MATH 234, Calculus for BusinessLecture 9, Textbook Sections 4.1, 4.2Derivatives of Sums, Differences, Products, QuotientsFebruary 20th, 2017Announcements (1) HW 4 is due Thursday, 2/23, 11:59pm; late HWs are not accepted. (2) No quiztomorrow, quiz next week.Remark. Recall from Lecture 7f0(a) =can be understood in 3 ways:(1)(2)(3)Remark. Recall from Lecture 8, the derivative of f(x) is• Constant Rule: If f(x) = k, then f0(x) =• Constant Multiple Rule: If f (x) = kg(x), then f0(x) =• Sum Rule:ddxf(x) + g(x)=• Difference Rule:f(x) − g(x)0=• Power Rule: If f(x) = xn, then f0(x) =Example 1. Find the derivative of f(x) = 5x1/2+ 7x2/3+ 2x32Solution.1Section 4.2, Derivatives of Products and QuotientsQuestion 1. Recall that• “the derivative of the sum is the sum of the derivatives”, and• “the derivative of the difference is the difference of the derivatives”Is it true that• “the derivative of the product is the product of the derivatives”, and• “the derivative of the quotient is the quotient of the derivatives”?Remark: If we replaced “derivative” with “limit”, then the statements are all true.Let us test the questions with some examples.Example 2. Let f(x) = x5. Note that f(x) = x3∗ x2, a product of two functions.Is the derivative of the product is the product of the derivatives?Solution.Example 3. Let f(x) = x5. Note that f(x) = x8/x3, a quotient of two functions.Is the derivative of the quotient is the quotient of the derivatives?Solution.Theorem 1 (The Product Rule). If f(x) = u(x) ∗ v(x), then f0(x) =.Example 4. Let f(x) = x5. Note that f(x) = x3∗ x2, a product of two functions.Use the product rule to compute f0(x).Solution.2Example 5. Let f(x) = (2x + 1)(3x + 2). Compute f0(x) in two ways• Multiply to get a quadratic function for f(x), and proceed as before.• Use the product rule.Solution.The Proof of the Product Rule is:3Example 6. A manufacturer of a product determines that, at t months,number of units sold = x(t) = t2− 12t + 40price per unit = p(t) = t2− 10t + 26At 4 months, is revenue (or income) increasing or decreasing?Solution.The graph of the revenue function, R(t) = x(t) ∗ p(t), is4Theorem 2 (The Quotient Rule). If f(x) =u(x)v(x), then f0(x) =Numerator mnemonic: “low dee high minus high dee low”Denominator mnemonic: “low square”The proof is part of Groupwork 8 (tomorrow).Example 7. Let f(x) = x5. Note that f(x) = x8/x3, a quotient of two functions.Use the quotient rule to compute f0(x).Solution.Example 8. Let f(x) =2x + 45x − 3. Compute f0(x).Solution.Definition. Let C(x) be the cost of producing x-units of a product.Recall, the marginal cost is the derivative of cost: C0(x).The average cost per item is C(x) =The marginal average cost is the :C(x)0=Definition. Let R(x) be the revenue of selling x-units of a product.Recall, the marginal revenue is the derivative of revenue: R0(x).The average revenue per item is R(x) = .The marginal average revenue is :R(x)0=5Example 9. Let C(x) =4x3+10x2+1and R(x) = 5x2+ 10.Compute the marginal average cost and the marginal average revenue.Solution.Question 2. Let P (x) denote the profit from producing and selling x-units.• What should be the definition of “average profit per item”?• What should be the definition of “marginal average
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