Question

The quantity, q, of a certain skateboard sold depends on the selling price, p, in dollars, so we write q = f(p). You are given that f(100) = 15200 and f '(100) = −90.

(a) What does f(100) = 15200 tell you about the sales of skateboards?

When the price of the skateboard is $_________ , then ______ skateboards will be sold.

What does f '(100) = −90 tell you about the sales of skateboards?

If the price increases from $100 to $101, the number of skateboards sold would _______ (increase/decrease) by roughly________ skateboards

(b) The total revenue, R, earned by the sale of skateboards is given by R = pq. Find R '(p).(chose)

-R '(p) = p 'q

-R '(p) = pf '(p) + f(p)   

- R '(p) = p + q

-R '(p) = q

-R '(p) = f '(p) + f(p)

(c) If the skateboards are currently selling for $100, what happens to revenue if the price is increased to $101?
The revenue  ---Select--- (increases/decreases) by roughly $ _______.

Answer 1

(a) Since we are given that demand function f(100) = 15200 and f'(100) = -90 it means that when price is 100, there are 15200 sjateboards being sold and an additional increase in price by $1 would roughly decrease the demand by 90. Hence,

When the price of the skateboard is $100, then 15200 skateboards will be sold

If the price increases from $100 to $101, the number of skateboards sold would decrease by roughly 90 skateboards

(b) Note the definition of revenue function, and take derivatives as

Hence, we get the result that

R '(p) = pf '(p) + f(p)

(c) We need to find incremental change in revenue when quantity is increased by one unit from 100. This is, by definition, simply the value of derivative of the Revenue function at q = 100, whose functional form we have obtained above.

Hence, By inserting p = 100 in above equation and using all given values,

So we get a positive change of $6200

The revenue increases by roughly $6200