (a + h) − f (a), measures the change in the value of f over the interval from a to a + h. The difference quotient is the change in f divided by the change in the independent variable, which we call x. Although the interval is no longer necessarily a time interval, we still talk about the average rate of change of f over the interval. If we want to emphasize the independent variable, we talk about the average rate of change of f with respect to x. Instantaneous Rate of Change: The Derivative We define the instantaneous rate of change of a function at a point in the same way that we defined instantaneous velocity: we look at the average rate of change over smaller and smaller intervals. This instantaneous rate of change is called the derivative of f at a, denoted by f ′ (a). The derivative of f at a, written f ′ (a), is defined as Rate of change of f at a = f ′ (a) = lim h→0 f (a + h) − f (a) h . If the limit exists, then f is said to be differentiable at a. To emphasize that f ′ (a) is the rate of change of f (x) as the variable x changes, we call f ′ (a) the derivative of f with respect to x at x = a. When the function y = s(t) represents the position of an object, the derivative s ′ (t) is the velocity. 86 Chapter Two KEY CONCEPT: THE DERIVATIVE Example 1 Eucalyptus trees, common in California and the Pacific Northwest, grow better with more water. Scientists in North Africa, analyzing where to plant trees, found that the volume of wood that grows on a square kilometer, in meters 3 , is approximated by 2 V (r) = 0.2r 2 − 20r + 600, where r is rainfall in cm per year, and 60 ≤ r ≤ 120.

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  Functions and Change

  A Modeling Approach to College Algebra

  • Bruce Crauder, Benny Evans, Alan Noell, , Bruce Crauder, Benny Evans, Alan Noell(Authors)
  • 2017(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  WCN 02-200-202 four.lffour.lffour.lf CHAPTER six.lf | RA T ES OF CHANGE Rates of Change for Other Functions Velocity is the rate of change in directed distance, and this same idea applies to any function. For a function f 5 f s x d , we can look at the rate of change in f with respect touni00A0 x , which tells how f changes for a given change in x . Rates of Change are so pervasive in mathematics, science, and engineering that they are given a special notation, most commonly d f d d d x d d or f 9 s x d , and a special name, the derivative of f with respect to x . We will use the notation d f d d d x d d but will consistently refer to it as the rate of change in f with respect to x . We should note that some applications texts use the notation D f D D D x for the rate of change. 1 uni25A0 Examples of Rates of Change The notation d f d d d x d d means the rate of change in f with respect to x . Specifically, d f d d d x d d tells how much f is expected to change if x increases by 1 unit. We note that d f d d d x is a func -tion of x ; in general, the rate of change varies as x varies. Some examples may help clarify things. one.lf. If S 5 S s t d gives directed distance for an object as a function of time t , then d S d t is the rate of change in directed distance with respect to time. This is velocity . It tells the additional distance we expect to travel in 1 unit of time. For example, if we are currently located S 5 1 0 0 miles south of Dallas, Texas, and if we are traveling south with a velocity d S d t of 50 miles per hour, then in 1 additional hour, we would expect to be 150 miles south of Dallas. two.lf. If V 5 V s t d is the velocity of an object as a function of time t , then d V d t is the rate of change in velocity with respect to time. This is acceleration . It tells the additional velocity we expect to attain in 1 unit of time.

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  Calculus and Ordinary Differential Equations

  • 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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  College Algebra

  Building Skills and Modeling Situations

  • Charles P. McKeague, Katherine Yoshiwara, Denny Burzynski(Authors)
  • 2013(Publication Date)
  • XYZ Textbooks

    (Publisher)

  EXAMPLE 6 © Dmitriy Shironosov/ iStockphoto 268 Chapter 4 Slope, Rates of Change, and Linear Functions Difference Quotients One of the important applications of calculus is finding Rates of Change. As you may recall, the slope of a line represents a rate of change. In the diagram below, the slope of the blue line is the average rate of change of the red curve from point P to point Q. The slope of the line passing through the points P and Q is given by the formula Slope of line through PQ = m = f (x) − f (a) ________ x − a The expression f (x) − f (a) ______ x − a is called a difference quotient. It represents the average rate of change of the function f from the point (a, f (a)) to the point (x, f (x)). If f (x) = 3x − 5, find f (x) − f (a) _________ x − a . SOLUTION f (x) − f (a) _________ x − a = (3x − 5) − (3a − 5) ________________ x − a = 3x − 3a _______ x − a = 3(x − a) _______ x − a = 3 If f (x) = x 2 − 4, find f (x) − f (a) ________ x − a and simplify. SOLUTION Because f (x) = x 2 − 4 and f (a) = a 2 − 4, we have f (x) − f (a) ________ x − a = (x 2 − 4) − (a 2 − 4) _______________ x − a = x 2 − 4 − a 2 + 4 ____________ x − a = x 2 − a 2 ______ x − a = (x + a)(x − a) ____________ x − a Factor and divide out common factor = x + a FIGURE 8 x y y = f(x) P Q Run = x - a Rise = f(x) - f(a) a x f(x) f(a) x - a Rise Run m = EXAMPLE 7 EXAMPLE 8 4.1 The Slope of a Line 269 There is a second form of the difference quotient. The diagram in Figure 9 is simi- lar to the one in Figure 8. The main difference is in how we label the points. From Figure 9, we can see another difference quotient that gives us the slope of the line through the points P and Q. Slope of line through PQ = m = f (x + h) − f (x) ____________ h Examples 9 and 10 use the same functions used in Examples 7 and 8, but this time the new difference quotient is used. If f (x) = 3x − 5, find f (x + h) − f (x) ____________ h .

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  Applied Calculus

  for Business, Life, and Social Sciences

  • Denny Burzynski(Author)
  • 2014(Publication Date)
  • XYZ Textbooks

    (Publisher)

  44 Chapter 1 Functions, Limits, and Rates of Change Solution To find the average rate of change in the surface area from r = 1 to r = 3, we find the slope of the line that passes through the points (1, S(1)) and (3, S(3)). Average rate of change = S(3) − S(1) __________ 3 − 1 = 4π . 3 2 − 4π . 1 2 _____________ 3 − 1 = 36π − 4π ________ 2 = 16π in 2 /in The surface area of the balloon increases an aver- age of 16π in 2 /in as the radius increases from 1 inch to 3 inches. Note that this is one situation in which reducing the units to in 2 /in = in makes the result less understandable. Difference Quotients and Average Rate of Change We can generalize the average rate of change of any function, and, at the same time, begin our work with a formula that is important in calculus. In the diagram below, the slope of the blue line (called a secant line) is the average rate of change of the red curve from point P to point Q. The slope of the line passing through the points P and Q is given by the following formula: Slope of line through P and Q = m = f (b ) − f (a) __________ b − a The expression f (b) − f (a) ______ b − a is called a difference quotient. It represents the average rate of change of the function f from the point (a, f (a)) to the point (b, f (b)). Throughout your study of calculus, you will see the difference quotient in a variety of forms. For instance, Example 8 uses the points (x 1 , y 1 ) and (x 2 , y 2 ) rather than (a, f (a)) and (b, f (b)). Either way, the difference quotient represents the slope of the secant line and the average rate of change between those points. Figure 6 r S(r) 100 50 150 200 1 2 3 4 S (3) - S(1) 3 - 1 Figure 7 x y y = f (x) P Q Run = b - a Rise = f (b) - f (a) a x f (b) f (a) b - a Rise Run m = Note A line that passes through two points on a graph is called a secant line. We will see much more of these lines in Section 1.6.

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  A 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)

  This average rate indicates the number of units of / per unit of φ in the interval (t, t + At). Definition. The limit v of the average rate v a v as A t -> 0, if it exists, is called the rate of change of function f(t) with respect to function (p(t) at the given point t*: v = lim ν αυ = lim A yA x. At-+0 At->0 Let us evaluate v. We have: Ay f(t + At)-f(t) ,. Δν Λ . At Έ . At hm -τ— = hm -;— = lim — j - — At-+o Ax At ^o Ax At->o q>(t + At) -cp{t) At At Thus, y f ff{t) V ~V~ w'(t) f(t)

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  Applied Calculus

  • Deborah Hughes-Hallett, Andrew M. Gleason, Daniel E. Flath, Patti Frazer Lock, Sheldon P. Gordon, David O. Lomen, David Lovelock, William G. McCallum, Brad G. Osgood, Andrew Pasquale, Jeff Tecosky-Feldman, Joseph Thrash, Karen R. Rhea, Thomas W. Tucker(Authors)
  • 2022(Publication Date)
  • Wiley

    (Publisher)

  Now we link this idea to derivatives. Analogous to the relative change, we look at the rate of change as a fraction of the original quantity. The relative rate of change of  =  () at  =  is defined to be Relative rate of change of  at  = ∕  =   ()  () . We see in Section 3.3 that an exponential function has a constant relative rate of change. If the independent variable is time, the relative rate is often given as a percent change per unit time. Example 8 Annual world soybean production, 17  =  (), in million tons, is a function of  years since the start of 2010. (a) Interpret the statements  (8) = 358 and   (8) = 16 in terms of soybean production. (b) Calculate the relative rate of change of  at  = 8; interpret it in terms of soybean production. Solution (a) The statement  (8) = 358 tells us that 358 million tons of soybeans were produced in the year 2018. The statement   (8) = 16 tells us that in 2018 annual soybean production was increasing at a rate of 16 million tons per year. (b) We have Relative rate of change of soybean production =   (8)  (8) = 16 358 = 0.045. In 2018, annual soybean production was increasing at a continuous rate of 4.5% per year. 17 http://www.sopa.org/statistics/world-soybean-production, accessed March 31, 2021. 120 Chapter 2 RATE OF CHANGE: THE DERIVATIVE Example 9 Solar photovoltaic (PV) cells are the world’s fastest growing energy source. At time  in years since 2018, peak PV energy-generating capacity worldwide was approximately  = 512 0.24 gigawatts. 18 Estimate the relative rate of change of PV energy-generating capacity in 2022 using this model with (a) Δ = 1 (b) Δ = 0.1 (c) Δ = 0.01 Solution Let  =  (). In 2022 we have  = 4. The relative rate of change of  in 2022 is   (4)∕ (4). We estimate   (4) using a difference quotient. (a) Estimating the relative rate of change using Δ = 1 at  = 4, we have ∕  =   (4)  (4) ≈ 1  (4)  (5) −  (4) 1 = 0.271 = 27.1% per year.

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