# 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.