# 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=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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Tuthornt
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