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Section 1.5

Section 1.5 1. By imagining tangents at he indicated points state whether the slope is positive, zero or negative at each point. P 2. P 1. P 3. P 2. P 1. 2. Use the red tangent lines shown to find the slopes of the curve at the points of tangency.

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Section 1.5

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  1. Section 1.5 1. By imagining tangents at he indicated points state whether the slope is positive, zero or negative at each point. P 2 P 1 P 3

  2. P 2 P 1 . 2. Use the red tangent lines shown to find the slopes of the curve at the points of tangency. . The tangent line at P 1 contains the points (1, 4) and (2, 6). The slope between these two points is 2. The tangent line at P 2 contains the points (5, 4) and (2, 6). The slope between these two points is -2/3.

  3. Find the average rate of change of f (x) = x 2 + x between the following • pairs of x-values. • x=1 and x=3 • x=1 and x=2 • x=1 and x=1.5 • x=1 and x=1.1 • x=1 and x= 1.01 • What number do your answers seem to be approaching The average rate of change is the change in the y values divided by the change in the x values. OR

  4. You may do this problem using the 4-step procedure. • Find the average rate of change of f (x) = x 2 + x between the following • pairs of x-values. • x=1 and x=3 Note h = 2 • x=1 and x=2 Note h = 1 • x=1 and x=1.5 Note h = 0.5 • x=1 and x=1.1 Note h = 0.1 • x=1 and x= 1.01 Note h = 0.01 • What number do your answers seem to be approaching Step 1: f (x + h) = (x + h) 2 + ( (x + h) = x 2 + 2xh + h 2 + x + h Step 2: f (x) = x 2 + x Step 3: f (x + h) – f (x) = 2xh + h 2 + h Note the beginning x value is 1 for all parts and only the h changes. Plug in x = 1 and the h values at the top of the page into step 4. 2x + h + 1 = 2 + h + 1 = 3 + h part a yields 5, b yields 4, c yields 3.5, d yields 3.1 and e yields 3.01

  5. 4. Find the average rate of change of f(x) = 5x +1 between the following • pairs of x-values. • x=3 and x=5 • x=3 and x=4 • x=3 and x=3.5 • x=3 and x=3.1 • x=3 and x=3.01 • What number do your answers seem to be approaching? The average rate of change is the change in the y values divided by the change in the x values. OR

  6. You may do this problem using the 4-step procedure. • 4. Find the average rate of change of f(x) = 5x +1 between the following • pairs of x-values. • x=3 and x=5 Note h = 2 • x=3 and x=4 Note h = 1 • x=3 and x=3.5 Note h = 0.5 • x=3 and x=3.1 Note h = 0.1 • x=3 and x=3.01 Note h = 0.01 • What number do your answers seem to be approaching? Step 1: f (x + h) = 5(x + h) + 1 = 5x + 5h + 1 Step 2: f (x) = 5x + 1 Step 3: f (x + h) – f (x) = 5h Note that by using this method you do not need to do the problem several time (parts a thru e) to get the value of 5.

  7. Find the instantaneous rate of change of f (x) = x 2 + x at x = 1. • Use the five step procedure. Step 1: f (x + h) = (x + h) 2 + (x + h) = x 2 + 2xh + h 2 + x + h Step 2: f (x) = x 2 + x Step 3: f (x + h) – f (x) = 2xh + h 2 + h f ’ (1) = 2 (1) + 1 = 3

  8. 6. Find the slope of the tangent of 2x 2 + x – 2 at x = 2 Graph this on your calculator and use “draw” “tangent” to get the answer. Remember the slope of the tangent is the number in front of the x in the tangent equation. In this case the slope is 9.

  9. Use the definition of derivative (5-step procedure) to find f ‘ (x) of • f (x) = 2x 2 – 3x + 5 Step 1: f (x + h) = 2(x + h) 2 - 3(x + h) + 5 =2 x 2 + 4xh + 2h 2 - 3x – 3h + 5 Step 2: f (x) = 2x 2 - 3x + 5 Step 3: f (x + h) – f (x) = 4xh + 2h 2 – 3h

  10. 8. Use the definition of derivative (5-step procedure) to find f ‘ (x) of • f (x) = 9x - 2 Step 1: f (x + h) = 9(x + h) – 2 = 9x + 9h - 2 Step 2: f (x) = 9x - 2 Step 3: f (x + h) – f (x) = 9h

  11. 9. Use the definition of derivative (5-step procedure) to find f ‘ (x) of • f (x) = 4 Step 1: f (x + h) = 4 Step 2: f (x) = 4 Step 3: f (x + h) – f (x) = 0

  12. 10. Use the definition of derivative (5-step procedure) to find f ‘ (x) of • f (x) = 2/x Step 1: f (x + h) = 2/(x + h) Step 2: f (x) = 2/x Step 3: f (x + h) – f (x) = 0

  13. 11. Use the definition of derivative (5-step procedure) to find f ‘ (x) of • f (x) = √x Step 1: f (x + h) = √(x + h) Step 2: f (x) = √x Step 3: f (x + h) – f (x) = √(x + h) - √x Hint Multiply the numerator or denominator of the difference quotient by √(x + h) + √x and then simplify.

  14. Find the equation to the tangent line to the curve f (x) = x 2 – 3x + 5 at x = 2, writing the equation in slope intercept form. • Use a graphing calculator to graph the curve together with the tangent line to • verify your answer. See problem 6.

  15. 13. In problem 8 you found the derivative of f (x) = 9x – 2. Explain why you answer makes sense. The derivative was 9. That makes sense because the derivative is the slope of the curve and the curve in problem 8 is a straight line with slope 9.

  16. Business: Temperature The temperature in an industrial pasteurization tank is f (x) = x 2 – 8x + 110 degrees centigrade after x minutes (for 0 ≤ x ≤ 12) • Find f’(x) by using the definition of the derivative. • Use your answer to part (a) to find the instantaneous rate of change of the temp. • after 2 minutes. Be sure to interpret the sign of your answer. • c. Use your answer to part (a) to find the instantaneous rate of change after 5 minutes. • Use the five-step procedure to get • f ’ (x) = 2x – 8. b. f ’ (2) = 2 (2) – 8 = - 4 After two minutes the temperature is decreasing at a rate of 4 degrees per minute. c. f ’ (5) = 2 (5) – 8 = 2 After five minutes the temperature is increasing at a rate of 2 degrees per minute.

  17. 15. Behavioral Science: Learning theory In a psychology experiment, a person • could memorize x words in 2x 2 -x seconds (for 0 ≤ x ≤ 10 ) • Find f’(x) by using the definition of the derivative. • Find F’(5) and interpret it as an instantaneous rate of change in the proper • units. • Use the five-step procedure to get • f ’ (x) = 4x – 1. b. f ’ (5) = 4 (5) – 1 = 19 it will take 19 seconds to memorize the next, sixth, word..

  18. 16. Social Science: Immigration The percentage if people in the United States • who are immigrants (that is, born elsewhere) for different decades is approximated by the function f (x) = ½ x 2 – 3.7 x + 12, where x stands for the number of decades since 1930 (so that, for example x = 6 would stand for 1990). • Find f ’(x) using the definition of the derivative. • Evaluate the derivative at x=1 and interpret the result • Find the rate of change of the immigrant percentage in the year 2000. • Use the five-step procedure to get • f ’ (x) = x – 3.7. b. f ’ (1) = 1 – 3.7 = - 2.7 During the 1940’s the percentage of immigrants was decreasing by 2.7 %. c. f ’ (7) = 7 – 3.7 = 3.3 During the 2000’s the percentage of immigrants was increasing by 3.3 %.

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