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1. Find the derivative of f (x) = x 4 .

Section 2.3. 1. Find the derivative of f (x) = x 4. 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.

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1. Find the derivative of f (x) = x 4 .

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  1. Section 2.3 1. Find the derivative of f (x) = x 4 . Use the power rule: If y = x n then y’ = nx n – 1

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

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

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

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

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

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

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

  10. 10. a. Find the equation of the tangent line to f (x) = x 2 + 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

  11. 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].

  12. 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

  13. 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]

  14. 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: • x = 8 and interpret your answer. • x = 64 and interpret your answer.

  15. 13. Business: Marginal Cost (12 continued) Use a calculator to find the actual cost of the 64th 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?

  16. 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 18-24-year old population in the United States • will follow the function • (in thousands), • wherex is the number of years after 2010. Find the rate of change of this population: • In the year 2030 and interpret your answer. • In the year 2010 and interpret your answer.

  17. 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 2000. Find the rate of change of this percentage in the year 2010 and interpret your answer.

  18. 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.

  19. 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. • Find the marginal utility function. MU(x). • Find MU(1), the marginal utility of the first dollar. • Find MU(1,000,000), the marginal utility of the millionth dollar. • U (x) = 100 x

  20. 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) = 0.831 x 2 – 18.1 x + 137.3 (for 10  x  16) • Find f(12) and f’(12) and interpret these numbers. [Hint: x = 12 corresponds • to a high school graduate.] • b. Find f(16) and f’(16) and interpret these numbers. [Hint: x = 16 corresponds • to a college graduate.]

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