sibuzwaW

2021-10-26

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 ${\underset{2}{\int }}^{13}{f}^{\prime }\left(x\right)dx$
2. Find the value of ${\underset{2}{\int }}^{8}\left(3-{f}^{\prime }\left(x\right)\right)dx$

### Answer & Explanation

Tuthornt

1. ${f}^{\prime }\left(4\right)=\frac{f\left(5\right)-f\left(3\right)}{5-3}$
=(-2-4)/2
=-3
2. The lengths of intervals are 1, 2, 3, and 5 respectively. Right end points:
$\underset{1}{x}=3,\underset{2}{x}=5,\underset{3}{x}=8,\underset{4}{x}=13$
Use Riemann Sum:
$\underset{4}{R}=f\left(3\right)×1+f\left(5\right)×2+f\left(8\right)×3+f\left(13\right)×5$
$\underset{4}{R}=4-2×2+3×3+6×5$
$\underset{4}{R}=39$
3. (C)
${\underset{2}{\int }}^{8}\left(3-{f}^{\prime }\left(x\right)\right)dx={\underset{2}{\left[3x-f\left(x\right)\right]}}^{8}$
$=3×8-f\left(8\right)-3×2+f\left(2\right)$
=24-3-6+1
=16

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