Question

Suppose g is a function which has continuous derivatives, and that g(8)=−4,g′(8)=3,g′′(8)=−4,g′′′(8)=4. (a) What is the Taylor polynomial of degree 2 for g near 8 ? P2​(x)= (b) What is the Taylor polynomial of degree 3 for g near 8 ? P3​(x)= (c) Use the two polynomials that you found in parts (a) and (b) to approximate g(8.1). With P2​,g(8.1)≈ With P3​,g(8.1)≈

Answer 1

To find the Taylor polynomials of degree 2 and 3 for the function g near 8, substitute the given derivatives into the formula. Then, to approximate g(8.1), plug in the value of x into the polynomials. P2(x) = -4 + 3(x - 8) - 4(x - 8)^2 / 2 and P3(x) = -4 + 3(x - 8) - 4(x - 8)^2 / 2 + 4(x - 8)^3 / 3!

Step-by-step explanation:

To find the Taylor polynomials of degree 2 and 3 for the function g near 8, we can use the formula:

P_n(x) = g(a) + g'(a)(x - a) + g''(a)(x - a)^2 + ... + g^n(a)(x - a)^n / n!

(a) To find P2(x), substitute a=8 and the given derivatives into the formula:

P2(x) = g(8) + g'(8)(x - 8) + g''(8)(x - 8)^2 / 2!

Substituting the given values into this formula, we get P2(x) = -4 + 3(x - 8) - 4(x - 8)^2 / 2.

(b) To find P3(x), substitute a=8 and the given derivatives into the formula:

P3(x) = g(8) + g'(8)(x - 8) + g''(8)(x - 8)^2 / 2 + g'''(8)(x - 8)^3 / 3!

Substituting the given values into this formula, we get P3(x) = -4 + 3(x - 8) - 4(x - 8)^2 / 2 + 4(x - 8)^3 / 3!

(c) To approximate g(8.1) using P2(x) and P3(x), substitute x=8.1 into each polynomial:

With P2, g(8.1) ≈ -4 + 3(8.1 - 8) - 4(8.1 - 8)^2 / 2.

With P3, g(8.1) ≈ -4 + 3(8.1 - 8) - 4(8.1 - 8)^2 / 2 + 4(8.1 - 8)^3 / 3!

Answer 2

To approximate g(8.1), the Taylor polynomials of degrees 2 and 3 for a function g near 8 are used. The degree 2 polynomial is -4 + 3(x-8) - 2(x-8)^2, and the degree 3 polynomial adds an additional term, resulting in -4 + 3(x-8) - 2(x-8)^2 + 4/6(x-8)^3.

Step-by-step explanation:

The question involves finding the Taylor polynomials of degree 2 and degree 3 for a function g near 8, and then using those polynomials to approximate g(8.1). The Taylor polynomial of degree n for a function f at a is given by Tn(x) = f(a) + f'(a)(x-a) + f''(a)/2!(x-a)2 + ... + f(n)(a)/n!(x-a)n.

(a) The Taylor polynomial of degree 2 for g near 8, P2(x), can be calculated using the given derivatives at x=8:
P2(x) = g(8) + g'(8)(x-8) + g''(8)/2!(x-8)2
So, P2(x) = -4 + 3(x-8) - 2(x-8)2.

(b) To include the third derivative for the Taylor polynomial of degree 3, the formula will be:
P3(x) = P2(x) + g'''(8)/3!(x-8)3
Thus, P3(x) = -4 + 3(x-8) - 2(x-8)2 + 4/6(x-8)3.

(c) To approximate g(8.1) using P2 and P3, we substitute x = 8.1 into both polynomials and calculate the results:
With P2, g(8.1) ≈ P2(8.1)
With P3, g(8.1) ≈ P3(8.1).