Mathematics
The Power Function
The power function is a mathematical concept that represents a relationship where one quantity is raised to the power of another. It is written in the form f(x) = x^n, where x is the base and n is the exponent. The power function is used to model various real-world phenomena and is a fundamental concept in algebra and calculus.
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11 Key excerpts on "The Power Function"
eBook - PDFMathematics
An Applied Approach
- Michael Sullivan, Abe Mizrahi(Authors)
- 2017(Publication Date)
- Wiley(Publisher)
[Recall that , and so on, are not defined in the system of real numbers.] COMMENT: It is important to distinguish a power function g(x) x n , n 2 an inte- ger, from an exponential function f (x) a x , a 0, a 1, a real. In a power function, the base is a variable and the exponent is a constant. In an exponential function, the base is a constant and the exponent is a variable. ◗ (2) 1/2 √2 , (3) 3/4 √ 4 (3) 3 √ 4 27 x 3 4 x 1 2 628 Chapter 11 Classes of Functions Some examples of exponential functions are Notice that in each example, the base is a constant and the exponent is a variable. You may wonder what role the base a plays in the exponential function f (x) a x . We use the following Exploration to find out. f(x) 2 x , F(x) 1 3 x TABLE 4 x f (x) 2 x 2 1 0 1 1 2 2 4 3 8 1 2 f (2) 2 2 1 2 2 1 4 TABLE 5 x g(x) 3x 2 2 g(2) 3(2) 2 4 1 1 0 2 1 5 2 8 3 11 EXPLORATION (a) Evaluate f (x) 2 x at x 2, 1, 0, 1, 2, and 3. (b) Evaluate g(x) 3x 2 at x 2, 1, 0, 1, 2, and 3. (c) Comment on the pattern that exists in the values of f and g. Result (a) Table 4 shows the values of f (x) 2 x for x 2, 1, 0, 1, 2, and 3. (b) Table 5 shows the values of g(x) 3x 2 for x 2, 1, 0, 1, 2, and 3. (c) In Table 4 we notice that each value of the exponential function f (x) a x 2 x could be found by multiplying the previous value of the function by the base, a 2. For example, and so on. Put another way, we see that the ratio of consecutive outputs is constant for unit increases in the inputs. The constant equals the value of the base of the exponential function a. For example, for the function f (x) 2 x , we notice that and so on. f (x 1) f (x) 2 x 1 2 x 2 f (1) f (0) 2 1 2, f (1) f (2) 1 2 1 4 2, 2 1 2 f (0) 2 f (1) 2 1 2 1, f (1) 2 f (0) f (1) 2 f (2) 2 1 4 1 2 ,
eBook - PDFAlgebra
Form and Function
- William G. McCallum, Eric Connally, Deborah Hughes-Hallett(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
The graph of () = 4 is U-shaped because the exponent, 4, is even, so it is Graph . That leaves Graph for ℎ() = 7 . This makes sense because the exponent, 7, is odd and Graph is chair-shaped. What Is a Power Function? In a power function the output is proportional to a power of the input. For example a circle with radius has area = 2 . Here is proportional to 2 , with constant of proportionality , so the area is a power function of the radius. In general: A power function is a function that can be written () = , for constants and . We call the coefficient and the exponent. In this section we consider only power functions where the exponent is a positive integer. Example 3 The stopping distance, in feet, of an Alfa Romeo on dry road is proportional to the square of its speed, mph, at the time the brakes are applied, and is given by The Power Function () = 0.04 2 . (a) Identify the coefficient and the exponent. (b) Evaluate (35) and (113) and interpret your answers in terms of stopping distances. Solution (a) The coefficient is 0.04 and and the exponent is 2. (b) We have (35) = 0.04 ⋅ 35 2 = 49 and (113) = 0.04 ⋅ 113 2 = 511. These statements tell us that it takes 49 ft for the car to stop if it is traveling at a speed of 35 mph and 511 feet if it is traveling at 113 mph. It takes longer to stop if it is traveling faster, which makes sense. Figure 4.3 shows this behavior on the graph of : As gets larger the values () get larger. 50 100 100 200 300 400 500 (35, 49) (113, 511) () , mph feet Figure 4.3: Stopping distance of an Alfa Romeo 50 50 (35, 49) (35, 61.25) Wet road Dry road , mph feet Figure 4.4: Stopping distance on wet road versus dry road, as a function of velocity The next example compares two different power functions with the same exponent but different coefficients.
- Andrei D. Polyanin, Alexei Chernoutsan(Authors)
- 2010(Publication Date)
- CRC Press(Publisher)
Chapter M2 Elementary Functions Basic elementary functions : power, exponential, logarithmic, trigonometric, and inverse trigonometric (arc-trigonometric or antitrigonometric) functions. All other elementary functions are obtained from the basic elementary functions and constants by means of the four arithmetic operations (addition, subtraction, multiplication, and division) and the operation of composition (composite functions). The graphs and the main properties of the basic as well as some other frequently occurring elementary functions of the real variable are described below. M2.1. Power, Exponential, and Logarithmic Functions M2.1.1. Power Function: y = x α ( α is an Arbitrary Real Number) ◮ Graphs of The Power Function. General properties of the graphs: the point ( 1 , 1 ) belongs to all the graphs, and y > 0 for x > 0 . For α > 0 , the graphs pass through the origin ( 0 , 0 ); for α < 0 , the graphs have the vertical asymptote x = 0 ( y → + ∞ as x → 0 ). For α = 0 , the graph is a straight line parallel to the x -axis. Consider more closely the following cases. Case 1 : y = x 2 n , where n is a positive integer ( n = 1 , 2 , ... ). This function is defined for all real x and its range consists of all y ≥ 0 . This function is even, nonperiodic, and unbounded. It crosses the axis Oy and is tangential to the axis Ox at the origin x = 0 , y = 0 . On the interval (– ∞ , 0 ) this function decreases, and it increases on the interval ( 0 , + ∞ ). It attains its minimum value y = 0 at x = 0 . The graph of the function y = x 2 (parabola) is given in Fig. M2.1 a . O O 1 1 1 2 1 1 2 2 1 3 2 1 1 2 2 3 3 x y x = y x = y x = y x = 2 2 1 3 x y y ( ) a ( ) b Figure M2.1. Graphs of The Power Function y = x n , where n is an integer. 15 16 E LEMENTARY F UNCTIONS Case 2 : y = x 2 n + 1 , where n is a positive integer. This function is defined on the entire x -axis and its range coincides with the y -axis.
eBook - PDF- David Pearson(Author)
- 1995(Publication Date)
- Butterworth-Heinemann(Publisher)
(7.2) It is difficult to imagine what the mathematical world would be like without the all pervasive exponential function, and its close cousin and inverse function the logarithm. If these functions were no longer available, a considerable body of mathematics, at all levels, would either disappear or at most struggle on with difficulty. The exponential function, for example, plays a central role in the precise definition of non-integral powers of x, in the evaluation of various limits, and in the solution of all manner of differential equations. Without the logarithmic function, we could no longer claim to be able to integrate all powers of x. Many of the applications of mathematics involve these functions in a crucial way. What is it that makes these functions so special? I shall try to answer this question, starting with the exponential function defined as a power series. We shall find that everything that can be said about the exponential and logarithm starts from this very simple beginning. After that, we shall look at some other examples of rather special functions, which are closely linked to the exponential function, and see how they too have important contributions to make to our subject. 7.1 The exponential function The exponential function exp(x) is defined for all x E IR by the power series x 2 x 3 x 4 exp(x) == 1 +x+ 2! +3T+ 4! + ... (7.1) We have already seen, by using the ratio test in Section 5.3, that this series is convergent for all values of x. The series has infinite radius of convergence, which implies in particular that there is no difficulty in differentiating term by term to determine the derivative d(exp(x))jdx. The derivative is given by the differentiated power series 2x 3x 2 4x 3 x 2 x 3 x 4 1 +2f+3f+4f+ ... == 1 +x+ 2! +3T+ 4! + ... Either you have carried out this derivative before, and know what the answer is, in which case you have probably by now got used to the result -or if not, this may be a minor miracle to you.
eBook - PDFBrief Calculus
An Applied Approach
- Michael Sullivan(Author)
- 2021(Publication Date)
- Wiley(Publisher)
The domain off is the set of all real numbers. We exclude the base a = l because this nction is simply the constant nction f(x) = 1 x = 1. We also need to exclude the bases that are negative, because, otherwise, we would have to exclude many values of x om the domain, such as x = � and x = i- [Recall that ( - 2) 1 12 = M, (-3) 314 = �(-3) 3 = �-27, and so on, are not defined in the system ofreal numbers.] COMMENT: It is important to distinguish a power nction g(x) = x", n 2 an inte- ger, om an exponential nction f(x) = a X , a> 0, a# l, a real. In a power nction, the base is a variable and the exponent is a constant. In an exponential nction, the base is a constant and the exponent is a variable. t 188 Chapter 2 Classes of Functions Some examples of exponential nctions are x) = 2 x , F(x) = ( + r Notice that in each example, the base is a constant and the exponent is a variable. You may wonder what role the base a plays in the exponential nction f(x) = a x . We use the llowing Exploration to find out. EXPLORATION (al Evaluate f(x) = 2 x at x = -2, -1, 0, l, 2, and 3. (bl Evaluate g(x) = 3x + 2 at x = -2, -1, 0, 1, 2, and 3. (cl Comment on the pattern that exists in the values off and g. Rest (al ble 4 shows the values off(x) = 2 x r x = -2, -1, 0, 1, 2, and 3. (bl Table 5 shows the values of g(x) = 3x + 2 r x = -2, -1, 0, l, 2, and 3. TABLE 4 X -2 -1 0 1 2 3 f(x) = 2 x 1 1 f(-2) = z-2 = = 4 1 2 1 2 4 8 TABLE 5 X g(x) = 3x + 2 -2 g(-2) = 3(-2) + 2 = -4 -1 -1 0 2 1 5 2 8 3 11 (cl In Table 4 we notice that each value of the exponential nction f(x) = a x = 2 x could be und by multiplying the previous value of the nction by the base, a = 2. For example, 1 1 1 f(-1) = 2 · f(-2) = 2 · 4 = 2 , f( O ) = 2 · f(- 1 ) = 2 · 2 = l, f(l) = 2- f( O ) = 2 · 1 = 2 and so on. Put another way, we see that the ratio of consecutive outputs is constant r unit increases in the inputs. The constant equals the value of the base of the exponential nction a.
eBook - ePub- Piotr Mikusi?????ski, Jan Mikusi?????ski;;;(Authors)
- 2017(Publication Date)
- WSPC(Publisher)
Chapter 7
ELEMENTARY FUNCTIONS
In this chapter we give rigorous definitions of elementary functions and establish their properties. By elementary functions we mean here exponential functions, logarithms, trigonometric and inverse trigonometric functions. In elementary calculus these functions are often considered without rigorous definitions. On the other hand, if we want to carefully define these function, we have to choose between different approaches. Our goal was to choose definitions that are natural and mathematically convenient.7.1The exponential function e
xWe now approach the didactic problem of introducing the exponential function whose value at x is ax. This function, perhaps the most important function in calculus, can be introduced in a great many ways. For instance it can be introduced as a generalization of the power anto nonnatural number values of the exponent n. First we introduce axfor negative exponents by declaring that a−n= 1/an. Next we define a1/nas the only positive solution of the equation xn= a.S:But there are values of a for which the equation has no solution, for example x2 = −1.T:Yes, and for that reason we assume that a > 0. We have to remember that the exponential function is an extension of the power with positive base.S:How do we know that a solution exists for a > 0 and that it is unique?T:This will be shown in the next section.S:And how do we define axfor other values of x?T:The exponents m/n are introduced by am/n= (a1/n)m. Finally, if x is an irrational number we approximate it by a sequence of rational numbers xn and then we prove that the sequence (ax) is convergent. The value of anxis defined as the limit of (axn).S:Since there are infinitely many sequences of rational numbers convergent to x, we may obtain infinitely many values for ax.T:No. One can prove that the limit does not depend on the choice of the sequence converging to x
eBook - ePubIntroduction to Differential Calculus
Systematic Studies with Engineering Applications for Beginners
- Ulrich L. Rohde, G. C. Jain, Ajay K. Poddar, A. K. Ghosh(Authors)
- 2012(Publication Date)
- Wiley(Publisher)
arithmetically (i.e., by simple interest) at the unit rate for the unit time will be 2, while the result of letting 1 grow by true compound interest at the unit rate for the unit time will be 2.71828. . ., which is the number e.Accordingly, we write 13a.3 Distinction Between Exponential and Power FunctionsThe expression 2x can be carelessly mistaken for the expression x 2 as typographically they are similar; however, the resemblance ends here. They in fact define entirely different functions. The function x 2 is an algebraic power function in which the base is a variable and the exponent is a constant . On the other hand, the function 2x is an exponential function in which the base is a constant and the exponent is a variable . The difference in their pattern of behavior is illustrated in Table 13a.1 .Table 13a.1 Comparative Values of the Function x 2 and 2xx x 2 2x 0 0 1 1 1 2 2 4 4 3 9 8 4 16 16 5 25 32 6 36 64 7 49 128 As can be seen from Table 13a.1 , the exponential function y = 2x increases more slowly for small values of x and is actually less than The Power Function y = x 2 between x = 2 and x = 4. However, y = 2x increases more and more rapidly as compared to y = x 2 . This is because the exponent in the exponential function increases with x (which means that the base is multiplied to itself more number of times), whereas for The Power Function the exponent remains constant and only the base increases with x. 3Another important difference between the two functions is as follows: Corresponding to the fact that 2x →0 as x →−∞, the graph of y = 2x has the line y = 0 (i.e., the x -axis) as a horizontal asymptote . In fact, every exponential function y = a x (a > 0, a ≠ 1) has the line y = 0 as a horizontal asymptote. By contrast, no power function (where α is a real number) has a horizontal asymptote.13a.4 The Value of eWe know that (1 + 1/n )n = e. A good number of values obtained for this expression, taking n = 2, n = 5, n = 10, and so on up to n
eBook - PDF- Brian Dennis(Author)
- 2016(Publication Date)
- Chapman and Hall/CRC(Publisher)
We then consider the curve in Figure 10.1 − 2 − 1 0 1 2 1 2 3 4 a y FIGURE 10.1 Graph of the function y x a = for varying values of a , with the value of x fixed at 2. 165 Exponential and Logarithmic Functions for all practical computing purposes as “smooth,” with no gaps for irrational values of the exponent. We thereby achieve real power. Raising x to a real power obeys all the algebraic exponent laws you learned for integer powers: x 0 1 = , x x 1 = , x x u u − = 1 / , x x x u v u v = + , ( ) x x u v uv = . In the above expressions, u and v are real numbers and x is any positive real number. Further, in mathematics 0 0 is usually defined to be 1, because the definition completes certain formulas in a sensible fashion. Try out your newly found powers at the R console. The square root of 2 and π can serve as examples of numbers known to be irrational. Remember that in R “ pi ” is a reserved name that returns the value of π to double precision: > 0^0 [1] 1 > pi [1] 3.141593 > x=2 > x^pi [1] 8.824978 > (x^pi) * (x^pi) [1] 77.88023 > x^(pi+pi) [1] 77.88023 > x^sqrt(2) [1] 2.665144 The Special Number e Take a look at the following power function: y = + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 1 1 x x . When the value of x is large, the term 1 / x inside the parentheses is small and the whole quantity in parentheses is close to the value 1. However, a large 166 The R Student Companion value of x also means that the exponent is large. The quantity in parentheses gets smaller as x gets bigger, but at the same time the quantity gets raised to a higher and higher power. As x becomes bigger, the function is pulled in two different ways. Will the function increase or decrease? The battle is on; who will be the winner? Curious? One thing about R is that its ease of use invites numerical experi-ments. It is a cinch to calculate values of the function for a range of increasing values of x . The number of commands needed just to calculate the function is small, so we could easily do it at the console.
eBook - PDF- Sebastian J. Schreiber, Karl J. Smith, Wayne M. Getz(Authors)
- 2014(Publication Date)
- Wiley(Publisher)
C H A P T E R 1 Modeling with Functions Figure 1.1 Mathematical models are used in Section 1.3 to identify Pocket Hercules as one of the all-time greatest weightlifters. MIKE HASKEY KRT/Newscom 1.1 Real Numbers and Functions 1.2 Data Fitting with Linear and Periodic Functions 1.3 Power Functions and Scaling Laws 1.4 Exponential Growth 1.5 Function Building 1.6 Inverse Functions and Logarithms 1.7 Sequences and Difference Equations Review Questions Group Projects Preview “Mathematicians do not study objects, but relations between objects.” Henri Poincare, 1854–1912. Although all readers taking a first course in calculus have a background in algebra, geometry, and trigonometry, the depth of exposure and choice of material covered can be quite variable. The material in this chapter is designed to provide a common framework upon which to build an introductory course in calculus for students who have a strong interest in the life sciences. In reviewing real numbers and functions, our intention is also to develop the notation we will use throughout the book. As students, you must become familiar with this notation if you want to be fluent in reading the mathematical text in this book. We also introduce data—and concepts around working with data—early on, because this component of the mathematical modeling process is critical to testing model predictions in the context of real-world problems (as discussed in the introduction to this book). We pay particular attention to power, exponential, and logarithmic functions since these all play a critical role in the development of differential and integral calculus. Trigonometric functions are important but less fundamental, and they have been dealt with extensively in precalculus mathematics courses. Thus, we provide only a brief review; we expect students who are rusty on this topic to go back and review trigonometry functions themselves.
eBook - PDFFunctions Modeling Change
A Preparation for Calculus
- Eric Connally, Deborah Hughes-Hallett, Andrew M. Gleason(Authors)
- 2019(Publication Date)
- Wiley(Publisher)
By the time = 20, the value of (20) = 2 20 is over six times as large as (20) = 20 4 . Figure 11.52 shows the exponential function () = 2 catching up to () = 4 . But what about a more slowly growing exponential function? After all, = 2 increases at a 100% growth rate. Figure 11.53 compares = 4 to the exponential function = 1.005 . Despite the fact that this exponential function creeps along at a 0.5% growth rate, at around = 7000, it overtakes The Power Function. In summary, Any positive increasing exponential function dominates any power function. Decreasing Exponential Functions and Decreasing Power Functions Just as an increasing exponential function eventually outpaces any increasing power function, an exponential decay function wins the race toward the -axis. In general: Any positive decreasing exponential function eventually approaches the horizontal axis faster than any positive decreasing power function. For example, let’s compare the long-term behavior of the decreasing exponential function = 0.5 with the decreasing power function = −2 . By rewriting = 0.5 = ( 1 2 ) = 1 2 and = −2 = 1 2 we can see the comparison more easily. In the long run, the smallest of these two fractions is the one with the largest denominator. The fact that 2 is eventually larger than 2 means that 1∕2 is eventually smaller than 1∕ 2 . 10 20 100,000 200,000 2 4 Figure 11.52: The exponential function = 2 dominates The Power Function = 4 5000 10,000 10 16 1.005 4 Figure 11.53: The exponential function = 1.005 dominates The Power Function = 4 428 Chapter 11 POLYNOMIAL AND RATIONAL FUNCTIONS 20 0.1 Close-up view ✛ = −2 ✛ = 0.5 Figure 11.54: Graphs of = −2 and = 0.5 100 0.0005 = −2 = 0.5 Far-away view Figure 11.55: Graphs of = −2 and = 0.5 Figure 11.54 shows = 0.5 and = −2 . Both graphs have the -axis as a horizontal asymp- tote.
eBook - PDF- Linda Almgren Kime, Judith Clark, Beverly K. Michael(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
9. If y = x −2 , as x becomes very large, then y approaches zero. 10. The domain of the function y x 5 3 = is (0, +∞). 11. The function y = x 10 eventually grows faster than the function y = 1.5 x . 12. The functions y = x 2 and y = x −1 intersect at the point (1, 1). 13. The graph of y x 2 5 = on a log‐log plot is a straight line with slope of −5. 14. Of the three functions f, g, and h in the accompanying figure, only function h could be a power function. x y h g f 15. If F = km −3 and k is a nonzero constant, then F is directly pro- portional to m. 16. If K(d) = 3d −2 , then K ( 2) 3 4 − = − . 17. The graphs of two power functions f x ( ) x 1 p = and g x ( ) x 1 q = are shown in the accompanying figure. For these functions, q < p. x y g f 18. If the number of gallons of gasoline (g) used to drive a fixed dis- tance is inversely proportional to the number of miles per gallon (m), then g k m = for k, a nonzero constant. 19. In The Power Function y x 1 p = graphed in the accompanying fig- ure, p is an even positive integer. x y y = 1 x p 20. If f (q) = −q 5 , then f (−2) = −32. 21. The graphs of The Power Functions A, B, and C, all of the form G = k ⋅ m p , are shown in the accompanying figure. For each of these functions, k < 0. m G A B B C II. In Problems 22–29, construct a function or functions with the specified properties. 22. A power function y = f (x) whose graph is steeper than the graph of y = 4x 6 when x > 1. 23. A power function y = g(x) whose graph is a reflection of the graph of y = −3.2x 4 across the horizontal axis. 24. A power function of even degree whose graph opens downward. 25. A function w = h(m) whose graph will eventually dominate the function w = 500m 10 . 26. A function M = f (Q) where M is inversely proportional to the cube of a quantity Q. 27. A function whose graph on a log‐log plot is a straight line. 28. A power function whose graph would be similar to the graph of y x 3 = − but would have a different power of x.
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