Problem:

 

Apple is selling iPhones for $99. Revenue for the sales of Iphones is given by the function R = p*q, where “p” represents the price of coffee bean bags and “q” represents the quantity of how many Iphones are sold. Based upon the data given by Apple (Apple - Press Info), the function for quantity is                            For next year’s iPhone, they’re thinking about lowering the price from $99 to $80 per iPhone. Using this equation, predict how the revenue will be changing when the price is $80.

 

Solution: 

 

First, we need to establish a goal. The goal of the problem is to find the change in the total revenue when the price of the iPhone is $80. The equation for revenue is given as R=p*q, where as "p" is price and "q" is quantity. We are also given the equation for quantity in terms of price. We want the equation to be revenue in terms of price, and this can be solved by substituting “q” for the quantity equation in terms of price.

 

We can solve for the derivative in order to find the change in the revenue when the price is $80. You can tell that the new revenue function is a function multiplied by another function. Using our knowledge of derivatives, we can use the product rule in order to find the derivative, and then solve using basic algebra. We are going to use that formula to find how much Apple’s revenue is changing by when the price of an Iphone is reduced to $80.

 

 

This is the equation we made based off of the data we collected, from the equation we had before which was                            before we made the equation in respect to price. With our new equation, we used the product rule formula to find the derivative, in other words, solving for “r” - the change in revenue.

 

 

 

Notice how “p” is being multiplied by “q” so we took the derivative of the first function (“p”) multiplied it by the original second function (“q”). Then added, the original first function (“p”) multiplied by the derivative of the second (“q”).

 

Note: The derivative of a single variable with no leading coefficients or exponents will always be equal to 1 because of the power rule.

 

Therefore:

 

 

 

 

 

 

 

Note: When finding the derivative of the second function, the chain rule is involved. We had to refer to the formula of multiplying the derivative of the first function by the original second function multiplied by the derivative of the second function.

 

Therefore:

 

 

 

 

 

 

We had to apply the chain rule because in our equation was an inside function of “e” raised to a power considering that we had a value for g(x). So, the outside function would be         This is for the process when using the chain rule. We had to evaluate the derivative of the outside function with the original inside function.

 

 

The derivative of     is still     .This is why the derivative of the inside function will just stay the same along with the outside function that your evaluation. Since the last step for the  chain rule is multiplying the original derivative of g(x), you would use the power rule so the variable attached to g(x) would be eliminated.

 

 

 

 

 

 

 

Since we were trying to solve when the price is $80 for an Iphone, we simply plugged in $80 into the variable for “p” every time it was in the equation.

 

 

 

 

 

 

Since the answer must be in millions, we multiplied the final answer by 1,000,000.

What it all means: This means that Apple’s revenue is increasing by $3,840,359,659.71 per Iphone sold if Apple was to charge $80 instead of $99.