Exponential function

Good Essays
In mathematics, the exponential function is the function ex, where e is the number (approximately 2.718281828) such that the function ex is its own derivative.[1][2] The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change (i.e. percentage increase or decrease) in the dependent variable. The function is often written as exp(x), especially when it is impractical to write the independent variable as a superscript. The exponential function is widely used in physics, chemistry, engineering, mathematical biology, economics and mathematics.
Exponential function
Representation e^x \,
Inverse \ln x \,
Derivative e^x \,
Indefinite Integral e^x + C \,
The graph of y = ex is upward-sloping, and increases faster as x increases. The graph always lies above the x-axis but can get arbitrarily close to it for negative x; thus, the x-axis is a horizontal asymptote. The slope of the tangent to the graph at each point is equal to its y coordinate at that point. The inverse function is the natural logarithm ln(x); because of this, some old texts[3] refer to the exponential function as the antilogarithm.
Sometimes the term exponential function is used more generally for functions of the form cbx, where the base b is any positive real number, not necessarily e. See exponential growth for this usage.
In general, the variable x can be any real or complex number, or even an entirely different kind of mathematical object; see the formal definition below. Part of a series of articles on
The mathematical constant e
Euler's formula.svg
Natural logarithm · Exponential function
Applications in: compound interest · Euler's identity & Euler's formula · half-lives & exponential growth/decay
Defining e: proof that e is irrational · representations of e · Lindemann–Weierstrass theorem
People John Napier · Leonhard Euler
Schanuel's conjecture
Contents [hide]
1 Formal definition
2

You May Also Find These Documents Helpful

  • Satisfactory Essays

    Exponential Function

    • 2732 Words
    • 11 Pages

    Practice Test Multiple Choice Identify the choice that best completes the statement or answers the question. ____ 1. Tell whether the function y = 2( 5 ) shows growth or decay. Then graph the function. a. This is an exponential growth function. c. This is an exponential decay function. x b. This is an exponential growth function. d. This is an exponential growth function. ____ 2. Graph the inverse of the relation. Identify the domain and range of the inverse. x y −1 4 1 2 3 1 5 0 7 1 a. c…

    • 2732 Words
    • 11 Pages
    Satisfactory Essays
  • Good Essays

    LOGARITHMIC AND EXPONENTIAL FUNCTIONS Inverse relations Exponential functions Exponential and logarithmic equations One logarithm THE LOGARITHMIC FUNCTION WITH BASE b is the function y = logb x. b is normally a number greater than 1 (although it need only be greater than 0 and not equal to 1). The function is defined for all x > 0. Here is its graph for any base b. Note the following: • For any base, the x-intercept is 1. Why? To see the answer, pass your mouse over the colored…

    • 3127 Words
    • 13 Pages
    Good Essays
  • Satisfactory Essays

    Exponential and Logarithmic Functions * Verify that the natural logarithm function defined as an integral has the same properties as the natural logarithm function earlier defined as the inverse of the natural exponential function. Integrals of Exponential and Logarithmic Functions Function | Integral | lnx | x ∙ lnx - x + c | logx | (x ∙ lnx - x) / ln(10) + c | logax | x(logax - logae) + c | ex | ex+c | ek∙x | 1 / k ∙ ek∙x + c | ax | ax / lna + c | xn | 1 / (n+1) ∙ xn+1 +…

    • 361 Words
    • 2 Pages
    Satisfactory Essays
  • Good Essays

    EXPONENTIAL AND LOGARITHMIC FUNCTIONS I.EXPONENTIAL FUNCTION A. Definition An exponential function is a function defined by f(x) = ax , where a > 0 and a ≠ 1. The domain of the function is the set of real numbers and the range is the set of positive numbers. B. Evaluating Exponential Functions 1. Given: f(x) = 2x, find a. f(3) = ____ b. f(5) = _____ c. f(-2) = ______ d. f(-4) = ______ 2. Evaluate f(x) = ( 1)x if 2 a. x = 2 ____ b. x = 4 _____ c. x = -3 ______ d…

    • 904 Words
    • 4 Pages
    Good Essays
  • Satisfactory Essays

    Cartesian Coordinate System 4. Straight Lines 5. Functions and their Graphs 6. The Algebra of Functions 7. Functions and Math Models 8. Limits Test 1 9. The Derivative 10. Basic Rules of Differentiation 11. The Product and Quotient Rules 12. The Chain Rule 13. Marginal Functions in Economics 14. Higher-Order Derivatives 15. Implicit…

    • 338 Words
    • 3 Pages
    Satisfactory Essays
  • Satisfactory Essays

    MATH133 Unit 5: Exponential and Logarithmic Functions Individual Project Assignment: Version 2A Show all of your work details for these calculations. Please review this Web site to see how to type mathematics using the keyboard symbols. IMPORTANT: See Question 1 in Problem 2 below for special IP instructions. This is mandatory. Problem 1: Photic Zone Light entering water in a pond, lake, sea, or ocean will be absorbed or scattered by the particles in the water and its intensity, I, will be attenuated…

    • 1387 Words
    • 9 Pages
    Satisfactory Essays
  • Good Essays

    Exponential Functions in Business Turgenbayeva Aiida ID 20092726 Variant 2 Kazakhstan Institute of Management, Economics and Strategic Research MSC1101 Mathematics for Business and Economics Instructor: Dilyara Nartova Section #2 Summer-I 2009 Abstract This project reflects my knowledge and understanding of the interest rate, its types, formula and its evaluation in order to determine the most profitable type of investment scheme for National Bank wishing to increase…

    • 908 Words
    • 4 Pages
    Good Essays
  • Good Essays

    Name:________________________________ Part 1 Exponential Functions Project There are three parts to this project. You must complete Part 1 (60 points), but you may choose to do either Part 2 or Part 3 (40 points each). You may also do all three parts for a total of 140 points; however, you must fully complete either Part 2 or Part 3 to get credit (NOT ½ of Part 2 and ½ of Part 3). This project is due on December 5th. Turning it in late forfeits your right to extra credit and there will be…

    • 2115 Words
    • 9 Pages
    Good Essays
  • Good Essays

    A function is a relation in which each element of the domain is paired with exactly one element in the range. Two types of functions are the exponential functions and the logarithmic functions. Exponential functions are the functions in the form of y = ax, where ''a'' is a positive real number, greater than zero and not equal to one. Logarithmic functions are the inverse of exponential functions, y = loga x, where ''a'' is greater to zero and not equal to one. These functions have certain differences…

    • 853 Words
    • 3 Pages
    Good Essays
  • Satisfactory Essays

    exponential

    • 331 Words
    • 2 Pages

    Exponential Smoothing Forecasting Method with Naïve start Formula: Ft = α (At-1) + (1 – α) (Ft – 1) where: Ft Forecast for time t Ft – 1 Past forecast; 1 time ahead or earlier than time t At-1 Past Actual data; 1 time ahead or earlier than time t α (read as alpha) as a smoothing constant takes the…

    • 331 Words
    • 2 Pages
    Satisfactory Essays