3.4 Derivatives as rates of change  (Page 5/15)

The position function $s\left(t\right)=t^3-8t$ gives the position in miles of a freight train where east is the positive direction and $t$ is measured in hours.

1. Determine the direction the train is traveling when $s\left(t\right)=0.$
2. Determine the direction the train is traveling when $a\left(t\right)=0.$
3. Determine the time intervals when the train is slowing down or speeding up.

The following graph shows the position $y=s\left(t\right)$ of an object moving along a straight line.

1. Use the graph of the position function to determine the time intervals when the velocity is positive, negative, or zero.
2. Sketch the graph of the velocity function.
3. Use the graph of the velocity function to determine the time intervals when the acceleration is positive, negative, or zero.
4. Determine the time intervals when the object is speeding up or slowing down.

The cost function, in dollars, of a company that manufactures food processors is given by $C\left(x\right)=200+\frac{7}{x}+\frac{x^2}{7},$ where $x$ is the number of food processors manufactured.

1. Find the marginal cost function.
2. Find the marginal cost of manufacturing 12 food processors.
3. Find the actual cost of manufacturing the thirteenth food processor.

The price $p$ (in dollars) and the demand $x$ for a certain digital clock radio is given by the price–demand function $p=10-0.001x.$

1. Find the revenue function $R\left(x\right).$
2. Find the marginal revenue function.
3. Find the marginal revenue at $x=2000$ and $5000.$

[T] A profit is earned when revenue exceeds cost. Suppose the profit function for a skateboard manufacturer is given by $P\left(x\right)=30x-0.3x^2-250,$ where $x$ is the number of skateboards sold.

1. Find the exact profit from the sale of the thirtieth skateboard.
2. Find the marginal profit function and use it to estimate the profit from the sale of the thirtieth skateboard.

[T] In general, the profit function is the difference between the revenue and cost functions: $P\left(x\right)=R\left(x\right)-C\left(x\right).$

Suppose the price-demand and cost functions for the production of cordless drills is given respectively by $p=143-0.03x$ and $C\left(x\right)=\mathrm{75,000}+65x,$ where $x$ is the number of cordless drills that are sold at a price of $p$ dollars per drill and $C\left(x\right)$ is the cost of producing $x$ cordless drills.

1. Find the marginal cost function.
2. Find the revenue and marginal revenue functions.
3. Find $R^{\prime }(1000)$ and $R^{\prime }(4000).$ Interpret the results.
4. Find the profit and marginal profit functions.
5. Find $P^{\prime }(1000)$ and $P^{\prime }(4000).$ Interpret the results.