Mathematics
Derivatives
In mathematics, derivatives represent the rate of change of a function at a given point. They are used to calculate slopes, velocities, and rates of change in various real-world applications. Derivatives are fundamental in calculus and are essential for understanding the behavior of functions and solving optimization problems.
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eBook - PDFCalculus
Late Transcendental
- 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
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Library Press(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter-3 Derivative The graph of a function, drawn in black, and a tangent line to that function, drawn in red. The slope of the tangent line is equal to the derivative of the function at the marked point. In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's instantaneous velocity. Conversely, the integral of the object's velocity over time is how much the object's position changes from the time when the integral begins to the time when the integral ends. The derivative of a function at a chosen input value describes the best linear approxi-mation of the function near that input value. For a real-valued function of a single real variable, the derivative at a point equals the slope of the tangent line to the graph of the function at that point. In higher dimensions, the derivative of a function at a point is a ________________________ WORLD TECHNOLOGIES ________________________ linear transformation called the linearization. A closely related notion is the differential of a function. The process of finding a derivative is called differentiation . The reverse process is called antidifferentiation . The fundamental theorem of calculus states that antidifferentiation is the same as integration. Differentiation and the derivative At each point, the derivative of is the slope of a line that is tangent to the curve. The line is always tangent to the blue curve; its slope is the derivative. Note derivative is positive where green, negative where red, and zero where black. Differentiation is a method to compute the rate at which a dependent output y changes with respect to the change in the independent input x .
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Learning Press(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter 3 Derivative The graph of a function, drawn in black, and a tangent line to that function, drawn in red. The slope of the tangent line is equal to the derivative of the function at the marked point. In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's instantaneous velocity. Conversely, the integral of the object's velocity over time is how much the object's position changes from the time when the integral begins to the time when the integral ends. The derivative of a function at a chosen input value describes the best linear approximation of the function near that input value. For a real-valued function of a single real variable, the derivative at a point equals the slope of the tangent line to the graph of the function at that point. In higher dimensions, the derivative of a function at a point is a ________________________ WORLD TECHNOLOGIES ________________________ linear transformation called the linearization. A closely related notion is the differential of a function. The process of finding a derivative is called differentiation . The reverse process is called antidifferentiation . The fundamental theorem of calculus states that antidifferentiation is the same as integration. Differentiation and the derivative At each point, the derivative of is the slope of a line that is tangent to the curve. The line is always tangent to the blue curve; its slope is the derivative. Note derivative is positive where green, negative where red, and zero where black. Differentiation is a method to compute the rate at which a dependent output y changes with respect to the change in the independent input x .
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)
- 2022(Publication Date)
- Wiley(Publisher)
58 CHAPTER 2 Henglein and Steets/Getty Images 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.
eBook - PDF- Geoffrey Berresford, Andrew Rockett(Authors)
- 2015(Publication Date)
- Cengage Learning EMEA(Publisher)
These three interpre-tations of the derivative will be used throughout this book. Marginal Analysis: Derivatives in Business and Economics There is another interpretation for the derivative, one that is particularly import-ant in business and economics. Suppose that a company has calculated its revenue, cost, and profit functions, as defined below. R ( x ) 5 a Total revenue (income) from selling x units b Revenue function C ( x ) 5 a Total cost of producing x units b Cost function P ( x ) 5 a Total profit from producing and selling x units b Profit function The term marginal cost means the additional cost of producing one more unit, C ( x 1 1) 2 C ( x ) , which may be written C ( x 1 1) 2 C ( x ) 1 , which is just the difference quotient C ( x 1 h ) 2 C ( x ) h with h 5 1. If many units are being produced, then h 5 1 is a relatively small number compared with x, so this difference quotient may be approximated by its limit as h S 0 , that is, by the derivative of the cost function. In view of this approximation, in calculus the marginal cost is defined to be the derivative of the cost function: MC ( x ) 5 C 9 ( x ) Marginal cost is the derivative of cost The marginal revenue function MR ( x ) and the marginal profit function MP ( x ) are similarly defined as the Derivatives of the revenue and cost functions. MR ( x ) 5 R 9 ( x ) Marginal revenue is the derivative of revenue MP ( x ) 5 P 9 ( x ) Marginal profit is the derivative of profit All three formulas can be summarized very briefly: “marginal” means “derivative of.” The use of Derivatives to find the changes in revenue, cost, or profit resulting from one additional unit is called marginal analysis . We now have three interpretations for the derivative: slopes , instantaneous rates of change , and marginals .
eBook - PDF- Deborah Hughes-Hallett, William G. McCallum, Andrew M. Gleason, Eric Connally, Daniel E. Flath, Selin Kalaycioglu, Brigitte Lahme, Patti Frazer Lock, David O. Lomen, David Lovelock, Guadalupe I. Lozano, Jerry Morris, David Mumford, Brad G. Osgood, Cody L. Patterson, Douglas Quinney, Karen R. Rhea, Ayse Arzu Sahin, Ad(Authors)
- 2017(Publication Date)
- Wiley(Publisher)
. . . . . . . . . . . . . . . . 115 Second and Higher Derivatives . . . . . . . . . . . . . . 115 What Do Derivatives Tell Us? . . . . . . . . . . . . . . . 115 Interpretation of the Second Derivative as a Rate of Change . . . . . . . . . . . . . . . . . . . . . . 117 Velocity and Acceleration . . . . . . . . . . . . . . . . . . 118 2.6 Differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 What Does It Mean for a Function to Be Differentiable? . . . . . . . . . . . . . . . . . . . . . 123 Some Nondifferentiable Functions . . . . . . . . . . . . 124 Differentiability and Continuity . . . . . . . . . . . . . 125 Chapter Two KEY CONCEPT: THE DERIVATIVE 84 Chapter 2 KEY CONCEPT: THE DERIVATIVE 2.1 HOW DO WE MEASURE SPEED? The speed of an object at an instant in time is surprisingly difficult to define precisely. Consider the statement: “At the instant it crossed the finish line, the horse was traveling at 42 mph.” How can such a claim be substantiated? A photograph taken at that instant will show the horse motionless—it is no help at all. There is some paradox in trying to study the horse’s motion at a particular instant in time, since by focusing on a single instant we stop the motion! Problems of motion were of central concern to Zeno and other philosophers as early as the fifth century B.C. The modern approach, made famous by Newton’s calculus, is to stop looking for a simple notion of speed at an instant, and instead to look at speed over small time intervals containing the instant. This method sidesteps the philosophical problems mentioned earlier but introduces new ones of its own. We illustrate the ideas discussed above with an idealized example, called a thought experiment. It is idealized in the sense that we assume that we can make measurements of distance and time as accurately as we wish.
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)
On finding y' from the first equation and substituting in the second, we arrive at an equation from which y can be expressed in terms of x and y. To find a higher order derivative of a function given parametrically, we have to differentiate the expression for the previous derivative as a function of the independent variable. Let y = f(t), x = (p(t). Derivatives AND DIFFERENTIALS 191 VVC I l c t V C Further, ,, d W(t)l y dx and since <-£' ά(-Ά)
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