Tangent Line using a limit

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