a. Find the linear approximating polynomial for the following function centered

Question

a. Find the linear approximating polynomial for the following function centered at the given point 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. e. Using the linear approximating polynomial to estimate. 1/0.98 is approximately . (Type an integer or a decimal.) Using the quadratic approximating polynomial to estimate, is approximately - (Type an integer or a decimal)

Explanation / Answer

f(x) = 1/(1-x) , a = 0

==>f(0) = 1/(1-0) = 1

==>f'(x) = 1/(1-x)2

==>f'(0) = 1/(1-0)2 = 1

f''(x) = 2/(1-x)3 ==> f''(0) = 2/(1-0)3 = 2

a) Linear approximating polynomial P1(x) = f(a)+f'(a)*(x-a)

==>P1(x) = 1+(1)(x-0) = 1+x

b) Quadratic approximating polynomial P2(x) = f(a)+f'(a)*(x-a)+(1/2)f''(a)*(x-a)2

==> P2(x) = 1+1*(x-1)+(1/2)(2)*(x-0)2

==> P2(x) = 1+x+x2

c) 1/0.98 ==> 1/(1-0.02)

==> x = 0.02

P1(0.02) = 1+0.02

==> P1(x) = 1.02

P2(0.02) = 1+0.02+(0.02)2 = 1+0.02+0.0004 = 1.0204

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