Instantaneous Rate of Change
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1 Instantaneous Rate of Change Lecture 13 Section 2.1 Robb T. Koether Hampden-Sydney College Wed, Feb 8, 2017 Robb T. Koether (Hampden-Sydney College) Instantaneous Rate of Change Wed, Feb 8, / 11
2 Reminder Reminder Test #1 is this Friday, February 10. Robb T. Koether (Hampden-Sydney College) Instantaneous Rate of Change Wed, Feb 8, / 11
3 Reminder Reminder Test #1 is this Friday, February 10. It will cover Chapter 1. Robb T. Koether (Hampden-Sydney College) Instantaneous Rate of Change Wed, Feb 8, / 11
4 Reminder Reminder Test #1 is this Friday, February 10. It will cover Chapter 1. Be there. Robb T. Koether (Hampden-Sydney College) Instantaneous Rate of Change Wed, Feb 8, / 11
5 Objectives Objectives The concept of instantaneous rate of change. The definition of the derivative. Using the definition to find a derivative. Robb T. Koether (Hampden-Sydney College) Instantaneous Rate of Change Wed, Feb 8, / 11
6 Instantaneous Rate of Change Definition (Instantaneous Rate of Change) The instantaneous rate of change of f (x) with respect to x at a point c is f (c + h) f (c) lim. h 0 h Robb T. Koether (Hampden-Sydney College) Instantaneous Rate of Change Wed, Feb 8, / 11
7 Instantaneous Rate of Change Instantaneous Rate of Change Let the cost function be C(x) = 20x where x is the number of units produced. Find the instantaneous rate of change of cost relative to production when production is x = 100. Robb T. Koether (Hampden-Sydney College) Instantaneous Rate of Change Wed, Feb 8, / 11
8 Instantaneous Rate of Change Instantaneous Rate of Change Let the cost function be C(x) = x x where x is the number of units produced. Find the instantaneous rate of change of cost relative to production when production (a) is x = 100. (b) is x = 200. (c) is x = 300. Robb T. Koether (Hampden-Sydney College) Instantaneous Rate of Change Wed, Feb 8, / 11
9 The Derivative Definition (The Derivative) The derivative of f (x) with respect to x is the function f f (x + h) f (x) (x) = lim. h 0 h Robb T. Koether (Hampden-Sydney College) Instantaneous Rate of Change Wed, Feb 8, / 11
10 The Derivative The Derivative Let the cost function be C(x) = x x where x is the number of units produced. (a) Find the derivative C (x) of cost with respect to production. (b) Evaluate C (x) at x = 100, x = 200, and x = 300. Robb T. Koether (Hampden-Sydney College) Instantaneous Rate of Change Wed, Feb 8, / 11
11 Tangent line to C(x) at x = 100 Tangent line to C(x) at x = Robb T. Koether (Hampden-Sydney College) Instantaneous Rate of Change Wed, Feb 8, / 11
12 Tangent line to C(x) at x = 200 Tangent line to C(x) at x = Robb T. Koether (Hampden-Sydney College) Instantaneous Rate of Change Wed, Feb 8, / 11
13 Tangent line to C(x) at x = 300 Tangent line to C(x) at x = Robb T. Koether (Hampden-Sydney College) Instantaneous Rate of Change Wed, Feb 8, / 11
14 Example Example Gordon owns a small manufacturing firm. He determines that when x thousand units of one of his products are produced and sold, the profit generated will be dollars. P(x) = 400x 2 + 6, 800x 12, 000 (a) Is production profitable when 9,000 units are produced? Robb T. Koether (Hampden-Sydney College) Instantaneous Rate of Change Wed, Feb 8, / 11
15 Example Example Gordon owns a small manufacturing firm. He determines that when x thousand units of one of his products are produced and sold, the profit generated will be dollars. P(x) = 400x 2 + 6, 800x 12, 000 (a) Is production profitable when 9,000 units are produced? (b) At what rate should Gordon expect profit to be changing with respect to the level of production x when 9,000 units are produced? Robb T. Koether (Hampden-Sydney College) Instantaneous Rate of Change Wed, Feb 8, / 11
16 Example Example Gordon owns a small manufacturing firm. He determines that when x thousand units of one of his products are produced and sold, the profit generated will be dollars. P(x) = 400x 2 + 6, 800x 12, 000 (a) Is production profitable when 9,000 units are produced? (b) At what rate should Gordon expect profit to be changing with respect to the level of production x when 9,000 units are produced? (c) Is the profit increasing or decreasing at this level of production? Robb T. Koether (Hampden-Sydney College) Instantaneous Rate of Change Wed, Feb 8, / 11
17 Example Graph of C(x) = 400x x Robb T. Koether (Hampden-Sydney College) Instantaneous Rate of Change Wed, Feb 8, / 11
18 Example Tangent Line to C(x) = 400x x Robb T. Koether (Hampden-Sydney College) Instantaneous Rate of Change Wed, Feb 8, / 11
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