Textbook Question
Where is the function continuous? Differentiable? Use the graph of f in the figure to do the following. <IMAGE>
c. Sketch a graph of f'.
2. Intro to Derivatives
Basic Graphing of the Derivative
2:41 minutesProblem 3.6.52
Textbook Question
A cost function of the form C(x) = 1/2x² reflects diminishing returns to scale. Find and graph the cost, average cost, and marginal cost functions. Interpret the graphs and explain the idea of diminishing returns.
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To find the average cost function, divide the cost function C(x) by x. The average cost function is given by AC(x) = C(x)/x = (1/2)x.
To find the marginal cost function, take the derivative of the cost function C(x) with respect to x. The marginal cost function is MC(x) = d(C(x))/dx = x.
Graph the cost function C(x) = (1/2)x², the average cost function AC(x) = (1/2)x, and the marginal cost function MC(x) = x on the same set of axes. The cost function is a parabola opening upwards, the average cost function is a straight line through the origin with a positive slope, and the marginal cost function is also a straight line through the origin with a slope of 1.
Interpret the graphs: The cost function shows that as production increases, the cost increases at an increasing rate. The average cost function shows that the average cost per unit increases linearly with production. The marginal cost function shows that the cost of producing one more unit increases linearly with the number of units produced.
Explain diminishing returns: Diminishing returns to scale occur when increasing production leads to a less than proportional increase in output. In this context, as more units are produced, the cost per additional unit (marginal cost) increases, reflecting inefficiencies that arise with larger scale production.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Cost Function
A cost function represents the total cost incurred by a firm in producing a certain level of output, denoted as C(x). In this case, C(x) = 1/2x² indicates that costs increase with the square of the output level, reflecting how production costs escalate as more units are produced.
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Marginal Cost
Marginal cost is the additional cost incurred from producing one more unit of output. It is derived from the cost function by taking the derivative, which in this case results in MC(x) = x. This concept is crucial for understanding how production decisions affect overall costs.
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Diminishing Returns
Diminishing returns refer to the principle that as more units of a variable input are added to a fixed input, the additional output produced from each new unit of input will eventually decrease. This concept is illustrated in the cost functions, where increasing production leads to higher costs at an increasing rate, indicating inefficiencies in scaling.
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