Instantaneous Rate Of Change

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Instantaneous Rate Of Change - Quiz

Can you interpret and/or estimate instantaneous rate of change in multiple representations?


Questions and Answers
  • 1. 

    The graph shows y = f(x) in purple. Which line has a slope equal to the rate expression shown? 

    • A.

      Line a

    • B.

      Line b

    • C.

      Line c

    • D.

      Line d

    Correct Answer
    B. Line b
    Explanation
    Line b has a slope equal to the rate expression shown in the graph. The slope of a line represents the rate of change between two points on the line. In this case, the slope of Line b matches the rate expression shown in the graph, indicating that the function f(x) has a constant rate of change along Line b. Therefore, Line b is the correct answer.

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  • 2. 

    Estimate the instantaneous rate using the graph of y = f(x).

    • A.

      0.02

    • B.

      –0.02

    • C.

      7.25

    • D.

      –7.25

    Correct Answer
    D. –7.25
    Explanation
    The correct answer is -7.25. This is because the graph of y = f(x) has a steep negative slope at the given point, indicating a high rate of change in the negative direction. Therefore, the instantaneous rate at this point is -7.25.

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  • 3. 

    Acceleration is the rate of change of velocity. If the velocity of a car driving on a straight road is given by v(t), which of the following expressions represent the car's instantaneous expression at t = 5 seconds? (Check all that apply.)

    • A.

      Expression 1:

    • B.

      Expression 2:

    • C.

      Expression 3:

    • D.

      Expression 4:

    • E.

      Expression 5:

    • F.

      Expression 6:

    Correct Answer(s)
    C. Expression 3:
    D. Expression 4:
    F. Expression 6:
    Explanation
    The expressions that represent the car's instantaneous acceleration at t = 5 seconds are Expression 3, Expression 4, and Expression 6.

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  • 4. 

    Estimate the instantaneous rate of change of H(x) = sin(ln x2) at x = –3

    • A.

      0.1

    • B.

      0.4

    • C.

      0.8

    • D.

      3.5

    • E.

      Undefined

    Correct Answer
    B. 0.4
    Explanation
    The instantaneous rate of change of a function at a specific point can be found by taking the derivative of the function and evaluating it at that point. In this case, the function H(x) = sin(ln x^2) needs to be differentiated. The derivative of sin(x) is cos(x), and the derivative of ln(x^2) is 2/x. Therefore, the derivative of H(x) is cos(ln x^2) * 2/x. Evaluating this derivative at x = -3 gives cos(ln 9) * 2/-3. Since cos(ln 9) is a constant value, the instantaneous rate of change at x = -3 is 2/(-3) = -2/3. None of the given answer choices match this value, so the correct answer is undefined.

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  • 5. 

    Which statement(s) correctly contrast average rate of change and instantaneous rate of change?

    • A.

      Average rate of change requires specifying two points on a graph while instantaneous rate of change requires only one. 

    • B.

      Average rate of change can be interpreted as a slope while instantaneous rate of change can not.

    • C.

      Average rate of change can be for any function while instantaneous rate of change is only for nonlinear functions.

    • D.

      Average rate of change can be calculated as a difference quotient while instantaneous rate of change is a limit of a difference quotient.

    Correct Answer(s)
    A. Average rate of change requires specifying two points on a graph while instantaneous rate of change requires only one. 
    D. Average rate of change can be calculated as a difference quotient while instantaneous rate of change is a limit of a difference quotient.
    Explanation
    The correct answer states that average rate of change requires specifying two points on a graph while instantaneous rate of change requires only one. This is because average rate of change is calculated by finding the slope between two points on a graph, while instantaneous rate of change is the rate of change at a specific point on a graph. Additionally, the correct answer states that average rate of change can be calculated as a difference quotient, which is the change in the dependent variable divided by the change in the independent variable, while instantaneous rate of change is the limit of a difference quotient as the change in the independent variable approaches zero.

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