5.1 Using the Mean Value Theorem

Cards (75)

  • What does the Mean Value Theorem (MVT) state for a function \( f(x) \) that is continuous on \([a, b]\) and differentiable on \((a, b)\)?
    There exists a point \( c \) in \((a, b)\) where \( f'(c) = \frac{f(b) - f(a)}{b - a} \)
  • One condition for the Mean Value Theorem is that the function must be continuous on the open interval \((a, b)\)
    False
  • What does it mean for a function to be continuous on a closed interval \([a, b]\)?
    There are no breaks or jumps
  • Differentiability of a function \( f(x) \) on the open interval \((a, b)\) means that the derivative f'(x) exists for every \( x \) in \((a, b)\)
  • The average rate of change of a function \( f(x) \) between two points \( (a, f(a)) \) and \( (b, f(b)) \) is the slope of the line connecting these points
  • To calculate the average rate of change, you first identify the values of \( a \) and \( b \) and then calculate \( f(a) \) and \( f(b)\)
    True
  • The average rate of change is the slope between two points
  • What is the formula for the average rate of change?
    f(b)−f(a)b−a\frac{f(b) - f(a)}{b - a}
  • The Mean Value Theorem requires a function to be continuous on a closed interval [a,b][a, b].

    True
  • The formula for the Mean Value Theorem is \( f'(c) = \frac{f(b) - f(a)}{b - a} \), where \( c \) lies in the open interval (a, b).
  • Steps to find the average rate of change:
    1️⃣ Identify the values of \( a \) and \( b \).
    2️⃣ Calculate \( f(a) \) and \( f(b) \).
    3️⃣ Apply the formula \(\frac{f(b) - f(a)}{b - a}\).
    4️⃣ Simplify the expression.
  • The average rate of change measures how much the function changes on average for each unit change in \( x \).
    True
  • What is the value of \( f(1) \) for the function \( f(x) = x^2 + 2x \)?
    3
  • The average rate of change measures how much the function changes on average for each unit change in x
  • Steps to find the average rate of change
    1️⃣ Identify the values of \( a \) and \( b \).
    2️⃣ Calculate \( f(a) \) and \( f(b) \).
    3️⃣ Apply the formula \(\frac{f(b) - f(a)}{b - a}\).
    4️⃣ Simplify the expression.
  • The Mean Value Theorem requires that a function \( f(x) \) is continuous on [a, b]
  • What is the value of \( f'(x) \) for \( f(x) = x^2 \)?
    2x
  • What condition must be checked after solving for \( c \) in the Mean Value Theorem?
    Ensure \( c \) lies in \((a, b)\)
  • What is the key reason why the Mean Value Theorem cannot be applied to \( f(x) = \frac{1}{x} \) on \([-1, 1]\)?
    Not continuous at \( x = 0 \)
  • The Mean Value Theorem graphically states that there exists a point \( c \) where the tangent line is parallel to the secant line.
  • The Mean Value Theorem has a straightforward graphical interpretation.
    True
  • Steps to apply the Mean Value Theorem graphically:
    1️⃣ Ensure continuity on \([a, b]\)
    2️⃣ Ensure differentiability on \((a, b)\)
    3️⃣ Find a point \( c \) where \( f'(c) = \frac{f(b) - f(a)}{b - a} \)
    4️⃣ Check that \( c \) lies within \((a, b)\)
  • The Mean Value Theorem requires a function to be continuous on \([a, b]\) and differentiable on \((a, b)\).

    True
  • What does it mean for a function to be differentiable on an interval?
    The derivative exists at every point
  • The average rate of change of a function measures how much the function changes per unit change in x.
  • What does the slope of the secant line between two points measure?
    Change in function per unit change in \( x \)
  • The Mean Value Theorem is expressed mathematically as \( f'(c) = \frac{f(b) - f(a)}{b - a} \)
  • What are the two key conditions for applying the Mean Value Theorem?
    Continuity and differentiability
  • A function must be differentiable on the closed interval \([a, b]\) for the Mean Value Theorem to apply
    False
  • What are the two conditions that a function \( f(x) \) must satisfy for the Mean Value Theorem to apply on the interval \([a, b]\)?
    Continuity on \([a, b]\) and differentiability on \((a, b)\)
  • What is the formula for calculating the average rate of change of a function \( f(x) \) between \( x = a \) and \( x = b \)?
    f(b)−f(a)b−a\frac{f(b) - f(a)}{b - a}
  • The average rate of change of \( f(x) = x^2 + 2x \) between \( x = 1 \) and \( x = 3 \) is 6
  • The average rate of change measures how much a function changes for each unit change in \( x \).

    True
  • The Mean Value Theorem states that if a function is continuous on a closed interval and differentiable on an open interval, there exists a point where the instantaneous rate of change equals the average rate of change.
  • Match the condition with its requirement for the Mean Value Theorem:
    Continuity ↔️ \( f(x) \) must be continuous on \([a, b]\)
    Differentiability ↔️ \( f(x) \) must be differentiable on \((a, b)\)
  • What is the instantaneous rate of change at point \( c \) equal to according to the Mean Value Theorem?
    f′(c)f'(c)
  • The formula for the average rate of change is f(b)−f(a)b−a\frac{f(b) - f(a)}{b - a}.
  • Find the average rate of change of \( f(x) = x^2 + 2x \) between \( x = 1 \) and \( x = 3 \).
    6
  • What is the average rate of change of \( f(x) = x^2 + 2x \) between \( x = 1 \) and \( x = 3 \)?
    6
  • The formula for average rate of change is f(b)−f(a)b−a\frac{f(b) - f(a)}{b - a}.

    True