Consider the function f (x) = ln (1 + x). (a) Enter the degree-n term in...
- Question 2 3 points Consider the function f (x) = ln (1+2). (a) Enter the degree-n term in the Taylor Series around x = 0. ((-1)^(n-1)*x^n)/n (b) Enter the error term En (2) which will also be a function of x and n. ((-1)^n*x^(n+1))/((n+1)*(1+z)^(n+1) (c) Find an upper bound for the absolute value of the error term when x > 0. It may help to remember that z is between x and 0. x^(n+1)/(n+1) 90 (d) Use this formula...
1. Consider the polynonial Pl (z) of degree 4 interpolating the function f(x) sin(x) on the interval n/4,4 at the equidistant points r--r/4, xi =-r/8, x2 = 0, 3 π/8, and x4 = π/4. Estimate the maximum of the interpolation absolute error for x E [-r/4, π/4 , ie, give an upper bound for this absolute error maxsin(x) P(x) s? Remark: you are not asked to give the interpolation polynomial P(r). 1. Consider the polynonial Pl (z) of degree 4...
5001 1 +- +-- 400 300 1200 100 -0 0.5 -100 Graph of rs 3. Let f and g be given by f(x)- xe and g(x)-(). The graph of f, the fifth derivatve of f is shown above for (a) write the first four nonzero terms and the general term of the Taylor series for e, about x = 0 . Write the first four nonzero terms and the general term of the Taylor series for f about x 0....
The nth-order Taylor polynomial for a function f(x) using the h notation is given as: Pa (x + h) = f(x) + f'(a)h + salt) 12 + () +...+ m (s) n." The remainder of the above nth-order Taylor polynomial is defined as: R( +h) = f(n+1)(C) +1 " hn+1, where c is in between x and c+h (n+1)! A student is using 4 terms in the Taylor series of f(x) = 1/x to approximate f(0.7) around x = 1....
Consider the function f(x) := v/x= x1/2. 6. (a) Give the Taylor polynomial P(x) of degree 5 about a1 of this function (b) Give the nested representation of the polynomial Qs()Ps((t)) where t -1 ((t)+1). (c) Using the nested multiplication method (also called Horner's algorithm), compute the approximation Ps (1.2) to V (give at least 12 significant digits of P(1.2)) (d) Without using the exact value of 12, compute by hand an upper bound on the absolute error V1.2 A(1.21...
1. The two-point forward difference quotient with error term is given by where ξ e ll, l + hl. In class we showed an additional error term appears to due to computer rounding error, e(r). Denoting (z) f(x) +e(x) as what the com- puter stores, and supposing f"(x)M and e() e where e, M are constants, we obtained an upper bound for the error between f(r) and the computed forward difference quotient 2c h Find the minimum value of the...
Question 1 Find the quartic Taylor series for the function f(x) (1+ based at the origin Also use the remainder term of the series to estimate the maximum possible error in using the quartic series to approximate f(x) on the interval [ -1, 1 Finally estimate (1.2)3, giving an appropriate error bound. Question 1 Find the quartic Taylor series for the function f(x) (1+ based at the origin Also use the remainder term of the series to estimate the maximum...
Complete all, especially part c and d (a) Glive the second-order Taylor polynomial T2 ( for the function () about a 16. 4+((X-16)/8)-(1/512) (X-16M2 b) Use Taylor's Theorem to give the Error Term E2(-f()T2) as a function of z and some z between 16 and az (((3/8) Z(-5/2)) (X-16) 3)/6 c) Estimate the domain of values z for which the error E2 () is less than 0.01. Enter a value p for which E2 ()I 0.01 for all 16 16+p,...
Please answer all, be explanatory but concise. Thanks. Consider the function f(x) = e x a. Differentiate the Taylor series about 0 of f(x). b. ldentify the function represented by the differentiated series c. Give the interval of convergence of the power series for the derivative. Consider the differential equation y'(t) - 4y(t)- 8, y(0)4. a. Find a power series for the solution of the differential equation b. ldentify the function represented by the power series. Use a series to...
Consider the following function. f[x) = x ln(3x), a = 1, n = 3, 0.8 lessthanorequalto x lessthanorequalto 1.2 Approximate f by a Taylor polynomial with degree n at the number a. T_3(x) = Use Taylor's Inequality to estimate the accuracy of the approximation f(x) = T_n(x) when x lies in the given Interval. (Round your answer to four decimal places.) |R_3 (x)| lessthanorequalto