a. Find the linear approximating polynomial for the following function centered at the given point a. b. Find the quadratic approximating polynomial for the following function centered at the given point a. c. Use the polynomials obtained in parts a. and b. to approximate the given quantity.

The linear approximating polynomial And quadratic approximating polynomial And approximate the given quantity of polynomials obtained in parts a. and b.

So, the given function is f(x) = 1/x

And

f'(x) = -1/x²

f''(x) = 2/x³

a = 1

Now, we find the linear approximating polynomial

So,

a. The linear approximating polynomial is:

p1(x) = f(a) + f'(a)*(x - a)

p1(x) = 1/1 + -1/1² * (x - 1)

p1(x) = 1 - x + 1

p1(x) = 2 - x

And Now, we find the quadratic approximating polynomial

So,

b. The quadratic approximating polynomial is:

p2(x) = p1(x) + 1/2 * f''(a)*(x - a)²

p2(x) = 2 - x + 1/2 * 2/1³ * (x - 1)²

p2(x) = 2 - x + (x - 1)²

p2(x) = 2 - x + x² - 2*x + 1

p2(x) = x² - 3*x + 3

And Now, we find the approximating polynomial value

So,

c. approximate 1/0.97 using p1(x)

p1(0.97) = 2 - 0.97 = 1.03

approximate 1/0.97 using p2(x)

p2(0.97) = 0.97² - 3*0.97 + 3 = 1.0309

So, The polynomial approximate values are

a. The linear approximating polynomial is p1(x) = 2 - x

b.The quadratic approximating polynomial is p2(x) = x² - 3*x + 3

c. The approximating polynomial value is p1(0.97) = 1.03; p2(0.97) = 1.0309.

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Question:

A. Find the linear approximating polynomial for the following function centered at the given point a. b. Find the quadratic approximating polynomial for the following function centeredat the given point a. c. Use the polynomials obtained in parts a. and b. to approximate the given quantity.

f(x)=1/x, a=1; approximate 1/0.97

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