Let f be a function that has derivatives of all orders for all real numbers. Ass
ID: 2888741 • Letter: L
Question
Let f be a function that has derivatives of all orders for all real numbers. Assume f(0) 5, f'(0) =-3, f"(0) = 1, and f"(0) = 4. (a) Write the third-degree Taylor polynomial for f about x = 0 and use it to approximate f(0.2). (b) Write the fourth-degree Taylor polynomial for g , where g (x) =f(x2), about x = 0. (c) Write the third-degree Taylor polynomial for h, where h (x)= | f(t) dt, about x= 0. (d) Let h be defined as in part (c). Given that f(l) 3, either find the exact value of h(1) or explain why it cannot be determined.Explanation / Answer
f(x) = f(0) + f '(0)x + f "(0)x2/2 + f'"(0)x3/6 + ... = 5 - 3x + x2/2 +2x3/3 + ..
a) f(x) = 5 - 3x + x2/2 + 2x3/3
f(0.2) = 5 - 3*0.2 + 0.22/2 + 2*0.23/3 =4.425
b) g(x) = f(x2) = 5 - 3x2 +x4/2
c) h(x) = f(x) dx = 5x- 3x2/2 + x3/6
d) h(x) in part c is anapproximate expression about x = 0, it can't use itto get exact value of h(1).
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