# How do you use the first and second derivatives to sketch #f(x) = sqrt(4 - x^2)#?

##### 1 Answer

The domain of the function is the interval

We can note that the function is even, that is

We can also note that

Calculate now:

which means that the function is differentiable only in the interior of the interval.

As:

the tangent to the graph at the limits of the interval of definition is vertical.

The only critical point where:

is

We can easily see that

The value of the local maximum is

The second derivative is always negative, so the function is concave down in its entire domain.

graph{sqrt(4-x^2) [-4.933, 4.932, -2.466, 2.467]}

If you let:

then:

so the graph is the part of the circle of center in the origin and radius