Write the formula to determine average rate of change.

\(\displaystyle \frac{\Delta f}{\Delta x}= \frac{f(x_{2})-f(x_{1})}{x_{2}-x_{1}}\)

Substitute the values and solve for the average rate of change.

\(\displaystyle \frac{-cos(\pi)-(-cos(\frac{\pi}{2}))}{\pi-\frac{\pi}{2}} = \frac{-(-1)+0}{\frac{\pi}{2}}=\frac{1}{\frac{\pi}{2}}=\frac{2}{\pi}\)

Write the formula for the average rate of change from the interval \(\displaystyle [{x_{1},x_{2}}]\).

\(\displaystyle \frac{f(x_2)-f(x_{1})}{x_2-x_1}\)

Solve for \(\displaystyle f(x_2)\) and \(\displaystyle f(x_1)\).

\(\displaystyle f(x_2=6)=2(6)-9=3\)

\(\displaystyle f(x_1=3)=2(3)-9=-3\)

Substitute the known values into the formula and solve.

\(\displaystyle \frac{f(x_2)-f(x_{1})}{x_2-x_1}=\frac{3-(-3)}{6-3}=\frac{6}{3}=2\)

Since the question is asking for the rate of change in terms of the perimeter, write the formula for the perimeter of the square and differentiate it with the respect to time.

\(\displaystyle P=4s\)

\(\displaystyle \frac{dP}{dt}= 4\frac{ds}{dt}\)

The question asks in terms of the perimeter. Isolate the term \(\displaystyle s\) by dividing four on both sides.

\(\displaystyle s=\frac{P}{4}\)

Write the given rate in mathematical terms and substitute this value into \(\displaystyle \frac{dP}{dt}\).

\(\displaystyle \frac{ds}{dt}=2\)

\(\displaystyle \frac{dP}{dt}= 4\frac{ds}{dt}=4(2)=8\)

Write the area of the square and substitute the side.

\(\displaystyle A=s^2=\left(\frac{P}{4}\right)^2=\frac{P^2}{16}\)

Since the area is changing with time, take the derivative of the area with respect to time.

\(\displaystyle \frac{dA}{dt}=\frac{P}{8} \cdot \frac{dP}{dT}\)

Substitute the value of \(\displaystyle \frac{dP}{dt}\).

\(\displaystyle \frac{dA}{dt}=\frac{P}{8} \cdot 8=P\)

In order to determine where the function is not changing, it is necessary to take the derivative and set the slope equal to zero. This will provide information on where the curve is not changing. Once we find the x value that gives the derivative a slope of zero, we can substitute the x-value back into the original function to obtain the point.

\(\displaystyle y=x^2-36x+6\)

\(\displaystyle y'=2x-36\)

\(\displaystyle 0=2x-36\)

\(\displaystyle 36=2x\)

\(\displaystyle x=18\)

Substitute this value back to the original equation to solve for \(\displaystyle y\).

\(\displaystyle y=18^2-36(18)+6 = 324-648+6 = -318\)

The point where the function is not changing is \(\displaystyle (18,-318)\).

Write the formula for average rate of change.

\(\displaystyle \frac{f(x_f)-f(x_i)}{x_f-x_i}\)

Determine the values of \(\displaystyle x_f\) and \(\displaystyle x_i\).

\(\displaystyle f(x_f)=f(x_i)=10\)

\(\displaystyle x_i=1\)

\(\displaystyle x_f=3\)

Substitute the known values.

\(\displaystyle \frac{f(x_f)-f(x_i)}{x_f-x_i}=\frac{10-10}{3-1}=0\)

We can solve by utilizing the formula for the average rate of change:\(\displaystyle \frac{f(x_2)-f(x_1)}{x_2-x_1}.\) Solving for \(\displaystyle f(x)\) at our given points:

\(\displaystyle f(x_1)=f(4)=5(4)+9=20+9=29\)

\(\displaystyle f(x_2)=f(6)=5(6)+9=30+9=39\)

Plugging our values into the average rate of change formula, we get:

\(\displaystyle \frac{(39-29)}{(6-4)}=\frac{10}{2}=5\)

We can solve by utilizing the formula for the average rate of change: \(\displaystyle \frac{f(x_2)-f(x_1)}{x_2-x_1}}\). Solving for \(\displaystyle f(x)\) at our given points:

\(\displaystyle f(x_1)=f(7)=9(7)+7=63+7=70\)

\(\displaystyle f(x_2)=f(8)=9(8)+7=72+7=79\)

Plugging our values into the average rate of change formula, we get:

\(\displaystyle \frac{(79-70)}{(8-7)}=\frac{9}{1}=9\)

We can solve by utilizing the formula for the average rate of change: \(\displaystyle \frac{f(x_2)-f(x_1) }{x_2-x_1}\) . Solving for \(\displaystyle f(x)\) at our given points:

\(\displaystyle f(x_1)=f(2)=3(2)+7=6+7=13\)

\(\displaystyle f(x_2)=f(3)=3(3)+7=9+7=16\)

Plugging our values into the average rate of change formula, we get:

\(\displaystyle \frac{(16-13)}{(3-2)}=\frac{3}{1}=3\)

We can solve by utilizing the formula for the average rate of change:

\(\displaystyle \frac{ f(x_2)-f(x_1)}{x_2-x_1}\).

Solving for \(\displaystyle f(x)\) at our given points:

\(\displaystyle f(x_1)=f(1)=4(1)+9=4+9=13\)

\(\displaystyle f(x_2)=f(2)=4(2)+9=8+9=17\)

Plugging our values into the average rate of change formula, we get:

\(\displaystyle \frac{17-13}{2-1}=\frac{4}{1}=4\)

We can solve by utilizing the formula for the average rate of change:

\(\displaystyle \frac{ f(x_2)-f(x_1)}{x_2-x_1}\).

Solving for \(\displaystyle f(x)\) at our given points:

\(\displaystyle f(x_1)=f(5)=6(5)-7=30-7=23\)

\(\displaystyle f(x_2)=f(6)=6(6)-7=36-7=29\)

Plugging our values into the average rate of change formula, we get:

\(\displaystyle \frac{29-23}{6-5}=\frac{6}{1}=6\)