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A homeowner sets the thermostat so that the temperature in the house begins to drop from $70\text{\xb0}\text{F}$ at $9$ p.m., reaches a low of $60\text{\xb0}$ during the night, and rises back to $70\text{\xb0}$ by $7$ a.m. the next morning. Suppose that the temperature in the house is given by $T\left(t\right)=0.4{t}^{2}-4t+70$ for $0\le t\le 10,$ where $t$ is the number of hours past $9$ p.m. Find the instantaneous rate of change of the temperature at midnight.
Since midnight is $3$ hours past $9$ p.m., we want to compute ${T}^{\prime}(3).$ Refer to [link] .
The instantaneous rate of change of the temperature at midnight is $\mathrm{-1.6}\text{\xb0}\text{F}$ per hour.
A toy company can sell $x$ electronic gaming systems at a price of $p=\mathrm{-0.01}x+400$ dollars per gaming system. The cost of manufacturing $x$ systems is given by $C\left(x\right)=100x+\mathrm{10,000}$ dollars. Find the rate of change of profit when $\mathrm{10,000}$ games are produced. Should the toy company increase or decrease production?
The profit $P\left(x\right)$ earned by producing $x$ gaming systems is $R\left(x\right)-C\left(x\right),$ where $R\left(x\right)$ is the revenue obtained from the sale of $x$ games. Since the company can sell $x$ games at $p=\mathrm{-0.01}x+400$ per game,
Consequently,
Therefore, evaluating the rate of change of profit gives
Since the rate of change of profit ${P}^{\prime}\left(\mathrm{10,000}\right)>0$ and $P\left(\mathrm{10,000}\right)>0,$ the company should increase production.
A coffee shop determines that the daily profit on scones obtained by charging $s$ dollars per scone is $P\left(s\right)=\mathrm{-20}{s}^{2}+150s-10.$ The coffee shop currently charges $\text{\$}3.25$ per scone. Find ${P}^{\prime}(3.25),$ the rate of change of profit when the price is $\text{\$}3.25$ and decide whether or not the coffee shop should consider raising or lowering its prices on scones.
${P}^{\prime}\left(3.25\right)=20>0;$ raise prices
For the following exercises, use [link] to find the slope of the secant line between the values ${x}_{1}$ and ${x}_{2}$ for each function $y=f\left(x\right).$
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