1.1 Introducing Calculus: Connecting Graphs and Rates of Change

    Cards (51)

    • What is the mathematical discipline that studies continuous change and rates of change?
      Calculus
    • Differential calculus focuses on accumulation and areas under curves.
      False
    • The key concept in integral calculus is the integral
    • The key concept in differential calculus is the derivative
    • In the formal definition of a limit, the symbols ε and δ represent small positive numbers
    • Calculus studies continuous change and rates of change
    • What is the primary tool used in integral calculus?
      Integration techniques
    • What is the defining characteristic of a continuous function?
      No jumps or breaks
    • What are asymptotes in a graph?
      Lines graph approaches
    • Local extrema are points where the function reaches a local maximum or minimum
    • Local extrema occur where the rate of change switches direction
    • Calculus deals with rates of change and continuous functions.

      True
    • What are local extrema of a function's graph?
      Peaks and valleys
    • What does an increasing function indicate about its rate of change?
      Positive slope
    • Match the graph feature with its relationship to the rate of change:
      Constant Portion ↔️ Zero rate of change
      Increasing Portion ↔️ Positive rate of change
      Decreasing Portion ↔️ Negative rate of change
    • The formula for average rate of change is (f(b) - f(a)) / (b - a)
    • The acceleration of a car speeding up from rest is a variable rate of change.

      True
    • Differential calculus deals with rates of change and slopes of curves
    • Match the branch of calculus with its key concept:
      Differential Calculus ↔️ Derivative
      Integral Calculus ↔️ Integral
    • Integration techniques are tools used in differential calculus.
      False
    • Limits are used to analyze the behavior of functions as they approach a specific point.

      True
    • The limit of f(x) = x + 2 as x approaches 3 is 5.
      True
    • What is the focus of differential calculus in terms of function behavior?
      Slopes of curves
    • The key concepts in the formal definition of a limit include ε, δ, |x - c|, and |f(x) - L|
    • A function is increasing if it rises from left to right
    • What is the definition of a continuous function in terms of its graph?
      No jumps or breaks
    • What is the graphical representation of asymptotes?
      Dashed lines
    • Match the graph feature with its relationship to the rate of change:
      Increasing Portion ↔️ Positive
      Decreasing Portion ↔️ Negative
      Constant Portion ↔️ Zero
    • What can differential calculus be used to find in physics?
      Velocity
    • Asymptotes are lines that the graph approaches but never touches
    • When a function is increasing, its rate of change is positive
    • What does the average rate of change measure over an interval?
      Function change
    • What is an example of a constant rate in real life?
      Steady car speed
    • What is an example of a constant rate of change in finance?
      Fixed interest rate
    • What is the primary focus of integral calculus?
      Areas under curves
    • Which real-life application uses differential calculus?
      Optimization
    • What is the focus of differential calculus?
      Rates of change
    • What does the formal definition of a limit describe?
      Value a function approaches
    • Match the definition of a limit with its description:
      Formal Definition ↔️ For every ε > 0, there exists δ > 0 such that if 0 < |x - c| < δ, then |f(x) - L| < ε
      Informal Definition ↔️ Describes the value a function approaches as x gets arbitrarily close to a point
    • Differential calculus is used to find integrals.
      False
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