Demand Suppose the demand for a certain item is given by...

Demand Suppose the demand for a certain item is given by $D(p)=-3 p^{2}-5 p+700,$ where $p$ represents the price of the item in dollars.
(a) Find the rate of change of demand with respect to price.
(b) Find and interpret the rate of change of demand when the price is $\$ 10 .$

Key Concepts

Marginal Analysis

Marginal analysis in economics refers to the examination of the effects of a small incremental change in an economic variable. When applied to a demand function, the computed derivative at a specific price, such as $10, illustrates the marginal change in demand, aiding in decision-making regarding pricing and market strategy.

Demand Function

A demand function represents the relationship between the price of a good and the quantity demanded by consumers. In economics, it is used to model how variations in price influence demand and helps in understanding consumer behavior in the marketplace.

Derivative (Rate of Change)

The derivative of a function measures its instantaneous rate of change with respect to a variable. In the context of a demand function, calculating the derivative with respect to price provides a mathematical tool to assess how small changes in price lead to changes in the quantity demanded.

Transcript

00:01 So in this question, demand of certain item is given with respect to price p.

00:06 We have to find the rate of change of demand with respect to price and find the demand of that item at when the price is $1 .10 and then we will interpret the result.

00:19 So the rate of change, we will use the formula, the definition of derivative, to find the rate of change, which is given by limit of age approaching zero.

00:31 So using the steps which we have learned before, we will first find the rate of change of demand with respect to the price.

00:44 So let's first plug in wherever here we are given, we're not given x, it's p.

00:50 So wherever we have p we will write.

00:54 So we can change it into p if that is easier to understand.

00:58 In place of x we can write p.

01:02 So let's find the rate of change.

01:04 Of the demand.

01:13 So, derivative of d.

01:15 First to find derivative of d, let's find f, p plus h.

01:24 P plus h is wherever i can, i find p, i will write p plus h squared minus 5p plus h and then simplify.

01:39 So here, if i use, if i foil it, i will get p squared plus 2 ph.

01:49 Plus 8 squared and then i can distribute 5 and not forget to change sign and then distribute negative 3 again will not forget to change sign if it's negative outside okay so now we will combine 5p plus h minus 5p so that's the next step so i'll rewrite whatever is given here in part a whatever we found so we'll rewrite that 6p h minus 3h squared minus 5p minus 5h plus 700 and then fp is given by the function here is given by this negative 3p squared because it's minus yes i'll not forget this is minus here so all the signs here will change at dp demand with the price so it will become 3 positive 3 p squared plus 5p minus then i'll see well i think okay so can i get rid of something i can get rid of three p square negative then five and then 700 and 700 so i'm left with negative negative 6 p h minus 3d square and minus 5 h i'll take out h common negative 6p minus 3 h and minus 5 the last step is divide everything by h and then find the limit.

03:41 So limit of h approaching 0...

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