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Post a total of 3 substantive responses over 2 separate days for full participation. This includes your initial post and 2 replies to classmates or your faculty member.
Due Thursday, July 4, by 11:59 pm, MST, (AZ time)
This week, we explore graphical characteristics of functions.
Respond to the following in a minimum of 175 words:
Identify an everyday example that can be modeled by one of the types of graphs you learned about this week.
Discuss what the extrema, concavity, critical points, and derivatives of that function will help you understand about the everyday example that you selected.
1st classmate:
Madison Muir
An everyday example that can be modeled by a quadratic graph is the profit made from selling a product, where profit depends on the number of units sold. This relationship can be represented by the quadratic function P(x) = -ax^2 + bx + c, where x represents the number of units sold, and P(x) represents the profit. (^2 means squared)
The extrema of this function, specifically the maximum point, represents the highest profit achievable. This maximum can be found by setting the first derivative to zero and solving for x. This x-value indicates the optimal number of units to sell to maximize profit.
The concavity of the function, determined by the second derivative, shows that the graph opens downward (since the coefficient of x ^2 (x squared) is negative). This indicates that there is a point of diminishing returns, where increasing sales beyond a certain point results in decreased profit due to factors like increased costs or market saturation.
The critical points are where the first derivative equals zero, identifying the number of units sold at maximum profit. Understanding these points helps in determining the sales target.
The derivatives of the function provide insights into the rate of change of profit. The first derivative represents the marginal profit, indicating how profit changes with each additional unit sold. The second derivative indicates the rate of change of the marginal profit, helping to identify the point at which increasing sales further reduces profit.
Analyzing these aspects allows businesses to optimize their sales strategy to achieve maximum profitability.
2nd classmate:
Caitlyn Oneil
An example of a quadratic function in business and economics is profit. Profit depends on the number of units sold. An example of a quadratic profit function would be P(x) = ax^2 + bx + c
In this equation, a, b, and c represent factors such as fixed costs, variable costs, and price.
The extrema of a quadratic function occurs at the vertex of the graph, the minimum or maximum.
For a quadratic function P(x) = ax^2 + bx + c
If a >0, the function has a minimum at x = −b/2a
If a <0, the function has a maximum at x = −b/2a
The concavity of a quadratic function is determined by the sign of a, therefore:
If a>0, the function is concave upwards
If a<0, the function is concave downwards
Critical points are the points where the derivative P′(x) is 0 or undefined
The derivative is P′(x) = 2ax + b
P′(x)=0
2ax+b=0
x= -b/2a is the critical point of the function
Derivatives - represents the rate of change of profit with respect to x
The derivative of P(x)= ax^2 + bx +c is P′(x)= 2ax + b
Examining these elements enables businesses to assess their sales strategies to maximize profits.
Understanding the slope of a tangent line is crucial because it gives the exact rate of change of the function at that exact spot. This information allows for finding critical points, extrema, and concavity.
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