Preview

Gods Gift to Calculators: the Taylor Series

Good Essays
Open Document
Open Document
573 Words
Grammar
Grammar
Plagiarism
Plagiarism
Writing
Writing
Score
Score
Gods Gift to Calculators: the Taylor Series
Gods Gift to Calculators: The Taylor Series

It is incredible how far calculators have come since my parents were in college, which was when the square root key came out. Calculators since then have evolved into machines that can take natural logarithms, sines, cosines, arcsines, and so on. The funny thing is that calculators have not gotten any
"smarter" since then. In fact, calculators are still basically limited to the four basic operations: addition, subtraction, multiplication, and division! So what is it that allows calculators to evaluate logs, trigonometric functions, and exponents? This ability is due in large part to the Taylor series, which has allowed mathematicians (and calculators) to approximate functions,such as those given above, with polynomials. These polynomials, called Taylor
Polynomials, are easy for a calculator manipulate because the calculator uses only the four basic arithmetic operators.

So how do mathematicians take a function and turn it into a polynomial function? Lets find out. First, lets assume that we have a function in the form y= f(x) that looks like the graph below.

We'll start out trying to approximate function values near x=0. To do this we start out using the lowest order polynomial, f0(x)=a0, that passes through the y-intercept of the graph (0,f(0)). So f(0)=ao.

Next, we see that the graph of f1(x)= a0 + a1x will also pass through x=
0, and will have the same slope as f(x) if we let a0=f1(0).

Now, if we want to get a better polynomial approximation for this function, which we do of course, we must make a few generalizations. First, we let the polynomial fn(x)= a0 + a1x + a2x2 + ... + anxn approximate f(x) near x=0, and let this functions first n derivatives match the the derivatives of f(x) at x=0. So if we want to make the derivatives of fn(x) equal to f(x) at x=0, we have to chose the coefficients a0 through an properly. How do we do this?
We'll write down the polynomial and its derivatives as

You May Also Find These Documents Helpful

  • Satisfactory Essays

    Pre Calc Checkup 4

    • 732 Words
    • 5 Pages

    1. If the function given holds true, in what year would the cost of mailing a first-class letter reach 60 cents?…

    • 732 Words
    • 5 Pages
    Satisfactory Essays
  • Satisfactory Essays

    Length and Cliff

    • 434 Words
    • 2 Pages

    5. Sketch the graphs of r(t) and A(t) on the same axes. Indicate which is which.…

    • 434 Words
    • 2 Pages
    Satisfactory Essays
  • Satisfactory Essays

    VI. Determining the Equation from a graph or table – Determine the equation for each function. Don’t forget to solve for the a-value.…

    • 269 Words
    • 3 Pages
    Satisfactory Essays
  • Satisfactory Essays

    hiihihi

    • 10962 Words
    • 41 Pages

    x − a, a∈ I , and the value of the polynomial expression at x = a (remainder theorem).…

    • 10962 Words
    • 41 Pages
    Satisfactory Essays
  • Satisfactory Essays

    The next problem, I will be finding the solution for g(x). I will replace the x in the f function with the g function. My function will look like this:…

    • 495 Words
    • 2 Pages
    Satisfactory Essays
  • Good Essays

    Compositions and Inverses

    • 696 Words
    • 3 Pages

    Next we are to compose two pairs of the functions into each other. First we…

    • 696 Words
    • 3 Pages
    Good Essays
  • Satisfactory Essays

    Mobile Phone

    • 215 Words
    • 1 Page

    1. Describe what the graph of this function would look like. What would the intercepts be? Would it have a positive or negative slope?…

    • 215 Words
    • 1 Page
    Satisfactory Essays
  • Satisfactory Essays

    Math Problems

    • 527 Words
    • 3 Pages

    2. Describe the transformations on the following graph of f (x)  log( x) . State the…

    • 527 Words
    • 3 Pages
    Satisfactory Essays
  • Powerful Essays

    Nt1110 Unit 3 Assignment

    • 479 Words
    • 2 Pages

    Computer operating systems as far back as early Unix have included interactive calculator programs such as dc and hoc, and calculator functions are included in almost all PDA-type devices (save a few dedicated address book and dictionary devices).…

    • 479 Words
    • 2 Pages
    Powerful Essays
  • Satisfactory Essays

    Mm207 Unit 2

    • 520 Words
    • 3 Pages

    Use the second given value as the reference value. Also, write a statement describing the result.…

    • 520 Words
    • 3 Pages
    Satisfactory Essays
  • Satisfactory Essays

    Pre Calc Checkup 3

    • 240 Words
    • 4 Pages

    Answer the following questions using what you've learned from this unit. Write your responses in the space provided, and turn the assignment in to your instructor.…

    • 240 Words
    • 4 Pages
    Satisfactory Essays
  • Good Essays

    Pt1420 Unit 6 Study Guide

    • 864 Words
    • 4 Pages

    27.|The tangent line to a curve at a point closely approximates the curve near the point. In fact, for x-values close enough to the point of tangency, the function and its tangent line are virtually indistinguishable. This problem explores this relationship. Write the equation of the tangent line to the curve at the indicated point. To check your answer, use a graphing utility to graph both the function and its tangent line, and repeatedly zoom in on the point of tangency until the function and the tangent line cannot be distinguished.|…

    • 864 Words
    • 4 Pages
    Good Essays
  • Good Essays

    When we went to the library to find a book for English class, the first book I checked out was entitled The Last Nazi. It was a book about the last known nazi that lived through the holocaust. It was interesting at first but then it was really hard for me to continue to read, because not only was it a historical non-fiction, it also included a large variety of words above my reading level. The second book I chose was, Calculus The Easy Way, one of the reasons I chose that book was because I thought it was going to help me in math, but It was not a good choice for my essay. The third book I chose was from the library, out of all those books I couldn’t find a good one. When I was ready to leave, a book on the checkout counter caught my eyes. The book title is Need. It's about teens living in Nottawa, Wisconsin who join the newest, hottest, networking site. It's a site that allows people to request anything that they NEED. However, access to this site is only by invitation only. Anything at all can be asked and your request will be fulfilled only if you accomplish a certain task that will be assigned by the site itself. It's a fast paced book, it takes social media stalking and bullying to a whole new level.…

    • 756 Words
    • 4 Pages
    Good Essays
  • Powerful Essays

    2. Which statement best supports the conclusion that society can control whether to adopt a new technology?…

    • 4331 Words
    • 23 Pages
    Powerful Essays
  • Good Essays

    A scientific calculator is a type of electronic calculator, designed to calculate problems in science, engineering, and mathematics. They have almost completely replaced slide rules in almost all traditional applications, and are widely used in both education and professional settings. In certain contexts such as higher education, scientific calculators have been superseded by graphing calculators, which offer a superset of scientific calculator functionality along with the ability to graph input data and write and store programs for the device. There is also some overlap with the financial calculator market.1…

    • 759 Words
    • 4 Pages
    Good Essays