Learn – Algebra/Precalculus – AROC and IROC

AROC

Definition

Exercises

Problem set

Find the AROC for the following functions in the interval [-10,10].

  1. f(x) = x
  2. g(x) = 2x-3
  3. h(x) = -5x + 4
  4. l(x) = -8x-3
Problem set

Find the AROC for the following functions in the interval [2,6].

  1. f(x) = x^2
  2. g(x) = 2x^2 + 3
  3. h(x) = x^3 -3
  4. t(x) = 2^x
  5. l(x) = 2^{x-2}-8
  6. l(x) = 4-3x^2
  7. f(x) = 8-x^3

Real-world interpretation

Exercises

Problem set
  1. The mean house price in a town was \$300000 in the year 2010, while the mean house price in the same town was \$500000 in the year 2020. Find the average rate of change of the mean house price in the town from 2010 through 2020.
  2. The cost of 15 pounds of rice is \$20, while the cost of 20 pounds of rick is \$24. Find the average rate of change of the cost of rice from 15 pounds of rice to 20 pounds of rice.
  3. A car consumed 7 gallons of gasoline to drive 140 miles, and the same car consumed 11 gallons of gasoline to drive 210 miles. Find the average rate of change of amount of gasoline consumed between driving 140 miles and driving 210 miles.
Problem set
  1. Chitra’s balance (in thousands of dollars) in her bank account is modeled by a function f(t), where t is time elapsed in years. Chitra observed that \mbox{AROC}^{f(t)}_{[2,9]} = 10. Give a layman’s interpretation of the AROC value for this real-world scenario.
  2. The population of a town is given by p(t), where t is the number of years elapsed from a certain date. If \mbox{AROC}^{p(t)}_{[10,20]} = 8000, explain in layman’s language how the population of the town changed over time.
  3. A ball is thrown up into the air from the top of a building, and the height of the ball (in meters) from the ground is modeled by the function h(t), where t is time elapsed in seconds. It was observed that \mbox{AROC}^{h(t)}_{[5,50]} = -5. Give a layman’s interpretation of the AROC value for this real-world scenario.
  4. A truck drove from New York to Los Angeles. The distance driven by the truck as a function of time elapsed (in hours) is given by the function d(t). In the end, the truck took 48 hours to complete the drive and it was observed that \mbox{AROC}^{d(t)}_{[0,48]} = 53. Give a layman’s interpretation of the \mbox{AROC}^{d(t)}_{[0,48]}.
  5. On a particular trip of length 400 miles, the gas remaining (in gallons) in the tank of a truck is modeled by the function g(m), where m gives the number of miles driven by the truck. If the truck consumed 1 gallon of gas for every 10 miles on average on the trip, give the value of \mbox{AROC}^{g(m)}_{[0,400]}.

Geometric interpretation

Exercises

Problem set

In each of the following problems, the graph of a function f(x) is shown. Using the graph, estimate the required AROC value.

  1. \mbox{AROC}^{f(x)}_{[0,5]}


  2. \mbox{AROC}^{f(x)}_{[1,6]}


  3. \mbox{AROC}^{f(x)}_{[-3,-2.5]}


  4. \mbox{AROC}^{f(x)}_{[-1,0]}


  5. \mbox{AROC}^{f(x)}_{[-4,4]}


  6. \mbox{AROC}^{f(x)}_{[-4,-2]}


Problem set
  1. For the following function f(x), identify which of the two is bigger: \mbox{AROC}^{f(x)}_{[-3,-2]}, \mbox{AROC}^{f(x)}_{[0,1]}.


  2. For the following function f(x), arrange the following in ascending order: \mbox{AROC}^{f(x)}_{[-3,-1.5]}, \mbox{AROC}^{f(x)}_{[-0.9,1]}, \mbox{AROC}^{f(x)}_{[0,3]}, \mbox{AROC}^{f(x)}_{[1,4]}


IROC

Definition

Exercises

Problem set

Find the IROC for the following functions using h = 0.001 at two different points: x=0, x=4. Use calculators on this problem set.

  1. y = 7x
  2. y = 3-4x
  3. y = x^2
  4. y = 3x^2
  5. y = -4-x^2
  6. y = 2x^3 - 3
  7. y = -5-x^3
  8. y = 2-x-x^2

Geometric interpretation

Exercises

Problem set

In each of the following problems, the graph of a function f(x) is shown. Using the graph, estimate the required IROC value.

  1. \mbox{IROC}^{f(x)}_{x = 1}


  2. \mbox{IROC}^{f(x)}_{x = 4}


  3. \mbox{IROC}^{f(x)}_{x=2.5}


  4. \mbox{IROC}^{f(x)}_{x=-0.5}


  5. \mbox{IROC}^{f(x)}_{x = -2}


Problem set
  1. For the following function f(x), identify which of the two is bigger: \mbox{IROC}^{f(x)}_{x=-2.5}, \mbox{IROC}^{f(x)}_{x=0.5}.


  2. For the following function f(x), arrange the following in ascending order: \mbox{IROC}^{f(x)}_{x=-4}, \mbox{IROC}^{f(x)}_{x=-1.1}, \mbox{IROC}^{f(x)}_{x=0}, \mbox{IROC}^{f(x)}_{x=3}, \mbox{IROC}_{x=4.2}


  3. For the following function f(x), arrange the following in ascending order: \mbox{IROC}^{f(x)}_{x=-1.5}, \mbox{IROC}^{f(x)}_{x=0}, \mbox{IROC}^{f(x)}_{x=3}, \mbox{AROC}^{f(x)}_{[-2,3]}.


Problem set
    In each of the following problems, certain AROC and IROC values of a function f(x) are given. Using that information, graph f(x) such that it satisfies the AROC and the IROC values.

    • \mbox{AROC}^{f(x)}_{[0,6]} = 0
    • \mbox{IROC}^{f(x)}_{x=0}=2
    • \mbox{IROC}^{f(x)}_{x=1} = -2
    • \mbox{IROC}^{f(x)}_{x=2}=2
    • \mbox{IROC}^{f(x)}_{x=3}=-2
    • \mbox{IROC}^{f(x)}_{x=4}=2
    • \mbox{IROC}^{f(x)}_{x=5}=-2
    • \mbox{IROC}^{f(x)}_{x=6}=2


    • \mbox{AROC}^{f(x)}_{[-3,3]} = 1
    • \mbox{IROC}^{f(x)}_{x=-3}=-3
    • \mbox{IROC}^{f(x)}_{x=-2} = -2
    • \mbox{IROC}^{f(x)}_{x=-1}=-1
    • \mbox{IROC}^{f(x)}_{x=0}=0
    • \mbox{IROC}^{f(x)}_{x=1}=1
    • \mbox{IROC}^{f(x)}_{x=2}=2
    • \mbox{IROC}^{f(x)}_{x=3}=4


    • \mbox{AROC}^{f(x)}_{[-3,3]} = 1
    • \mbox{IROC}^{f(x)}_{x=-3}=4
    • \mbox{IROC}^{f(x)}_{x=-2} = 2
    • \mbox{IROC}^{f(x)}_{x=-1}=1
    • \mbox{IROC}^{f(x)}_{x=0}=0
    • \mbox{IROC}^{f(x)}_{x=1}=-1
    • \mbox{IROC}^{f(x)}_{x=2}=-2
    • \mbox{IROC}^{f(x)}_{x=3}=-3

Natural base of an exponential function

Exercises

Problem set

Using e = 2.71828, find the IROC for the following functions using h = 0.001 at the three different points: x=0, x=4, x=10. Use calculators on this problem set.

  1. y = e^x
  2. y = 2e^x
  3. y = 0.5e^x
  4. y = -3e^x
  5. y = e^{2x}
  6. y = e^{7x}
  7. y = e^{-0.5x}
  8. y = 4e^{0.5x}
Problem set

Problems requiring where IROC is proportional to function value (Heat transfer, Newton’s law of cooling?)