Let *f*(*x*) = sin(*x*). By taking derivatives, find a function *g*(*x*) of the form *g*(*x*) = *a* + *bx* + *cx*^{2} that has the same value, slope, and second derivative} as *f* when *x* = 0.

Graph *f* and *g* on the same axes.

Answer

We want *g*(0) and *f*(0) to be the same. We can see that

*g*(0) = *a* + *b*(0) + *c*(0)^{2} = *a*

and we know that

*f*(0) = sin(0) = 0.

In order for *g*(0) and *f*(0) to be the same, we must have *a* = 0.

Next, we want *g*'(0) and *f*'(0) to be the same. We can compute

*g*'(*x*) = *b* + 2*cx*

and we know *f*'(*x*) = cos *x*. To have these derivatives be the same at 0, we need

*g*'(0) = *b* + 2*c*(0) = *b *

and

*f*'(0) = cos(0) = 1

to be the same, so we must have *b* = 1.

Finally, we want the second derivatives to be the same. We calculate the second derivatives:

*g*^{(2)}(*x*) = 2*c*

and

*f*^{(2)}(*x*) = -sin(*x*)

so we need

2*c* = *g*^{(2)}(0) = *f*^{(2)}(0) = -sin(0) = 0

which means *c* = 0. Now that we know *a*, *b*, and *c*, we can write

*g*(*x*) = *a* + *bx* + *cx*^{2} = 0 + *x* + 0*x*^{2} = *x*.

If we graph *g*(*x*) and *f*(*x*) on the same graph, *g* does look like a good approximation of *f* close to zero:

In this case, it just happened that the second derivative of *f* was zero, so we ended up with *g* being a line anyway.