Question

at a price of x dollars, the supply function for a music player is q = 80e^{0.003x}, where q is in thousands of units.
how many music players will be supplied at a price of 150? (round to the nearest thousand.)
thousand units
find the marginal supply
marginal supply(x) =
which is the best interpretation of the derivative?
the rate of change of the quantity supplied as the price increases
the rate of change of the price as the quantity supplying increases
the price at a given supply of units
the quantity supplied if the price increases
the number of units that will be demanded at a given price
add work

Explanation:

Step1: Find quantity supplied at $x = 150$

Substitute $x = 150$ into $q=80e^{0.003x}$. So $q = 80e^{0.003\times150}=80e^{0.45}$. Using a calculator, $e^{0.45}\approx1.56831$, then $q = 80\times1.56831 = 125.4648\approx125$ (rounded to nearest thousand).

Step2: Find marginal supply

Differentiate $q = 80e^{0.003x}$ with respect to $x$. Using the rule $\frac{d}{dx}(e^{ax})=ae^{ax}$, we have $\frac{dq}{dx}=80\times0.003e^{0.003x}=0.24e^{0.003x}$. So Marginal supply$(x)=0.24e^{0.003x}$.

Step3: Interpret derivative

The derivative of the supply - function with respect to price $x$ represents the rate of change of the quantity supplied as the price increases.

Answer:

125
$0.24e^{0.003x}$
The rate of change of the quantity supplied as the price increases