Instantaneous Rate of Change

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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, 2017 1 / 11

Reminder Reminder Test #1 is this Friday, February 10. Robb T. Koether (Hampden-Sydney College) Instantaneous Rate of Change Wed, Feb 8, 2017 2 / 11

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, 2017 2 / 11

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, 2017 2 / 11

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, 2017 3 / 11

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, 2017 4 / 11

Instantaneous Rate of Change Instantaneous Rate of Change Let the cost function be C(x) = 20x + 8000 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, 2017 5 / 11

Instantaneous Rate of Change Instantaneous Rate of Change Let the cost function be C(x) = x 2 + 20x + 8000 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, 2017 6 / 11

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, 2017 7 / 11

The Derivative The Derivative Let the cost function be C(x) = x 2 + 20x + 8000 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, 2017 8 / 11

Tangent line to C(x) at x = 100 Tangent line to C(x) at x = 100 200 000 150 000 100 000 50 000-50 000 100 200 300 400-100 000 Robb T. Koether (Hampden-Sydney College) Instantaneous Rate of Change Wed, Feb 8, 2017 9 / 11

Tangent line to C(x) at x = 200 Tangent line to C(x) at x = 200 200 000 150 000 100 000 50 000-50 000 100 200 300 400-100 000 Robb T. Koether (Hampden-Sydney College) Instantaneous Rate of Change Wed, Feb 8, 2017 10 / 11

Tangent line to C(x) at x = 300 Tangent line to C(x) at x = 300 200 000 150 000 100 000 50 000-50 000 100 200 300 400-100 000 Robb T. Koether (Hampden-Sydney College) Instantaneous Rate of Change Wed, Feb 8, 2017 11 / 11

Example 2.1.5 Example 2.1.5 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, 2017 12 / 11

Example 2.1.5 Example 2.1.5 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, 2017 12 / 11

Example 2.1.5 Example 2.1.5 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, 2017 12 / 11

Example 2.1.5 Graph of C(x) = 400x 2 + 6800x 12000 15 000 10 000 5000-5000 2 4 6 8 10 12 14-10 000 Robb T. Koether (Hampden-Sydney College) Instantaneous Rate of Change Wed, Feb 8, 2017 13 / 11

Example 2.1.5 Tangent Line to C(x) = 400x 2 + 6800x 12000 15 000 10 000 5000-5000 2 4 6 8 10 12 14-10 000 Robb T. Koether (Hampden-Sydney College) Instantaneous Rate of Change Wed, Feb 8, 2017 14 / 11

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