## Introduction to Curve Sketching

Again, Curve Sketching in Calculus makes us “appreciate the math” and helps visualize what is going on with function and their derivatives. It’s not my favorite part of calculus, but it’s a necessary evil :).

Before we get started, here are some basic graphs to look at. The graphs below show example original functions, and their derivatives (**slopes**). Notice how when we take the **derivative**, we go from a **quadratic** (**2**^{nd} degree) to a **line** (**1**^{st} degree), and from a **cubic** (**3**^{rd} degree) to a **quadratic** (**2**^{nd} degree). This makes sense, since the we are always going down a degree when we take a derivative. Also notice what happens where the derivative doesn’t exist (the last graph, an absolute value function).

## Relative Extrema and the First Derivative Test

### Extrema and Critical Numbers

We first talked about **Extrema of Functions** in the **Advanced Functions** section here. **Extrema** is just a fancy word for finding the **lowest (minimum) **or** highest (maximum) **$ \boldsymbol {y}$**-value** in a function or interval of a function; we will use Extrema to find **Critical Numbers**.

We can talk about **absolute extrema**, or **relative (local) extrema. **Think of the **absolute** extrema as the absolute lowest or highest point in the **whole domain** of the function, and the **relative (local) extrema** as the lowest or highest for a part of the graph. Technically, relative extrema must be the minimum or maximum of a point from **both sides of **$ \boldsymbol {x}$, so they can’t be endpoints; they are just “valleys” or “hills”. Note that not every function has a lowest (minimum) or highest (maximum) point in an interval or even the whole domain (like the function $ y=x$), so there may not be any extrema. The **endpoints** of a function may be the lowest or highest points (thus the absolute, not relative extrema); these are called the **endpoint extrema**.

When we find the minimum and maximum values in an interval, we can use this information to find where a function is **decreasing** and **increasing**, since at a minimum or maximum value, the function takes a turn from “down to up” or “up to down”.

Here is a graph that shows some examples of **absolute**/**relative extrema**; note also the **endpoint extrema** points:

**Critical numbers or critical points **exist where a function has a minimum or maximum, whether or not the function is differentiable at that point. And it turns out that if a function is differentiable at a certain point, and that point is a minimum or maximum, the **derivative at that point is 0**. Here is the formal definition of a **critical number**:

**critical number**of $ f$.

### Increasing and Decreasing Functions, and the First Derivative Test

We talked about **critical points** (**critical numbers**) of a function (minimums or maximums), where the **first** **derivative is 0** (or not defined). Now let’s talk about the derivative when the function is increasing (going upward from left to right), or decreasing (going downward from left to right).

When a **function** is **increasing**, the **derivative (slope) is positive**. When a **function** is **decreasing**, the **derivative (slope) is negative**. When a **function** is **constant**, or staying the same, the **derivative (slope) is** **0**.

Here are the guidelines for finding intervals for which a function is increasing or decreasing: For a function $ f$ that is continuous on interval $ [a,b]$ and differentiable on interval $ (a,b)$, to find the intervals for which $ f$ is increasing or decreasing:

- Find the
**critical points**(minimums or maximums) in $ (a,b)$, and use these numbers to find test intervals. - For each of these test intervals, find the
**sign of the derivative**at one test value. - If $ {f}’\left( x \right)>0$, then $ f$ is
**increasing**on $ [a,b]$, if $ {f}’\left( x \right)<0$, then $ f$ is**decreasing**on $ [a,b]$, and if $ {f}’\left( x \right)=0$, then $ f$ is**constant**on $ [a,b]$.

Based on these guidelines, here is the **First Derivative Test**, which allows us to **find relative minimums** and **maximums **(also known as** local minimums **and

**).**

**local maximums****First Derivative Test**

Assume that $ c$** **is a

**critical number**of a function that is continuous on an open interval, and $ f$ is differentiable on the interval, except possibly at $ c$.

- If $ {f}’\left( x \right)$ changes from
**negative to positive**at critical point $ x=c$, then $ f$ has a relative**minimum**at $ x=c$. - If $ {f}’\left( x \right)$ changes from
**positive to****negative**at critical point $ x=c$, then $ f$ has a relative**maximum**at $ x=c$. - If $ {f}’\left( x \right)$ is positive on both sides of $ x=c$, or negative on both sides of $ x=c$, then that point is neither a relative minimum or relative maximum.

Let’s think about why this makes sense. If we have a point where a function goes from falling to rising (negative to positive slope), that point would be a minimum. Similarly, if we have a point where a function goes from rising to falling (positive to negative slope), that point would be a maximum:

Here are some problems; notice that we are using **sign charts** to determine the intervals that the function is decreasing or increasing. Sometimes we need to use values that aren’t even in the domain of the function (like in **vertical asymptotes**) in the sign charts; theoretically, these aren’t critical numbers, since they don’t exist in the original function.

Here’s a First Derivative Problem with a **trigonometric function**:

## Concavity and the Second Derivative

I like to think of concavity as “cup up” or “cup down”. Think of **concave upwards** of a cup that can hold water at all points, and **concave downward** is a cup that empties water out at all point. It turns out that when a graph is **concave upward** (cup up), its slope (first derivative) is increasing, so its **second derivative is positive**. When a graph is **concave downward** (cup down), its slope is decreasing, so its **second derivative is negative**.

A **point of inflection (POI)** is exactly where the concavity changes from **concave up** to **concave down** or **concave down** to **concave up**. It turns out that a graph **crosses its tangent line** at a POI.

Here’s an illustration of **concavity**:

The **Second Derivative Test **can also be used in curve sketching to find **relative minima** and **relative maxima**, and is the following:

**Second Derivative Test**

Let $ f$ be a function whose **second derivative** exists on an open interval that contains $ c$:

- If $ {{f}^{\prime \prime}(c)}>0$, $ f$ has a relative
**minimum**at $ x=c$. (Think “cup up”) - If $ {{f}^{\prime \prime}(c)}<0$, $ f$ has a relative
**maximum**at $ x=c$. (Think “cup down”)

Note that the second derivative test does not necessarily work when the **first and** **second derivatives are 0** **or undefined; the concavity of the function must change at a point of inflection**. It also doesn’t say what happens at the endpoints of a function.

You might see problems like this on **concavity**, **points of inflection**, and the **Second Derivative Test**. I think the best way to tackle these problems is to create a **sign chart** using points where the first derivative is **0** or undefined (critical values) and also where the second derivative is **0** or undefined:

Here’s a concavity problem with a **trigonometric function**:

Here are more **Second Derivative Test** problems:

Here’s one more problem where the **Second Derivative Test does not apply at certain points**:

## Curve Sketching: General Rules

Here are some general curve sketching rules:

- Find
**critical numbers**(numbers that make the first derivative**0**or undefined). - Put the
**critical numbers**in a**sign chart**to see where the first derivative is positive or negative (plug in the first derivative to get signs). - Where
**first derivative**is**positive**, the function is**increasing**; where it’s**negative**, the function is**decreasing**(remember that you can**combine**two consecutive intervals only if the**original function is defined**for that critical number). - To get
**relative minimums**and**relative maximums**, see how the derivative is changing. If it’s changing from**negative to positive**, it’s a**minimum**, and if it’s changing from**positive to negative**, it’s a**maximum**. To get the coordinates of the point at these places, plug the $ x$-value into the original function to get the $ y$-value. - Get the
**second derivative**, and find the values where it’s either**0**or undefined.**(Make sure these points are defined for both the function and first derivative).** - Put those values in a
**sign chart**to see where the second derivative is positive or negative (plug in the second derivative to get signs). - Where
**second derivative**is**positive**, the graph is**concave up**, where the**second derivative**is**negative**, the graph is**concave down**. Remember that you can**combine**two consecutive intervals only if the**original function is defined**for that for the first and second derivative.**A****point of inflection (POI)**occurs when the second derivate changes sign. - Use other points (can use a
) to help graph!*t*-chart

Also, these tips may help:

- If finding
**absolute extrema**, find the**critical numbers**and**endpoints**, then plug into the original function to find $ y$-**values**. Compare all these values: the**largest**is the**absolute maximum**and the**smallest**is the**absolute minimum**. - When graphing, it might be helpful to identify any
**asymptotes**or**removable discontinuities**(holes) by seeing what makes the denominator of the original function**0**. Remember that a hole happens when you can cross out a factor in both the numerator and denominator (see**Drawing Rational Graphs**in the**Graphing Rational Functions, including Asymptotes**section).

Here are some other hints that may help with the relationship of $ f$**, **$ {f}’$ and $ {{f}^{\prime \prime}}$. Again, remember that the relationship of $ f$ to $ {f}’$ is the same as $ {f}’$ to $ {{f}^{\prime \prime}}$; similarly, the relationship of $ {f}’$ to $ f$** **is the same as $ {{f}^{\prime \prime}}$ to$ {f}’$. Also, I like to use the **PMS** acronym (sorry 🙂 ) to “travel” back and forth among the curves of the **function**, **derivative**, and **second derivative**:

Here’s a nice visual for curve sketching that shows the relationship of a function, derivative, and second derivative for the four curves shown:

Here are some problems that you may see:

Here’s one where we might have the information in a table.

**Sketch a possible graph:**

Here are more types of **curve sketching problems** you may see:

**Learn these rules, and practice, practice, practice!**

Click on Submit (the arrow to the right of the problem) to solve this problem. You can also type in more problems, or click on the 3 dots in the upper right hand corner to drill down for example problems.

If you click on “Tap to view steps”, you will go to the **Mathway** site, where you can register for the **full version** (steps included) of the software. You can even get math worksheets.

You can also go to the **Mathway** site here, where you can register, or just use the software for free without the detailed solutions. There is even a Mathway App for your mobile device. Enjoy!

On to **Optimization**** **– you are ready!