Homework_6_Solutions - Math 1110 Name: Homework 6 Due 10/6...

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Unformatted text preview: Math 1110 Name: Homework 6 Due 10/6 or 10/7 in class Please print out these pages. Write your answers to the “text exercises” on separate paper and staple it to these pages. You should include computational details. These problems will be assessed for completeness. Always write neatly and legibly. Please answer the “presentation problems” in the spaces provided. Include full explanations and write your answers in complete, mathematically and grammatically correct sentences. Your answers will be assessed for style and accuracy, and you will be given written feedback on these problems. GRADES Text exercises / 20 Pres probs / 20 Staple Text exercises. Please do the following problems from the book. §3.3 #12, 14, 18, 24, 32, 42, 52, 56, 70, 78 §3.4 #4, 10, 13, 16, 22, 26 §3.5 #8, 14, 16, 23, 28, 38, 40, 47, 52, 57, 62 /1 Math 1110 (Fall 2011) HW6 Presentation Problems 2 Question 1. (10 points) If it costs a manufacturer C(x) dollars to produce x items, then the average cost of production is A(x) = C(x) dollars per item. Recall that the marginal cost is the first x derivative of the cost. (a) In economics, the marginal cost at x is used as a good approximation of how much it costs to manufacture one additional item, given that you are already manufacturing x items. Explain (in one or two complete sentences!) why this makes sense. We will assume that the cost function is differentiable near the point x. We want to estimate C ( x + 1 ) − C ( x) C ( x + 1 ) − C ( x) = . This is the slope of the secant line from (x, C(x)) to (x + 1 1, C(x + 1). Recall that if a function is differentiable at a point, then close to that point the tangent line is a good approximation of the function. Further, the slope of the tangent line is a good approximation of the slope of secant lines through nearby points. In particular, this means that C ￿ (x) is a good approximation of the cost to manufacture one additional item. (b) The value of x which minimizes the average cost is the value for which the derivative of the average cost is 0. Show that this value makes the average cost equal to the marginal cost. First we find the derivative of average cost using the quotient rule. A ￿ ( x) = Now we set A ￿ (x) equal to 0. C ￿ ( x) x − C ( x ) x2 C ￿ ( x) x − C ( x ) x2 C ( x) Which means 0 = C ￿ (x)x − C(x), and C ￿ (x) = = A(x). We conclude that at any value x where the derivative of average cost equals 0, average cost equals marginal cost. In particular, at the value of x which minimizes average cost, average cost and marginal cost are equal. 0= Math 1110 (Fall 2011) HW6 Presentation Problems 3 Question 2. (10 points) Consider the function f(x) whose graph is shown below. y=1 y=0 x=0 x=1 The graph of the function of f(x). At which value(s) of x pictured above is this function NOT differentiable? The function is not differentiable at x = −1, 0, or 2. Please sketch the graph of the derivative, f ￿ (x), on the axes provided. For values of x ≤ −1, what is most important is where the derivative is positive, negative, and 0. For x > −1, your graph should be exact. y=1 y=0 x=0 x=1 Homework 6 Book Problem Answers: Section 3.3: Section 3.4: Section 3.5: ...
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This note was uploaded on 03/01/2012 for the course MATH 1110 at Cornell University (Engineering School).

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