Flashcards on Derivative of a function
Derivative of a Function: A Comprehensive Analysis for Students
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Derivative of a Function
9 cards
Card 1
Question: What does the derivative of a function quantify?
Answer: The instantaneous rate of change of a function with respect to its independent variable.
Card 2
Question: What is the geometric interpretation of the derivative?
Answer: It represents the slope of the tangent line to the graph of a function at a specific point.
Card 3
Question: How is the slope of a secant line connecting two points (x₀, f(x₀)) and (x₀+h, f(x₀+h)) on a function f(x) calculated?
Answer: m_sec = (f(x₀+h) − f(x₀)) / h
Card 4
Question: What happens to the secant line as h approaches zero?
Answer: The secant line rotates and approaches the tangent line at the point (x₀, f(x₀)).
Card 5
Question: How is the derivative of a function f(x) at a point x₀ formally defined?
Answer: f′(x₀) = lim (h→0) [(f(x₀+h) − f(x₀)) / h]
Card 6
Question: What is the formula lim (h→0) [(f(x₀+h) − f(x₀)) / h] known as?
Answer: The definition of the derivative.
Card 7
Question: When is a function f said to be differentiable at x₀?
Answer: If the limit in the formal definition of the derivative exists.
Card 8
Question: What does the derivative f′(x) represent as a function itself?
Answer: It gives the slope of the tangent line to f(x) at any point x where the derivative exists.
Card 9
Question: What are the units of the derivative?
Answer: The units of the dependent variable divided by the units of the independent variable.