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- (a) Show that the function defined by
is not equal to its Maclaurin series.
- (b) Graph the function in part (a) and comment on its behavior near the origin.
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Chapter 11 Solutions
Multivariable Calculus
- Q3. (a) Write down the function corresponding to the graph given below. Is the function even or odd or neither even nor odd? 3 -3 3 3 (b) Write the Fourier series for the function defined in part (a)arrow_forward(a) Show that the function defined by -1/x² /½2 if x + 0 f(x) = if x = 0 is not equal to its Maclaurin series.arrow_forwardFind a power series for f(x)=3/2x-1 c=2arrow_forward
- The question is to find a power series for the function centered at c f(x) = 5/(5-x) , c = -3arrow_forwardFIND THE TAYWR SERIES FUR f (x) = 7 ABOUT x=-3 X4arrow_forwarda) Write the given function as a power series: (cos(x)-1)/x. Your power series should start with n=1 and not n=0. b) Evaluate the indefinite integral as an infinite series: integral of ((cos(x)-1)/x); evaluate from n=1 to n=infinity.arrow_forward
- Without computing any Fourier series, address the following statements regarding the functions below:arrow_forwardObserve the function X f(x) = (1+2x)² In order to find the power series for this function, complete the following steps: 1 1-x a. Start with the series Σ. Replace x with (−2x) in this series and k=0 write the corresponding power series for = 1 1+2x b. Take derivative of the series from part (a) above and relate it to the power series for the function 1 (1+2x)²· c. Multiply both sides of the resulting series from above with x, and obtain the series for Write the first four non-zero terms of this series. X (1+2x)² d. What is the radius of convergence for this series? What is the interval of convergence?arrow_forward3 (b) Write power series of f(x) = about x = 0. 1- 4.rarrow_forward
- 6. Let f(x) = L − 2x, 0 < x < L. (a) Find the cos series for f(x) with period 2L. (b) Sketch the graph of the function to which the series converges over 3 periods.arrow_forward13.) Find a power series for the function f(x) = = 5 2-6x3*arrow_forwardEXAMPLE 10 (a) Evaluate the integral below as an infinite series. Je-z² (b) Evaluate the integral below correct to within an error of 0.0001. 6.0³. SOLUTION (a) First we find the Maclaurin series for f(x) = ex. Although it's possible to use the direct method, let's find it simply by replacing x with -x² in the series for e* given in Table 1 in the book. Thus for all values of x, ex² = Σ = 1 ∞o (-x²)" n=0 n! 1! 4 Sex²dx = (1- = C + X- = Now we integrate term by term. = Σ (-1)^ n=0 x² 1! 2x³ 2(1) 2! e-² dx = [x - x2n n! 1/24 + 1/24 + X 2! + +6 3! +5 x² 5 2 7 3! + x7 (-1)^2n+1 (-1) x² n! 2n+1 This series converges for all x because the original series for e*² converges for all x. (b) The fundamental Theorem of Calculus gives 6.³ 2n + 1 + ... 10.5 X ).n! 5.2! 7.3! 1/5376 + 1/110592 - 1/5376 + 1/110592 0.4278 x The Alternating Series Estimation Theorem shows that the error involved in this approximation is less than 0.0001. Activate Windows Go to Settings to activate Windows.arrow_forward
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