estimating function values with taylor and maclaurin series\ndue no due date points 100 submitting an…

estimating function values with taylor and maclaurin series\ndue no due date points 100 submitting an external tool available until may 12 at 11:59pm\nquestion\nconsider the function ( f(x)=x^{3}-x ). find the second degree taylor polynomial for ( f ) centered at ( x = 1 ) and use it to estimate the value of ( f(1.2) ).\nround your answer to three decimal places.\nprovide your answer below:

estimating function values with taylor and maclaurin series\ndue no due date points 100 submitting an external tool available until may 12 at 11:59pm\nquestion\nconsider the function ( f(x)=x^{3}-x ). find the second degree taylor polynomial for ( f ) centered at ( x = 1 ) and use it to estimate the value of ( f(1.2) ).\nround your answer to three decimal places.\nprovide your answer below:

Answer

Explanation:

Step1: Find the first and second derivatives of (f(x))

The function is (f(x)=x^{3}-x). The first derivative (f^{\prime}(x) = 3x^{2}-1). The second derivative (f^{\prime\prime}(x)=6x).

Step2: Evaluate (f(x)), (f^{\prime}(x)), and (f^{\prime\prime}(x)) at (x = 1)

  • (f(1)=1^{3}-1=0).
  • (f^{\prime}(1)=3\times1^{2}-1 = 2).
  • (f^{\prime\prime}(1)=6\times1=6).

Step3: Write the second - degree Taylor polynomial (P_{2}(x))

The Taylor polynomial centered at (a) is (P_{n}(x)=\sum_{k = 0}^{n}\frac{f^{(k)}(a)}{k!}(x - a)^{k}). For (n = 2) and (a = 1), (P_{2}(x)=f(1)+f^{\prime}(1)(x - 1)+\frac{f^{\prime\prime}(1)}{2!}(x - 1)^{2}). Substitute (f(1) = 0), (f^{\prime}(1)=2), (f^{\prime\prime}(1)=6) into the formula: (P_{2}(x)=0 + 2(x - 1)+\frac{6}{2}(x - 1)^{2}=2(x - 1)+3(x - 1)^{2}).

Step4: Estimate (f(1.2)) using (P_{2}(x))

Substitute (x = 1.2) into (P_{2}(x)): (P_{2}(1.2)=2(1.2 - 1)+3(1.2 - 1)^{2}). First, calculate ((1.2 - 1)=0.2). Then (2\times0.2+3\times(0.2)^{2}=0.4 + 3\times0.04). (=0.4+0.12=0.520).

Answer:

(0.520)