Exponential and Logarithmic functions Exponential functions Definition Take a > 0 and not equal to 1 . Then, the function defined by f : R -> R : x -> ax is called an exponential function with base a. Graph and properties Let f(x) = an exponential function with a > 1. From the graphs we see that The domain is R The range is the set of strictly positive real numbers The function is continuous in its domain The function is increasing if a > 1 and decreasing if 0 < a < 1 The x-axis is a horizontal asymptote Examples: y = 3x ; y = 0.5x ; y = 100.2x-1 Logarithmic functions Definition and basic properties Take a > 0 and not equal to 1 . are either increasing or decreasing, the inverse functions are defined. loga(x) log10(x) is written as log(x) So, From this we see that the domain of the logarithmic function is the set of strictly positive real numbers, and the range is R. log2(8) = 3 ; log3(sqrt(3)) = 0.5 ; log(0.01) = -2 From the definition it follows immediately that Example: log(102x+1) = 2x+1 Graph log2(x) ; log(2x+4) ; log0.5(x) Thus,
THE CALCULUS PAGE PROBLEMS LIST Problems and Solutions Developed by : D. A. Kouba And brought to you by : Beginning Differential Calculus : Problems on the limit of a function as x approaches a fixed constant limit of a function as x approaches plus or minus infinity limit of a function using the precise epsilon/delta definition of limit limit of a function using l'Hopital's rule ... Beginning Integral Calculus : Problems using summation notation Problems on the limit definition of a definite integral Problems on u-substitution Problems on integrating exponential functions Problems on integrating trigonometric functions Problems on integration by parts Problems on integrating certain rational functions, resulting in logarithmic or inverse tangent functions Problems on integrating certain rational functions by partial fractions Problems on power substitution Problems on integration by trigonometric substitution ... Sequences and Infinite Series :
Karl's Calculus Tutor - Box 6.0: Exponential and Log Identities Adding the Exponents: If b is any positive real number then bx by = bx+y for all x and y. This is the single most important identity concerning logs and exponents. Since ex is only a special case of an exponential function, it is also true that ex ey = ex+y Multiplying the Exponents: If b is any positive real number then (bx)y = bxy for all x and y. (ex)y = exy Converting to roots to exponents: The nth root of x is the same as x1/n for all positive x. _ √x = x1/2 In addition: __ √ex = ex/2 Converting to ex form: If b is any positive real number then bx = ex ln(b) for all x. xx = ex ln(x) or if you have f(x)x: f(x)x = ex ln(f(x)) or if you have xf(x): xf(x) = ef(x) ln(x) or if you have f(x)g(x): f(x)g(x) = eg(x) ln(f(x)) As an example, suppose you had (x2 + 1)1/x. e(1/x) ln(x2 + 1) ex is its own derivative: The derivative of ex is ex. ex is always positive: You can put in any x, positive or negative, and ex will always be greater than zero. logb(xy) = logb(x) + logb(y) This includes k logb(x) = logb(xk) and
The AP Calculus BC Exam Exam Content In 1956, 386 students took what was then known as the AP Mathematics Exam. By 1969, still under the heading of AP Mathematics, it had become Calculus AB and Calculus BC. The Calculus BC exam covers the same differential and integral calculus topics that are included in the Calculus AB exam, plus additional topics in differential and integral calculus, and polynomial approximations and series. This is material that would be included in a two-semester calculus sequence at the college level. If students take the BC exam, they cannot take the AB exam in the same year because the exams share some questions. Multiple-Choice Questions For sample multiple-choice questions, refer to the Course Description. AP Calculus Course Description, Effective Fall 2012 (.pdf/2.28MB) AP Calculus Multiple-Choice Question Index: 1997, 1998 and 2003 AP Calculus Exams(.xls/144KB) Free-Response Questions Below are free-response questions from past AP Calculus BC Exams. 2013: Free-Response Questions
Homework and Study Help - Free help with your algebra, biology, environmental science, American government, US history, physics and religion homework Can I take a course at HippoCampus for credit? How do I enroll in a course at HippoCampus? Are there any fees to take your courses? How do I make a comment or ask a question? How do I get individual help with my homework assignment? What are the preferred texts? How can I use HippoCampus in my classroom? How can I use HippoCampus in my home school? Can I use the resources you have available for my homeschoolers? Do you know of any wet lab resources to accompany HippoCampus content? Is there a script, app, or something that can be used to track student use of HippoCampus? Can I share my HippoCampus content with my fellow teachers? Can I download the video? Can I change the size of the video window? Why won't the Environmental Science animations play? What if my page scroll bars or "submit" button are not showing? I can't find closed captioning. Where does the content from your site come from? There is an error in the multimedia presentation. How do I report a course errata item? No. AP Course Ledger
Differential Equations Differential Equations (Math 3301) Here are my online notes for my differential equations course that I teach here at Lamar University. Despite the fact that these are my “class notes”, they should be accessible to anyone wanting to learn how to solve differential equations or needing a refresher on differential equations. I’ve tried to make these notes as self contained as possible and so all the information needed to read through them is either from a Calculus or Algebra class or contained in other sections of the notes. A couple of warnings to my students who may be here to get a copy of what happened on a day that you missed. Because I wanted to make this a fairly complete set of notes for anyone wanting to learn differential equations I have included some material that I do not usually have time to cover in class and because this changes from semester to semester it is not noted here. Here is a listing and brief description of the material in this set of notes. Basic Concepts
MATH1200 Calculus The author of this page, Kevin Hutchinson would welcome any comments or suggestions for its improvement. Lecturer Lecturer: Dr Kevin Hutchinson Office: Room 8, Mathematics Department, Second Floor, Science Lecture Theatre Building Phone: 716 2577 e-mail: kevin.hutchinson@ucd.ie Office Hours: Tuesday 4-5 pm, Thursday 3-4 pm, or by appointment. Syllabus The syllabus for this course is entirely determined by the classnotes and assigned homework. The question of whether any given topic is part of the syllabus is resolved by determining whether it occurs anywhere in the classnotes or homeworks. For an outline of the course syllabus, aims and objectives, click here. Texts The primary `text' for this course is the classnotes provided below. Homeworks Calculus homework will be assigned throughout the year, once every fortnight during term. Here is the first homework for the current year (2004/5): Homework 1 (2004) Here is the third homework for the current year (2004/5): Homework 3 (2004)
Calculus, Contemporary Calculus, Hoffman Contemporary Calculus Dale HoffmanBellevue Collegedhoffman@bellevuecollege.edu A free on-line calculus text Many of these materials were developed for the Open Course Library Project of the Washington State Colleges as part of a Gates Foundation grant. The goal of this project was to create materials that would be FREE (on the web) to anyone who wanted to use or modify them (and not have to pay $200 for a calculus book). The textbook sections, in color, are available free in pdf format at the bottom of this page. The links below are to pdf files. Chapter 0 -- Review and Preview Chapter 1 -- Functions, Graphs, Limits and Continuity Chapter 2 -- The Derivative Chapter 3 -- Derivatives and Graphs Chapter 4 -- The Integral Chapter 5 -- Applications of Definite Integrals Chapter 6 -- Introduction to Differential Equations Chapter 7 -- Inverse Trigonometric Functions Chapter 8 -- Improper Integrals and Integration Techniques
Calculus III - Higher Order Partial Derivatives Just as we had higher order derivatives with functions of one variable we will also have higher order derivatives of functions of more than one variable. However, this time we will have more options since we do have more than one variable.. Consider the case of a function of two variables, since both of the first order partial derivatives are also functions of x and y we could in turn differentiate each with respect to x or y. The second and third second order partial derivatives are often called mixed partial derivatives since we are taking derivatives with respect to more than one variable. , then we will differentiate from left to right. , it is the opposite. Let’s take a quick look at an example. Notice that we dropped the from the derivatives. Now let’s also notice that, in this case, . Clairaut’s Theorem Now, do not get too excited about the disk business and the fact that we gave the theorem for a specific point. So far we have only looked at second order derivatives. .