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
According to the statement
we have given that the f(x) = 1/x
And we have to find that polynomial approximate values which are written below
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.
Learn more about the polynomial approximate values here brainly.com/question/2595842
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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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