Derivatives - Slopes of Tangent lines (vs Secant lines)
- Use c slider to move point A to desired location on given f(x) curve. Alternatively, type in a value for c or drag point A in the graphics view.
- Visually, what does the slope of the tangent line (the "derivative" of f at point A) appear to be?
- To determine the slope analytically, we'd need to know the coordinates of two points on the tangent line to calculate "rise" Δy over "run"Δx.
- Because we only know the coordinates of one point on the tangent line (point A), we consider a secant line instead.
- Toggle on the "secant line" checkbox to view the line that passes through two points on f(x), points A and B.
- Drag the Δx slider (or type in a value) and observe how the secant line becomes closer and closer to matching the tangent line as Δx gets smaller.
- What happens when Δx = 0? Why can we not calculate a slope then?
- Despite this problem, do you have a sense of what the slope should be when Δx=0?
- How might we use the concept of "limits" to determine this "derivative" analytically?