Break-even
The break-even point is the point at which revenue equals cost. In other words, it’s when these two functions are equal, so we can find the break even point at the intersection of the cost and revenue functions.
Algebraically, you could set R(x) = C(x) and solve for x. Or, you could set the profit function equal to zero and solve for x. In calculus though, we usually skip the algebra and head straight for a graph, which can be much easier for ugly-looking functions.
If there are two points or more, you’re generally only concerned with the first point: the point at which the break-even point is first achieved.
Example question: A phone app has fixed production costs of $70 for the cover design and marginal distribution costs of 40 cents per copy. If each app sells for 50 cents per download, what is the break even point?
Step 1: Find the Revenue function. The company is selling the app for $0.50, so the revenue function is: R(x) = 0.50x.
Step 2: Find the Cost function. The question states that the marginal cost is $0.40 per copy, and the fixed costs are $70. This means that the cost function is: C(x) = 0.40x + 70.
Step 3: Graph the two functions and find the break even point intersection.
Step 4: Find the Intersection. Exactly how you find the intersection (the crossing point) depends on your graphing calculator. I’m using Desmos here, and to find the intersection just click on the point where the two graphs cross and it will show you the coordinates.
Step 5: Identify the “x” value in the coordinates from Step 4. This is the break even point: the “x” units that must be sold to break even.
In this case, the break even point is 700 units.
Line of Best Fit
- A line of best fit is a line through a set of data points that best represents the relationship between the data.
- This line can be placed “by eye” by trying to have the line as close as possible to all points, and having a similar number of points above and below the line.
- For better accuracy, we can use Least Squares Regression to calculate the line.
- The aim is to calculate the values of the slope (m) and y-intercept (b) in the equation of a line (y = mx + b).
- To find the line of best fit for N points:
- Step 1: For each (x,y) point calculate x2 and xy.
- Step 2: Sum all x, y, x2 and xy, which gives us Σx, Σy, Σx2 and Σxy (Σ means “sum up”).
- Step 3: Calculate Slope m: m = N Σ(xy) − Σx Σy / N Σ(x2) − (Σx)2
- Step 4: Calculate Intercept b: b = Σy − m Σx / N
- Step 5: Assemble the equation of a line: y = mx + b