Flashcards on The derivative as a limit
The Derivative as a Limit: Understanding Instantaneous Change
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Calculus Derivatives
10 cards
Card 1
Question: What does the derivative measure at its core?
Answer: The instantaneous rate of change of a function at a specific point.
Card 2
Question: How can the derivative be understood geometrically?
Answer: As the slope of a tangent line to a curve.
Card 3
Question: How can the derivative be understood physically?
Answer: As instantaneous velocity or acceleration.
Card 4
Question: What is the formula for the average rate of change (difference quotient) between two points (x₀, f(x₀)) and (x₀ + h, f(x₀ + h))?
Answer: (f(x₀ + h) − f(x₀)) / h
Card 5
Question: What does 'h' represent in the difference quotient?
Answer: The horizontal distance between the two points.
Card 6
Question: What is the formal definition of the derivative of a function f(x) at a point x₀?
Answer: f'(x₀) = lim (h→0) [(f(x₀ + h) − f(x₀)) / h]
Card 7
Question: What does f'(x₀) represent if the limit exists?
Answer: The instantaneous rate of change of f(x) with respect to x at x₀, or the slope of the tangent line to the graph of f(x) at (x₀, f(x₀)).
Card 8
Question: What is the formula for the right-hand derivative?
Answer: lim (h→0⁺) [(f(x₀ + h) − f(x₀)) / h]
Card 9
Question: What is the formula for the left-hand derivative?
Answer: lim (h→0⁻) [(f(x₀ + h) − f(x₀)) / h]
Card 10
Question: What condition must be met for a function to be differentiable at x₀ regarding one-sided derivatives?
Answer: Both the right-hand derivative and the left-hand derivative must exist and be equal at x₀.