Consider the function f(x)=2x^{3}-3x

a) Find the average rate of change between the points on the function when x=1 and x=2.

b) On a graph, a line that connects any two points is called a secant line. Find the equation of the secant line that connects the points on the function when x=1 and x=2.

c) In calculus, we find the slope between a point on the graph when x=a and another point that is a small distance away. We use x=a+h for the second point. Find the average rate of change between the points x=a and x=a+h.

AdamTaurus
Oct 25, 2017

#3**+2 **

a) average rate of change =

\(\frac{\text{change in f(x)}}{\text{change in x}}\,=\,\frac{f(1)-f(2)}{1-2}\,=\,\frac{(\,2(1)^3-3(1)\,)\,-\,(\,2(2)^3-3(2)\,)}{1-2}\,=\,\frac{(-1)-(10)}{-1}\,=\,\frac{-11}{-1}\,=\,11\)

like helperid said.

b) We want an equation of a line that has a slope of 11 and passes through the point (1, f(1) )

And (1, f(1) = (1, -1)

So.. the equation in point-slope form is y + 1 = 11(x - 1)

And in slope intercept form, it is y = 11x - 12

c) Find the average rate of change between the points x = a and x = a + h .

Thie is the same as part a , just with letters in place of numbers.

average rate of change =

\(=\,\frac{\text{change in f(x)}}{\text{change in x}}\\~\\ =\,\frac{f(a+h)-f(a)}{(a+h)-(a)} \\~\\ =\,\frac{(\,2(a+h)^3-3(a+h)\,)\,-\,(\,2(a)^3-3(a)\,)}{h} \\~\\ =\,\frac{2(a+h)^3-3(a+h)-2a^3+3a}{h} \)

I don't know if you need it more simplified than that.

hectictar
Oct 25, 2017

#1

#2**+3 **

Could you explain how to do these though? Just the answers does me no good.

AdamTaurus
Oct 25, 2017

#3**+2 **

Best Answer

a) average rate of change =

\(\frac{\text{change in f(x)}}{\text{change in x}}\,=\,\frac{f(1)-f(2)}{1-2}\,=\,\frac{(\,2(1)^3-3(1)\,)\,-\,(\,2(2)^3-3(2)\,)}{1-2}\,=\,\frac{(-1)-(10)}{-1}\,=\,\frac{-11}{-1}\,=\,11\)

like helperid said.

b) We want an equation of a line that has a slope of 11 and passes through the point (1, f(1) )

And (1, f(1) = (1, -1)

So.. the equation in point-slope form is y + 1 = 11(x - 1)

And in slope intercept form, it is y = 11x - 12

c) Find the average rate of change between the points x = a and x = a + h .

Thie is the same as part a , just with letters in place of numbers.

average rate of change =

\(=\,\frac{\text{change in f(x)}}{\text{change in x}}\\~\\ =\,\frac{f(a+h)-f(a)}{(a+h)-(a)} \\~\\ =\,\frac{(\,2(a+h)^3-3(a+h)\,)\,-\,(\,2(a)^3-3(a)\,)}{h} \\~\\ =\,\frac{2(a+h)^3-3(a+h)-2a^3+3a}{h} \)

I don't know if you need it more simplified than that.

hectictar
Oct 25, 2017