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Section 2.5

Section 2.5. 1. Find the first four derivatives of f (x) = x 4 - 2x 3 – 3x 2 - 5x - 7. 2. Find the first four derivatives of . . 3. Find the first and second derivatives of and evaluate the second derivative at x = 3. . . 4. Find the first and second derivatives of

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Section 2.5

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  1. Section 2.5 1. Find the first four derivatives of f (x) = x 4 - 2x 3 – 3x2 - 5x - 7

  2. 2. Find the first four derivatives of

  3. . 3. Find the first and second derivatives of and evaluate the second derivative at x = 3.

  4. . 4. Find the first and second derivatives of and evaluate the second derivative at x = 3.

  5. 5. Find the first and second derivatives of

  6. 6. Find the first and second derivatives of

  7. 7. Find the first and second derivatives of f (x) = r 2.

  8. 8. Find the first and second derivatives of f (x) = (x 2 – x + 1)(x 3 – 1)

  9. After t hours a freight train is s (t) = 18t 2 – 2t 3 miles due north of its starting point (for 0 ≤ t ≤ 9). • Find its velocity at time t = 3 hours. • Find its velocity at time t = 7 hours. • Find its acceleration at time t = 1 hour. a. Velocity = s’ (t) = 36 t – 6 t 2 . And s‘ (3) = 36 (3) – 6 (9) = 54 miles per hour. • Velocity = s’ (t) = 36 t – 6 t 2 . And s‘ (7) = 36 (7) – 6 (49) = - 42 miles per hour. • 2.5 #33 c. acceleration = s” (t) = 36 – 12 t . And s” (1) = 36 – 12 = 24 miles per hour per hour.

  10. If a steel ball is dropped from the top of the Taipei 101, the tallest building in the world, its height above the ground t seconds after it is dropped will be • s (t) = 1667 – 16t 2 feet. • a. How long will it take to reach the ground? • b. Use your answer in part a to find the velocity at which it will strike the ground. • c. Find the acceleration at any time t. a. To find when the steel ball will reach the ground, we need to determine what value of t produces s(t)=0. Thus, set s(t)=0 and solve the equation by find the x-intercepts (zeros) using our calculator. The steel ball will reach the ground after 10.2 seconds. b. The velocity is the derivative of the distance function in part a. Use your calculator to find the derivative at x = 10.2 from part a. The velocity will be 326.4 feet per second. c. The acceleration is the second derivative of the given distance formula or s’ (t) = -32t and s” (t) = - 32 feet per second per second.

  11. ECONOMICS: National Debt The national debt of a South American country t years from now is predicted to be D (t) = 65 + 9 t 4/3 billion dollars. • Find D’ (8) and D” (8) and interpret your answers. D’ (t) = 12 t 1/3 and D’ (8) = 24 billion dollars per year. This is the amount the national debt is expected to increase between the 8th and 9th year. D” (t) = 4 t - 2/3 and D” (8) = 1 billion dollars per year. The rate of growth in the national debt is expected to increase by 1 billion dollars per year after the 8th year.

  12. GENERAL: Wind-chill Index the wind-chill index for a temperature of 32 degrees Fahrenheit and a wind speed of x miles per hour is • W (x) = 55.628 – 22.07 x 0.16. • Graph the wind-chill and find the wind-chill index for wind speeds of x = 15 and x = 30 miles per hour.. • Notice from your graph that the wind-chill index has a first derivative that is negative and a second derivative that is positive. What does this mean about how successive 1-mph increases in wind speed will affect the wind-chill index? • W (15) = 21.6 • W (30) = 17.6 • b. Each 1-mph increase in wind speed lowers the wind-chill index. As wind speed increases, the rate with which the wind-chill index decreases slows.

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