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Understanding the Tangent Line: Limits and Slope Concepts

This engaging animation illustrates the concept of the tangent line to a curve, focusing on the fundamental slope formula. As we explore the behavior of secant lines that approach a specific point, we demonstrate how the slope of these lines converges to that of the tangent line as the distance (h) approaches zero. By analyzing the limit expression for the tangent line's slope, we visualize how calculus allows us to find instantaneous rates of change. This animation serves as a practical tool for grasping these foundational mathematical ideas.

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Understanding the Tangent Line: Limits and Slope Concepts

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  1. Tangent Line using a limit • The tangent line to a curve is based on the fundamental formula of the slope of a line. • Additionally we consider the slope of various secant lines as they get closer to the point at which we want our tangent line. • This short animation emphasizes that idea. Click to continue

  2. f(x+h) f(x) h x x+h If we let h be some distance from x, then x2 becomes x+h and f(x2) becomes f(x+h) Click to continue

  3. f(x2) – f(x1) Slope of a line where m = x2 – x1 f(x + h) f(x2) f(x2) becomes f(x+h) f(x+h) – f(x) x2 becomes x + h f(x1) f(x) x2 – x1 x x1 x2 x + h If we leth = x2 – x1 … h then slope m becomes…

  4. Slope of a tangent line = lim f(x+h) – f(x) h→0 h We see that as h approaches zero… Tangent line at x The slope of the secantline approaches the slope of the tangentline h x Pick a smaller h Click to continue

  5. The concept is to let h approach Zero and by doing so, the slope of the Secant line will approach the slope of the Tangent line. Slope of a tangent line = lim f(x+h) – f(x) h→0 h The End

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