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 +
Premium Derivative Natural logarithm Mathematics
21. Optimization 2 Test 3 22. Exponential Functions 23. Logarithmic Functions 24. Compound Interest 25. Differentiation of Exponential Functions 26. Differential of Logarithmic Functions 27. Exponential Functions as Mathematical Models
Premium Derivative Calculus
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
Premium Natural logarithm Derivative
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
Free Compound interest
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
Premium English-language films Derivative Diagram
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
Premium Statistics Normal distribution Standard deviation
Solving Exponential and Logarithmic Equations Exponential Equations (variable in exponent position) 1. Isolate the exponential portion ( base exp onent ): Move all non-exponential factors or terms to the other side of the equation. 2. Take ln or log of each side of the equation. • Make sure to use ln if the base is “e”. Then remember that ln e = 1 . • Make sure to use log if the base is 10. • If the base is neither “e” nor “10”‚ use either ln or log‚ your choice.. 3. Bring the power (exponent)
Premium Polynomial
Global Warming: An exponential threat We currently live in a highly globalized world from every point of view (technological‚ scientific‚ cultural‚ economic‚ communicative‚ etc.) hence‚ one of the most negative effects that generated globalization has been the growing ecological imbalance that has harmed the planet. The global warming has become of the most dangerous threats and challenges to face in the 21st century. Many companies and factories are currently one of the main factors which
Premium Global warming Carbon dioxide Recycling
Introduction According to the International Program Center‚ U.S. Census Bureau‚ the total population of the World‚ projected to 03/27/08 at 19:37 GMT (EST+5) is 6‚657‚527‚872. (US Census Bureau) This rapid growth in population means little to most people living in this today’s world but it’s a phenomenon that should be a concern to all. It took from the start of human history to the industrial revolution around 1945 for the population to grow to 2 billion. If we then look at the figures after
Premium World population Overpopulation Population growth
The concept of exponential growth is when a value increases by a multiplicative factor per unit of time. This explains population growth. The concept of linear growth is when a value increases by a constant. This clarifies how food production grows. These concepts apply to
Premium Industrial Revolution City Economics