# Second Derivatives and Beyond

### Topics

## Introduction to Second Derivatives And Beyond - At A Glance:

There are three types of points to find on the function, and the great thing is that you already know how to find all of them.

**Intercepts:**To find the*y*-intercept of a function, plug*x*= 0 into the function and see what we find. To find the*x*-intercepts, also known as roots, we set the function equal to zero and solve. Sometimes we'll also find vertical asymptotes in this step (find where*f*is undefined).

**Critical points:**To find the critical points, set*f*'(*x*) = 0 (or undefined) and solve.

**Inflection points:**To find the inflection points, set*f*"(*x*) = 0 (or undefined), solve, and check each solution to see if it's a real inflection point.

In summary, we're finding where *f*, *f *', and *f *" are zero or undefined. These will mostly be dots, but there may be asymptotes or holes where *f* is undefined.

Here's the only thing you need to do that we didn't do earlier: after finding the *x*-value of a CP or IP, plug that *x*-value back into the original function *f* to find the corresponding *y*-value. In order to graph a point, we need to know both coordinates.

#### Exercise 1

Find and graph all intercepts, vertical asymptotes, critical points, and inflection points of the function. Label each point with its exact coordinates.

*f* (*x*) = *e*^{x}

#### Exercise 2

Find and graph all intercepts, vertical asymptotes, critical points, and inflection points of the function. Label each point with its exact coordinates.

*f* (*x*) = ( *x* - 2 )^{3}

#### Exercise 3

Find and graph all intercepts, vertical asymptotes, critical points, and inflection points of the function. Label each point with its exact coordinates.

*f* (*x*) = (1-*x*)*e*^{x}

#### Exercise 4

*f* (*x*) = *x*^{2}*e*^{x} - 4*e*^{x}