1.1 Introducing Calculus: Connecting Graphs and Rates of Change

• 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