1. Find the derivative of f (x) = x 4. Section 2.1 Use the power rule: If y = x n then y’ = nx n – 1.

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Presentation transcript:

1. Find the derivative of f (x) = x 4. Section 2.1 Use the power rule: If y = x n then y’ = nx n – 1

2. Find the derivative of f (x) = x 1/2. Use the power rule: If y = x n then y’ = nx n – 1

3. Find the derivative of Use the power rule: If y = x n then y’ = nx n – 1

4. Find the derivative of f (x) = 4x 2 - 3x + 2. Use the power rule: If y = x n then y’ = nx n – 1

5. Find the derivative of Use the power rule: If y = x n then y’ = nx n – 1

6. Find the derivative of Use the power rule: If y = x n then y’ = nx n – 1

7. Find the derivative of Use the power rule: If y = x n then y’ = nx n – 1

8. Find the derivative of the following function at x = - 2.

9. Find the derivative of the following function at x = - 3. f ‘ (x) = 3x 2 so f ‘ (- 3) = 3 ( - 3) 2 = 27

10. a. Find the equation of the tangent line to f (x) = x 2 – 2x + 2 at x = 3. b. Graph the function and the tangent line on the window [-1,6] by [-10,20]. Use your calculator and the draw – tangent button. OR

10. a. Find the equation of the tangent line to at x =3. b. Graph the function and the tangent line on the window [-1,6] by [-10,20]. 2x - 2 Use the slope 4 and the point (3, 5) You need to calculate the 5.

11. a. Find the equation of the tangent line to f (x) = x 3 - 3x 2 + 2x - 2 at x = 2. b. Graph the function and the tangent line on the window [-1,4] by [-7,5]. Use your calculator and the draw – tangent button. OR

11. a. Find the equation of the tangent line to at x =2. b. Graph the function and the tangent line on the window [1,4] by [-7,5] Use the slope 2 and the point (2, - 2) You need to calculate the - 2.

12. Business: Software Costs Businesses can buy multiple licenses for PowerZip data compression software at a total cost of approximately: C (x) = 24x 2/3 dollars for x licenses. Find the derivative of this cost function at: a.x = 8 and interpret your answer. b.x = 64 and interpret your answer. The cost of the 65 th license will be about $4 The cost of the 9 th license will be about $8

13. Business: Marginal Cost (12 continued) Use a calculator to find the actual cost of the 64 th license by evaluating C(64)-C(63) for the cost function in 12. Is your answer close to the $4 that you found for part (b) of that exercise?

14. Business: Marketing to Young Adults Companies selling products to young adults often try to predict the size of that population in the future years. According to the predictions by the Census Bureau, the year old population in the United States will follow the function (in thousands), where x is the number of years after Find the rate of change of this population: a.In the year 2030 and interpret your answer. b.In the year 2010 and interpret your answer. The derivative of the function is P ‘ (x) = - x x a.P’ (20) = – 300 = 300 In 2030, the population will be increasing by 300 thousand per year. b. P’ (0) = – 300 = In 2010, the population will be decreasing by 300 thousand per year.

15. General: Internet Access The percentage of U.S. households with broadband Internet access is approximated by, where x is the number of years after the year Find the rate of change of this percentage in the year 2010 and interpret your answer. OR in the next year, 2011, the percentage of households with broadband internet access will increase by about 10%.

16. Psychology: Learning Rates A language school has found that it’s students can memorize P(t) = 24  t, phrases in t hours of class (for 1  t  10). Find the instantaneous rate of change of this quality after 4 hours of class and interpret your answer. In the next hour, the fifth hour, the student will memorize 6 phrases.

17. Economics: Marginal Utility Generally, the more you have of something, the less valuable each additional unit becomes. For example, a dollar is less valuable to a millionaire than to a beggar. Economists define a person’s “utility function” U(x) for a product as the “perceived value” of having x units of that product. The derivative of U(x) is called marginal utility function, MU(x)=U’(x). Suppose that a person’s utility function for money is given by the function below. That is, U(x) is the utility (perceived value) of x dollars. a.Find the marginal utility function. MU(x). b.Find MU(1), the marginal utility of the first dollar. c.Find MU(1,000,000), the marginal utility of the millionth dollar. U (x) = 100  x

18. General: Smoking and Education According to a study, the probability that a smoker will quit smoking increases with the smoker’s educational level. The probability (expressed as a percent) that a smoker with x years of education will quit is approximated by the equation f (x) = x 2 – 18.1 x (for 10  x  16) a.Find f(12) and f’(12) and interpret these numbers. [Hint: x = 12 corresponds to a high school graduate.] a.b. Find f(16) and f’(16) and interpret these numbers. [Hint: x = 16 corresponds to a college graduate.]