Flashcards on Tangent to a graph and the geometric meaning of the derivative
Tangent to a Graph & Geometric Meaning of the Derivative
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Tangent Lines and Derivatives
12 cards
Card 1
Question: What is the geometric representation of a tangent line?
Answer: A tangent line represents the best linear approximation of a curve at a given point.
Card 2
Question: How does a tangent line differ from a secant line?
Answer: A tangent line touches the curve at only one point in its immediate vicinity, sharing the same instantaneous direction as the curve at that specific p
Card 3
Question: What is the formula for the slope of a secant line passing through points P(x₀, f(x₀)) and Q(x₀ + Δx, f(x₀ + Δx))?
Answer: m_sec = (f(x₀ + Δx) − f(x₀)) / Δx
Card 4
Question: How is the slope of the tangent line defined mathematically?
Answer: The slope of the tangent line (m_tan) is defined as the limit of the slope of the secant line as Δx approaches zero: m_tan = lim (Δx→0) [(f(x₀ + Δx) −
Card 5
Question: What does the derivative of a function at a specific point represent geometrically?
Answer: The derivative of a function at a specific point gives the exact slope of the tangent line to the function's graph at that point.
Card 6
Question: What is the point-slope form of the equation of a tangent line?
Answer: y − f(x₀) = f′(x₀)·(x − x₀)
Card 7
Question: In the equation y − f(x₀) = f′(x₀)·(x − x₀), what does f′(x₀) represent?
Answer: f′(x₀) is the slope of the tangent line at x₀, which is the value of the derivative of f(x) evaluated at x₀.
Card 8
Question: If f(x) = x² and x₀ = 2, what is the y-coordinate of the point of tangency?
Answer: f(2) = 2² = 4. The y-coordinate is 4.
Card 9
Question: What is the slope of the tangent line to f(x) = x² at x = 2?
Answer: The derivative f′(x) = 2x, so f′(2) = 2·(2) = 4. The slope is 4.
Card 10
Question: What is the equation of the tangent line to f(x) = x² at the point where x = 2?
Answer: y = 4x − 4