Math Insight

Applet: Quadratic approximation to a function

The function $f(x)$ (thick green curve) is approximated by a quadratic function $Q(x)$ (thin blue curve) and linear tangent line $L(x)$ (thin red line) at the point $x=a$. The quadratic function is equal to the linear function plus $c(x-a)^2$, where $c$ is specified by typing a value in its box. One can enter an expression involving derivatives like $f''(a)$ in the box for $c$. The idea is to find a value for $c$ that gives the best quadratic approximation as you zoom in around the point at $x=a$ (purple point). You can change $a$ by dragging the magenta point or typing a value in the box. You can also type in a new function for $f(x)$. Values of $f$, its derivatives, and the linear and quadratic approximations are shown. You can use the buttons at the top to zoom in and out as well as pan the view.

Applet file: quadratic_approximation.ggb

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