Find the average rate of change of f: x \longmapsto 2x^2-4x+1 between -2 and 3.
The average rate of change of any function f between x=a and x=b is:
\dfrac{f\left(b\right)-f\left(a\right)}{b-a}
Here, we have:
f\left(-2\right)=2\left(-2\right)^{2}-4\left(-2\right)+1=17
f\left(3\right)=2\left(3\right)^{2}-4\left(3\right)+1=7
Therefore, the average rate of change of f between x=-2 and x=3 is:
\dfrac{f\left(3\right)-f\left(-2\right)}{3-\left(-2\right)}=\dfrac{7-17}{5}=-2
The average rate of change of f between x=-2 and x=3 is -2.
Find the average rate of change of f: x \longmapsto x^2-5x+7 between 2 and 5.
The average rate of change of any function f between x=a and x=b is:
\dfrac{f\left(b\right)-f\left(a\right)}{b-a}
Here, we have:
f\left(2\right)=\left(2\right)^{2}-5\left(2\right)+7=1
f\left(5\right)=\left(5\right)^{2}-5\left(5\right)+7=7
Therefore, the average rate of change of f between x=2 and x=5 is:
\dfrac{f\left(5\right)-f\left(2\right)}{5-2}=\dfrac{7-1}{3}=2
The average rate of change of f between x=2 and x=5 is 2.
Find the average rate of change of f: x \longmapsto 3x^2-4 between -1 and 1.
The average rate of change of any function f between x=a and x=b is:
\dfrac{f\left(b\right)-f\left(a\right)}{b-a}
Here, we have:
f\left(-1\right)=3\left(-1\right)^{2}-4=-1
f\left(1\right)=3\left(1\right)^{2}-4=-1
Therefore, the average rate of change of f between x=-1 and x=1 is:
\dfrac{f\left(1\right)-f\left(-1\right)}{1-\left(-1\right)}=\dfrac{-1-\left(-1\right)}{2}=0
The average rate of change of f between x=-1 and x=1 is 0.
Find the average rate of change of f: x \longmapsto -x^2+4x+7 between -2 and 0.
The average rate of change of any function f between x=a and x=b is:
\dfrac{f\left(b\right)-f\left(a\right)}{b-a}
Here, we have:
f\left(-2\right)=-\left(-2\right)^{2}+4\left(-2\right)+7=-5
f\left(0\right)=7
Therefore, the average rate of change of f between x=-2 and x=0 is:
\dfrac{f\left(0\right)-f\left(-2\right)}{0-\left(-2\right)}=\dfrac{7-\left(-5\right)}{2}=6
The average rate of change of f between x=-2 and x=0 is 6.
Find the average rate of change of f: x \longmapsto 5x^2+x-10 between -1 and 2.
The average rate of change of any function f between x=a and x=b is:
\dfrac{f\left(b\right)-f\left(a\right)}{b-a}
Here, we have:
f\left(-1\right)=5\left(-1\right)^{2}+\left(-1\right)-10=-6
f\left(2\right)=5\left(2\right)^{2}+\left(2\right)-10=12
Therefore, the average rate of change of f between x=-1 and x=2 is:
\dfrac{f\left(2\right)-f\left(-1\right)}{2-\left(-1\right)}=\dfrac{12-\left(-6\right)}{3}=6
The average rate of change of f between x=-1 and x=2 is 6.
Find the average rate of change of f: x \longmapsto -2x^2+3x-10 between -1 and 5.
The average rate of change of any function f between x=a and x=b is:
\dfrac{f\left(b\right)-f\left(a\right)}{b-a}
Here, we have:
f\left(-1\right)=-2\left(-1\right)^{2}+3\left(-1\right)-10=-15
f\left(5\right)=-2\left(5\right)^{2}+3\left(5\right)-10=-45
Therefore, the average rate of change of f between x=-1 and x=5 is:
\dfrac{f\left(5\right)-f\left(-1\right)}{5-\left(-1\right)}=\dfrac{-45-\left(-15\right)}{6}=-5
The average rate of change of f between x=-1 and x=5 is -5.
Find the average rate of change of f: x \longmapsto -x^2+x between -4 and 4.
The average rate of change of any function f between x=a and x=b is:
\dfrac{f\left(b\right)-f\left(a\right)}{b-a}
Here, we have:
f\left(-4\right)=-\left(-4\right)^{2}+\left(-4\right)=-20
f\left(4\right)=-\left(4\right)^{2}+4=-12
Therefore, the average rate of change of f between x=-4 and x=4 is:
\dfrac{f\left(4\right)-f\left(-4\right)}{4-\left(-4\right)}=\dfrac{-12-\left(-20\right)}{8}=1
The average rate of change of f between x=-4 and x=4 is 1.