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garik1379 [7]
3 years ago
9

Find the degree 3 Taylor polynomial P3(x) of function f(x)=(-3x+15)^(3/2) at a=2

Mathematics
1 answer:
OLEGan [10]3 years ago
4 0

Take up to the third-order derivative:

f(x)=(-3x+15)^{3/2}

f'(x)=\dfrac32(-3x+15)^{1/2}(-3)=-\dfrac92(-3x+15)^{1/2}

f''(x)=-\dfrac94(-3x+15)^{-1/2}(-3)=\dfrac{27}4(-3x+15)^{-1/2}

f'''(x)=-\dfrac{27}8(-3x+15)^{-3/2}(-3)=\dfrac{81}8(-3x+15)^{-3/2}

Evaluate each derivative at x=a=2:

f(2)=9^{3/2}=27

f'(2)=-\dfrac929^{1/2}=-\dfrac{27}2

f''(2)=\dfrac{27}4\dfrac1{9^{1/2}}=\dfrac94

f'''(2)=\dfrac{81}8\dfrac1{9^{3/2}}=\dfrac38

Then the Taylor polynomial is

P_3(x)=f(2)+f'(2)(x-2)+\dfrac{f''(2)}2(x-2)^2+\dfrac{f'''(2)}6(x-2)^3

P_3(x)=27-\dfrac{27}2(x-2)+\dfrac98(x-2)^2+\dfrac1{16}(x-2)^3

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