# Science:Math Exam Resources/Courses/MATH101/April 2010/Question 05 (c)

MATH101 April 2010
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### Question 05 (c)

An unknown function ${\displaystyle f(x)}$ has the following values: ${\displaystyle f(0)=1}$, ${\displaystyle f(1)=2}$, ${\displaystyle f(2)=0}$, ${\displaystyle f(3)=-2}$, ${\displaystyle f(4)=-4.}$

It is known that the fourth derivative ${\displaystyle f^{(4)}(x)}$ lies between ${\displaystyle -3}$ and ${\displaystyle 2}$ on the interval ${\displaystyle [0,4]}$. What is the largest possible value that ${\displaystyle \int _{0}^{4}f(x)dx}$ could have? You may use the fact that if ${\displaystyle \left|f^{(4)}(x)\right|\leq K}$ on the interval ${\displaystyle [a,b]}$, then the error in using ${\displaystyle S_{n}}$ to approximate ${\displaystyle \int _{a}^{b}f(x)dx}$ has absolute value less than or equal to ${\displaystyle K(b-a)^{5}/180n^{4}}$.

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