The demand function for the Luminar desk lamp is given by...

The demand function for the Luminar desk lamp is given by $$p=f(x)=-0.1 x^{2}-0.4 x+35$$ where $x$ is the quantity demanded in thousands and $p$ is the unit price in dollars. a. Find $f^{\prime}(x)$ b. What is the rate of change of the unit price when the quantity demanded is 10,000 units $(x=10) ?$ What is the unit price at that level of demand?

The demand function for the Luminar desk lamp is given by
$$p=f(x)=-0.1 x^{2}-0.4 x+35$$
where $x$ is the quantity demanded in thousands and $p$ is the unit price in dollars.
a. Find $f^{\prime}(x)$
b. What is the rate of change of the unit price when the quantity demanded is 10,000 units $(x=10) ?$ What is the unit price at that level of demand?

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Key Concepts

Demand Function

A demand function represents the relationship between the price of a good and the quantity demanded. In economic analysis, this function helps determine how changes in price affect consumer purchasing behavior, encapsulating the idea that demand typically decreases as price increases.

Differentiation

Differentiation is a process in calculus used to determine the derivative of a function, which gives the rate at which the function's output changes with respect to its input. This technique is essential for analyzing how small changes in variables impact outcomes in various fields, including economics.

Derivative

The derivative is a fundamental concept in calculus that measures the rate of change of a function with respect to one of its variables. It quantifies how a small change in the input leads to a change in the output, offering crucial insights into the behavior of functions.

Instantaneous Rate of Change

The instantaneous rate of change is the value of the derivative at a specific point in the function. It provides a precise measure of how quickly the function's value is changing at that particular moment, which is particularly useful in economics to understand how variables such as price respond to changes in demand.

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