If f(x)=x^3−x, what is the average rate of change of f(x) over the interval [1, 5]? a)29.5 b)30 c)43 d)120

1 answer:

7 0

$\bf slope = m = \cfrac{rise}{run} \implies 
\cfrac{ f(x_2) - f(x_1)}{ x_2 - x_1}\impliedby 
\begin{array}{llll}
average~rate\\
of~change
\end{array}\\\\
-------------------------------\\\\
f(x)= x^3-x \qquad 
\begin{cases}
x_1=1\\
x_2=5
\end{cases}\implies \cfrac{f(5)-f(1)}{5-1}
\\\\\\
\cfrac{[5^3-5]~-~[1^3-1]}{4}\implies \cfrac{125-5~~-~~0}{4}\implies \cfrac{120}{4}\implies 30$

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The quantity demanded x each month of Russo Espresso Makers is 250 when the unit price p is $138. The quantity demanded each mon

Stells [14]

Answer:

(a) $D(q)=\frac{-1}{25} q+148$

(b) $S(q)=\frac{1}{50}q+58$

Step-by-step explanation:

(a) For the demand equation D(q) we have

P1: 138 Q1: 250

P2: 108 Q2: 1000

We can find m, which is the slope of the demand equation,

$m=\frac{p_{2} -p_{1} }{q_{2} -q_{1} }=\frac{108-38}{1000-250} =\frac{-30}{750}=\frac{-1}{25}$

and then we find b, which is the point where the curve intersects the y axis.

We can do it by plugging one point and the slope into the line equation form:

$y=mx+b\\\\D(q)=mq+b\\\\138=\frac{-1}{25}(250) +b\\\\138=-10+b\\\\138+10=b=148$

With b: 148 and m: -1/25 we can write our demand equation D(q)

$D(q)=\frac{-1}{25} q+148$

(b) to find the supply equation S(q) we have

P1: 102 Q1: 2200

P2: 102 Q2: 700

Similarly we find m, and b

$m=\frac{p_{2} -p_{1} }{q_{2} -q_{1} }=\frac{72-102}{700-2200} =\frac{-30}{-1500}=\frac{1}{50}$

$y=mx+b\\\\D(q)=mq+b\\\\72=\frac{1}{50}(700) +b\\\\72=14+b\\\\72-14=b=58\\$

And we can write our Supply equation S(q):

$S(q)=\frac{1}{50}q+58$

(c) Now we may find the equilibrium quantity q* and the equilibrium price p* by writing D(q)=S(q), which means the demand equals the supply in equilibrium:

$D(q)=S(q)\\\\\frac{-1}{25}q+148=\frac{1}{50}q+58\\\\$