Temperature. The temperature (in degrees Celsius) in an artificial habitat is made to change according to the equation
Use a Taylor series at 0 to approximate the average temperature over the time interval [0, 2] to within ±0.005.
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Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)
- The formula for the amount A in an investmentaccount with a nominal interest rate r at any timet is given by A(t)=a(e)rt, where a is the amount ofprincipal initially deposited into an account thatcompounds continuously. Prove that the percentageof interest earned to principal at any time t can becalculated with the formula I(t)=ert1.arrow_forwardLet f(x) = 1 + x 1 X Find the power series representation for the function f(x) by completing the following steps: a. First, express the fraction 1¹ as a power series. = X b. Now, express the fraction as a power series. 1-x 1+x x c. The function f(x) 1-x 1-x + 1 is the sum of the two series from parts (a) and (b). Express the function f(x) as a power series. d. What is the interval of convergence and the radius of convergence for this power series?arrow_forward3. Find a Taylor series for the function f(x) = ln(x) about x = 0.arrow_forward
- 00 f(x) = Σχ* k=0 = 1 1 and S(x) = n-1 Ext. k=0 The remainder in truncating the power series after n terms is R₁ = f(x) = S(x), which depends on x. a. Show that R₁(x) = x" /(1-x). b. Graph the remainder function on the interval x < 1, for n = 1, 2, and 3. Discuss and interpret the graph. Where on the interval is R, (x)| largest? Smallest? c. For fixed n, minimize |R₁(x)| with respect to x. Does the result agree with the observations in part (b)? d. Let N(x) be the number of terms required to reduce |R₁(x)| to less than 106. Graph the function N(x) on the interval x < 1. Discuss and interpret the graph.arrow_forwardConsider the function: f(x) = log(3)(2x-1) Using the first five terms of the Taylor series, approximate the value of f(2). Compare it with the actual value of f(2).arrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage