Suppose that the profit for a company is increasing at a rate of
where the company has been in operation for t years. What is the total change in profit over the first three years?
Continue reading “How Do You Find Profit from Marginal Profit?”
Category: Applied Calculus
How Do You Calculate A Rate From A Function?
In many business problems, we are interested in knowing the rate at which some quantity is changing. If this quantity is modeled by a function, the rate is modeled by the derivative of the function.
In the problem below, we are given a model of revenue for Verizon, a national wireless carrier. To find the rate, we’ll need to take the derivative of the revenue function.
From 2006 through 2011, Verizon Wireless grew steadily. As the number of connections increased, so did the revenue from those connections.A cubic model for the revenue over this period is
where c is the number of connections in millions.
At what rate is the revenue increasing when there are 80,000,000 connections?
Questions that ask about rate or refer to “marginal” are questions about the derivative. In this case, it is referring to the value of the derivative at c = 80. The derivative of the revenue function is
The value of the derivative at 80 million connections is
To find the units on this rate, recall that the derivative is also the slope of the tangent line at c = 80. The slope of this line has vertical units of billions of dollars and horizontal units of millions of connections. This may be simplified to
Since a billion divided by a million is one thousand. So means that an additional connection at this level results in an increase in revenue of 0.31265 thouusand dollars or $321.65.
How Do You Find The Equation Of A Tangent Line?
A tangent line to a function is a line that looks most like the function at a point. In common terms, it just grazes the function.
To find its equation, we need to locate the point where the two meet as well as the slope of the function at that point. Then we can use the slope-intercept form or point-slope form of a line to get the equation.
Find the equation of the tangent line to
at x = 3.
Since this problem is asking for the equation of a line, let’s start with the point-slope form
This requires a point (x1, y1) and slope m. We’ll use the function to get the point and the derivative to get the slope of the tangent line.
Find the point: We are given a point x = 3. To find the corresponding y value, put the x value into the function
Find the slope of the tangent line: We need h′(3) to get the slope of the tangent line. We’ll use the Power Rule to take the derivative,
The slope of the tangent is
Write the equation of the tangent line: Putting the point (3, 10) and the slope 9 into the line yields
If you are asked to write this in slope-intercept form, you’ll need to solve this for y to give
If you graph h(x) and the tangent line together, it should be obvious that your tangent line is correct (ie. tangent).
How Do I Find Area Between Curves?
Sometimes a seemingly easy problem can get fairly complicated with the addition of a few extra requirements. For instance, suppose we want to find the area between the functions y = x2 – 4 and y = 3x. If we graph the two functions, we see that they appear to cross at x = -1 and x = 4.
We can verify these two points by setting the functions equal to each other and solving for x.
The area between these curves lies above the parabola and below the line.
To find the area of the shaded region, take the definite integral from x = -1 to x = 4 of the higher function minus the lower function,
We can evaluate this integrand using the Fundamental Theorem of Calculus,
Let’s now modify this problem by finding the area enclosed by the functions y = x2 – 4 and y = 3x as well as the lines x = -5 and x = 1. Graph each of these equations.
The region enclosed by these graphs is more complicated since the functions cross at x = -1.
To find the area of the enclosed region, we need to break it into two parts. The first part runs from x = -5 to the point of intersection at x = -1. The area of this part is
The second part extends from x = -1 to x = 1 and has area
The area enclosed by the functions y = x2 – 4 and y = 3x as well as the lines x = -5 and x = 1 is the sum of these parts or 206/3. Adding the vertical lines on either side of the point of intersection requires the use of two definite integrals since the parabola is higher on the left side of x = -1 and the line is higher on the right side of x = -1.
Thus the area between the curves is 184/3 + 22/3 or 206/3.
Death and Piecewise Functions
Although this may seem a little gruesome, it is not uncommon for businesses to give discounts for volume sales. In this case, a mortician charges less per pound for bodies weighing more than a certain amount.
The local mortician charges by the pound for embalming according to the following table:
Find a piecewise linear function that models the cost as a function of weight.