1. Sec 2.6 – Marginals and Differentials 2012 Pearson Education, Inc. All rights reserved Let C(x), R(x), and P(x) represent, respectively, the total cost, revenue, and profit from the production and sale of x items. The marginal cost at x, given by C (x), is the approximate cost of the (x + 1) th item: C (x) ≈ C(x + 1) – C(x), or C(x + 1) ≈ C(x) + C (x). The marginal revenue at x, given by R (x), is the approximate revenue from the (x + 1) th item: R (x) ≈ R(x + 1) – R(x), or R(x + 1) ≈ R(x) + R (x). The marginal profit at x, given by P (x), is the approximate profit from the (x + 1) th item: P (x) ≈ P(x + 1) – P(x), or P(x + 1) ≈ P(x) + P (x).

2. Sec 2.6 – Marginals and Differentials Example 1: Given the following cost and revenue functions, find each of the following: a) Total profit, P(x). b) Total cost, revenue, and profit from the production and sale of 50 units of the product. c) The marginal cost, revenue, and profit when 50 units are produced and sold. a)

3. Sec 2.6 – Marginals and Differentials b) c) The approximate cost of the 51 st unit will be $6200. The approximate revenue from the sale of the 51 st unit will be $6340. The approximate profit from the 51 st unit will be $140.

4. Sec 2.6 – Marginals and Differentials

5. For f a continuous, differentiable function, and small ∆x.

6. Sec 2.6 – Marginals and Differentials DEFINITION: For y = f (x), we define dx, called the differential of x, by dx = ∆x and dy, called the differential of y, by dy = f (x)dx.

7. Sec 2.6 – Marginals and Differentials Examples

8. Sec 2.6 – Marginals and Differentials a) b)

9. An Implicit function is one where the variable “y” can not be easily solved for in terms of only “x”. Examples: Sec 2.7 – Implicit Differentiation and Related Rates

10. To differentiate implicitly : e) Divide both sides of the equation to isolate dy/dx. a) Differentiate both sides of the equation with respect to x (or whatever variable you are differentiating with respect to). b) Apply the rules of differentiation as necessary. Any time an expression involving y is differentiated, dy/dx will be a factor in the result. c) Isolate all terms with dy/dx as a factor on one side of the equation. d) Factor out dy/dx.

11. Sec 2.7 – Implicit Differentiation and Related Rates

12. Find the equation of the tangent and normal lines for the following implicit function at the given point. Sec 2.7 – Implicit Differentiation and Related Rates

15. 1. Read the problem, pull out essential information and identify a formula to be used. 2. Sketch a diagram if possible. 3. Write down any known rate of change & the rate of change you are looking for. 4. Be careful with signs…if the amount is decreasing, the rate of change is negative. 5. Pay attention to whether quantities are constant or varying. 6. Set up an equation involving the appropriate quantities. 7. Differentiate with respect to t (could be other variables) using implicit differentiation. 8. Plug in known items (you may need to find some quantities using geometry). 9. Solve for the item you are looking for, most often this will be a rate of change. 10. State your final answer with the appropriate units. Related rates problems involve finding a rate at which a quantity changes by relating that quantity to other quantities whose rates of change are known.

16. 5) A balloon is being inflated at a rate of 10 cubic centimeters per second. How fast is the radius of a spherical balloon changing at the instant the radius is 5 centimeters? Sec 2.7 – Implicit Differentiation and Related Rates

17. 7) A 25-foot ladder is leaning against a wall. The bottom of the ladder is being pulled away from the wall at a rate of 3 feet per second. How fast is the top of the ladder moving at the instant the bottom of the ladder is 15 feet away from the wall?

18. Sec 2.7 – Implicit Differentiation and Related Rates Example