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15 wrz 2024 · Use this Interactive Element to help you understand the difference between secant lines, the tangent line, rates of change, and the instantaneous rate of change of a function at a point. Interact: Move the point \(P\) close to the anchor point.
26 wrz 2024 · The tangent line to \(f(x)\) at a is the line passing through the point \((a,f(a))\) having slope \(m_{tan}=lim_{x→a}\frac{f(x)−f(a)}{x−a}\) provided this limit exists. Equivalently, we may define the tangent line to \(f(x)\) at a to be the line passing through the point \((a,f(a))\) having slope \(m_{tan}=lim_{h→0}\frac{f(a+h)−f(a)}{h}\)
A tangent line just touches a curve at a point, matching the curve's slope there. (From the Latin tangens "touching", like in the word "tangible".) A secant line intersects two or more points on a curve. (From the Latin secare "cut or sever") They are lines, so extend in both directions infinitely.
How tangent lines are a limit of secant lines, and where the derivative and rate of change fit into all this.
12 sty 2000 · slope of a secant line between the points (a,f(a)) and (a+h,f(a+h)). This slope is given by f(a+h)−f(a) h. This approximation gets better as h → 0, so we will shrink h to zero to obtain the slope of the tangent line. Exercise 1. If f(x) = x3 − 3x, what is the slope of the tangent line at x = 4? What is the equation of the tangent line at ...
tangent and secant lines is greatest where the graph of f(x) is curved. If the graph of y = f(x) is sharply curved, the value of x must be very close to 0 for the secant line to be close to the tangent line.
We defined the tangent line as a limit of secant lines. We also know that as Δx approaches 0 the secant’s slope Δf approaches the slope of the tangent line.
