At a price of x dollars, the demand, in thousands of units,...

At a price of $x$ dollars, the demand, in thousands of units, for a certain turntable is given by the demand function $$ q=240 e^{-0.003 x} $$ a) How many turntables will be bought at a price of $\$ 250 ?$ Round to the nearest thousand. b) Graph the demand function for $0 \leq x \leq 400$. c) Find the marginal demand, $q^{\prime}(x)$. d) Interpret the meaning of the derivative.

At a price of $x$ dollars, the demand, in thousands of units, for a certain turntable is given by the demand function
$$
q=240 e^{-0.003 x}
$$
a) How many turntables will be bought at a price of $\$ 250 ?$ Round to the nearest thousand.
b) Graph the demand function for $0 \leq x \leq 400$.
c) Find the marginal demand, $q^{\prime}(x)$.
d) Interpret the meaning of the derivative.

Key Concepts

Exponential Functions

Exponential functions model processes where the rate of change of a quantity is proportional to the current amount. In economic models, they often describe demand or decay processes, capturing how a response decreases (or increases) exponentially with changes in another variable, such as price.

Demand Functions

A demand function expresses the relationship between the price of a product and the quantity consumers are willing to purchase. It is a fundamental concept in economics, helping to understand consumer behavior and guide pricing strategies by typically reflecting inverse relationships where higher prices lead to lower demand.

Graphing Functions

Graphing functions is a method used to visually represent the relationship between independent and dependent variables over a specified domain. In the context of demand functions, graphing reveals trends such as exponential decay, and helps in analyzing how changes in price influence demand quantities over a range of values.

Differentiation

Differentiation is the process of finding the derivative of a function, which measures the rate at which the function’s value changes with respect to its variable. When applied to exponential functions, differentiation typically uses the chain rule to account for the function’s composite structure, providing insights into the sensitivity of the output to small variations in the input.

Marginal Analysis

Marginal analysis in economics uses the derivative of a demand function to quantify the change in demand resulting from a small change in price. The derivative, often referred to as marginal demand, helps determine the incremental impact on sales, making it a key tool in decision-making processes and in understanding the dynamics of consumer behavior.

Transcript

00:01 Hey guys, in this problem we're going to be looking at a demand function for a certain music player, donated by an exponential function q, which is equal to 240e to the negative 0 .003 power.

00:14 And this will give us the demand in thousands, and our x will be dollars.

00:22 So for part a, it asks how many music players will be bought at a price of $250.

00:27 So we can simply plug in 250 for x, plug into our calculator, and solve.

00:35 And for this, i got 113, and this will be in thousands.

00:54 Part b asked to graph the demand function for the interval 0, which is less than or equal to x, which is less than or equal to 400.

01:05 So i am going to just make some quick asymptotes here.

01:11 If i can, let's make a little straight ascentote.

01:15 Nice pretty asymptote.

01:22 And we can go by increments of 100 for our x -axis.

01:30 400.

01:32 400.

01:36 Make another asymptote.