if x=2 , f(x)=1 if x=3 , f(x)=4 if x=5 , f(x)=-2 if x=8 , f(x)=3

sibuzwaW 2021-10-26 Answered
if x=2 , f(x)=1
if x=3 , f(x)=4
if x=5 , f(x)=-2
if x=8 , f(x)=3
if x=13 , f(x)=6
f is twice variable for all real numbers.
1. Find f'(4)
2. Approximate \(\displaystyle{\underset{{{2}}}{{\int}}}^{{13}}{f}'{\left({x}\right)}{\left.{d}{x}\right.}\)
2. Find the value of \(\displaystyle{\underset{{{2}}}{{\int}}}^{{8}}{\left({3}-{f}'{\left({x}\right)}\right)}{\left.{d}{x}\right.}\)

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Expert Answer

Tuthornt
Answered 2021-10-27 Author has 13622 answers
1. \(\displaystyle{f}'{\left({4}\right)}=\frac{{{f{{\left({5}\right)}}}-{f{{\left({3}\right)}}}}}{{{5}-{3}}}\)
=(-2-4)/2
=-3
2. The lengths of intervals are 1, 2, 3, and 5 respectively. Right end points:
\(\displaystyle{\underset{{{1}}}{{{x}}}}={3},{\underset{{{2}}}{{{x}}}}={5},{\underset{{{3}}}{{{x}}}}={8},{\underset{{{4}}}{{{x}}}}={13}\)
Use Riemann Sum:
\(\displaystyle{\underset{{{4}}}{{{R}}}}={f{{\left({3}\right)}}}\times{1}+{f{{\left({5}\right)}}}\times{2}+{f{{\left({8}\right)}}}\times{3}+{f{{\left({13}\right)}}}\times{5}\)
\(\displaystyle{\underset{{{4}}}{{{R}}}}={4}-{2}\times{2}+{3}\times{3}+{6}\times{5}\)
\(\displaystyle{\underset{{{4}}}{{{R}}}}={39}\)
3. (C)
\(\displaystyle{\underset{{{2}}}{{\int}}}^{{8}}{\left({3}-{f}'{\left({x}\right)}\right)}{\left.{d}{x}\right.}={\underset{{{2}}}{{{\left[{3}{x}-{f{{\left({x}\right)}}}\right]}}}}^{{8}}\)
\(\displaystyle={3}\times{8}-{f{{\left({8}\right)}}}-{3}\times{2}+{f{{\left({2}\right)}}}\)
=24-3-6+1
=16
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