1. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 1 of 115 § 1.7 More About Derivatives

2. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 2 of 115  Differentiating Various Independent Variables  Computing Second Derivatives  Second Derivatives Evaluated at a Point  Marginal Cost  Marginal Revenue  Marginal Profit Section Outline

3. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 3 of 115 Differentiating Various Independent VariablesSOLUTION Find the first derivative. EXAMPLE We first note that the independent variable is t and the dependent variable is T. This is significant inasmuch as they are considered to be two totally different variables, just as x and y are different from each other. We now proceed to differentiate the function. This is the given function. We begin to differentiate. Use the Sum Rule. Use the General Power Rule.

4. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 4 of 115 Differentiating Various Independent Variables Finish differentiating. CONTINUED Simplify.

5. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 5 of 115 Second DerivativesSOLUTION Find the first and second derivatives. EXAMPLE This is the given function. This is the first derivative. This is the second derivative.

6. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 6 of 115 Second Derivatives Evaluated at a PointSOLUTION Compute the following. EXAMPLE Compute the first derivative. Compute the second derivative. Evaluate the second derivative at x = 2.

7. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 7 of 115 Marginal Cost

8. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 8 of 115 Marginal CostSOLUTION Let C(x) be the cost (in dollars) of manufacturing x bicycles per day in a certain factory. Assume C(50) = 5000 and Estimate the cost of manufacturing 51 bicycles per day. EXAMPLE We will first use the additional cost formula for manufacturing 1 more bicycle per day beyond the cost of producing 50 bicycles per day. We already know it costs $5000 to produce 50 bicycles per day since C(50) = 5000. So we wish to determine how much more, beyond that $5000, it costs to produce 51 bicycles. Therefore, we estimate the cost of manufacturing 51 bicycles to be $5045.

9. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 9 of 115 Marginal Revenue & Marginal Profit

10. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 10 of 115 Marginal RevenueSOLUTION Suppose the revenue from producing (and selling) x units of a product is given by R(x) = 3x – 0.01x 2 dollars. EXAMPLE (a) Since we are looking for the marginal revenue at a production level of 20, and we have an equation for R(x), we will simply find (a) Find the marginal revenue at a production level of 20. (b) Find the production levels where the revenue is $200. This is the given revenue function. This is the marginal revenue function. Evaluate the marginal revenue function at x = 20. Therefore, the marginal revenue at a production level of 20 is 2.6.

11. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 11 of 115 Marginal Revenue (b) To find the production levels where the revenue is $200 we need to use the revenue function and replace revenue, R(x), with 200 and then solve for x. This is the given revenue function. Replace R(x) with 200. Therefore, the production levels for which revenue is $200 are x = 100 and x = 200 units produced. CONTINUED Get everything on the left side of the equation. Multiply everything by 100. Factor. Solve for x.