# Derivative applications

Contents

The reason we study calculus, and the reason it was invented, is for its many uses in real-world problems. In particular, derivatives let us optimize functions and study their rates of change.
Here we cover optimization, rates of change, L'Hopital's rule, the mean value theorem, and more!

## Equations of normal and tangent lines

A derivative at a point in a curve can be viewed as the slope of the line tangent to that curve at that point. Given this, the natural next question is what the equation of that tangent line is. In this tutorial, we'll not only find equations of tangent lines, but normal ones as well.

7:17

Equation of tangent line example 1

Use the derivative to find the equation for a tangent line.

5:59

y-intercept of tangent line example

A practice problem for finding the y-intercept of a tangent line to f(x) = 1/x

5:27

Equation of normal line

The "normal" line of a curve at a point is perpendicular to the tangent line. Here we practice finding the equation for a normal line.

Exercise

Applications of derivatives: Tangent and normal lines

Practice finding tangent and normal lines.

## Motion along a line

Derivatives can be used to calculate instantaneous rates of change. The rate of change of position with respect to time is velocity and the rate of change of velocity with respect to time is acceleration. Using these ideas, we'll be able to analyze one-dimensional particle movement given position as a function of time.

10:09

Total distance traveled by a particle

Given a function representing the position of a function over time, how can you find the total distance traveled?

5:20

Analyzing particle movement based on graphs

Learn how to analyze a particles motion given the graph of its position over time.

8:57

When is a particle speeding up

Given a function representing a particles position as a function of time, how can you tell when it is speeding up?

Exercise

Applications of derivatives: Motion along a line

Practice analyzing a particle's position, velocity and acceleration.

## Critical points and graphing with calculus

One of the reasons calculus was invented was to be able to optimize functions. When you have some function modeling a real world situation, you often want to find its maximum or minimum.
In this tutorial, you will see how information about the derivative of a function can give powerful ways to mathematically describe the "shape" of a function.

7:53

Minima, maxima and critical points

Here we see how the highest and lowest points on a graph relate to the derivative.

5:51

Finding critical numbers

Critical numbers of a function are inputs where the derivative equals zero. Here we practice finding some.

Exercise

Critical numbers

Practice finding critical numbers.

5:25

Testing critical points for local extrema

Once you find a critical point, how can you tell if it is a minimum, maximum or neither?

9:42

Identifying minima and maxima for x^3 - 12x + 2

Some practice with finding extrema of a function.

## Absolute and relative maxima and minima

When you're looking for the maximum or minimum of a function, a good way to start is by finding points where the derivative equals zero. However, you won't always get the maximum possible value of the function; you might just end up with a point which is maximum *relative* to those points around it.
In this tutorial, you will learn about the extreme value theorem, and what it tells us about relative maxima and minima.

7:58

Extreme value theorem

The extreme value theorem tells us that continuous functions must attain a maximum and a minimum on any closed interval where it is defined.

5:30

Introduction to minimum and maximum points

Sal explains all about minimum and maximum points, both absolute and relative.

3:08

Identifying relative minimum and maximum values

How can you tell if a point is a miximum/minimum of a function?

Exercise

Extreme values from graphs

Practice identifying maximum and minimum values by looking at the graph of a function.

6:56

Applying extreme value theorem

Watch the extreme value theorem in action.

Exercise

Extreme value theorem

Practice using the extreme value theorem.

## Concavity and inflection points

One of the neat things about calculus is that it gives us a mathematical way to describe the shape of curves. In this tutorial, you will learn about concavity and inflection points, which describe quantitatively how a curve curves. This will be useful for finding maxima and minima.

9:54

Concavity, concave upwards and concave downwards intervals

What is concavity?

2:23

Recognizing concavity exercise

Sal walks through an exercise where you are asked to recognize the concavity of a function in certain regions.

Exercise

Recognizing concavity

Practice your understanding of concavity.

2:34

Inflection points

Inflection points are where a function changes in concavity.

20:31

Graphing using derivatives

Because the derivative and second derivative give us information about the "shape" of a function, we can use this information to sketch the graph of a function.

25:08

Another example graphing with derivatives

More practice using the derivative and second derivative to draw a function's graph.

Exercise

Concavity and the second derivative

Practice your understanding of concavity and its relationship with the second derivative.

Exercise

Second derivative test

Practice using the second derivative test for extrema

## Optimization with calculus

Choosing which topic in calculus is the most useful is like asking a parent to choose their favorite child. What you're supposed to say is that there isn't one. That said, between you and me, optimization is quite possible the most important topic from calculus that you should remember.

7:35

Minimizing sum of squares

What is the minimum possible value of x^2+y^2 given that their product has to be fixed at xy = -16

9:50

Optimizing box volume graphically

If you are making a box out of a flat piece of cardboard, how do you maximize the volume of that box?

9:00

Optimizing box volume analytically

Finishing up the last video by working through the formulas.

11:27

Optimizing profit at a shoe factory

Who knows, you may end up running a shoe factory one day. So it might not be a bad idea to know how to maximize profits.

12:40

Minimizing the cost of a storage container

With all the storage you might have to handle for your shoe factory, I bet you'd also like to be able to minize the cost of storage.

6:19

Expression for combined area of triangle and square

A contrived, but fun, minimization practice problem.

5:31

Minimizing combined area

More practice with minimization.

Exercise

Optimization

Practice those optimization skills!

## Applying differentiation in different fields

So, you've probably been told that calculus is useful, but this fact can be hard to believe and remember when you're deep in the weeds of a particularly nasty derivative. In this tutorial, we begin to just scratch the surface as we apply derivatives in fields as disperse as biology and economics.

4:40

Derivative and marginal cost

In economics, the idea of marginal cost can be nicely captured with the derivative.

7:46

Approximating incremental cost with derivative

If you can come up with a function to model cost, its derivative will let you estimate marginal cost.

5:00

Modeling a forgetting curve

Let's face it, sometimes it feels like we forget much of what we learn. In this example we model that forgetful phenomenon.

Exercise

Applications of differentiation in biology, economics, physics, etc.

Practice using derivatives in a variety of different fields.

## Related rates

Have you ever wondered how fast the area of a ripple of a pond is increasing based on how fast the ripple is? What about how fast a volcano's volume is increasing? This tutorial on related rates will satiate your curiosity and then some!
Solving related rates problems using calculus

7:43

Rates of change between radius and area of circle

What's the relationship between how fast a circle's radius changes, and how fast its area changes?

8:17

Related rates: Balloon example

In this example, you are analying the rate of change of a ballon's altitude based on the angle you have to crane your neck to look at it.

11:32

Related rates: water pouring into a cone

As you pour water into a cone, how does the rate of change of the depth of the water relate to the rate of change in volume.

5:49

Related rates: Falling ladder

You're on a ladder. The bottom of the ladder starts slipping away from the wall. Amidst your fright, you realize this would make a great related rates problem...

6:53

Related rates: Approaching cars

As two cars approach the same intersection from different roads, how does the rate of change of the distance between them change?

10:39

Related rates: Shadow of a bird

And owl, his shadow, the mouse he's hunting, and more related rates practice.

Exercise

Related rates

## Mean value theorem

If over the last hour on the highway, you averaged 60 miles per hour, then you must have been going exactly 60 miles per hour at some point. This is the gist of the mean value theorem (which generalizes the idea for any continuous, differentiable function).

6:37

Mean value theorem

The mean value theorem tells us when certain values for the derivative must exist.

4:50

Example of the mean value theorem

See what the mean value theorem looks like in practice.

7:12

Getting a ticket because of the mean value theorem

Even if a cop never spots you while you are speeding, he can still infer when you must have been speeding...

4:14

Maximizing function at value

In this example, we find the maximum possible value of a function based on a contraint put on its derivative.

Exercise

Mean value theorem

Practice using the mean value theorem.

16:48

Mean value theorem

Intuition behind the Mean Value Theorem

## L'Hôpital's rule

Limits have done their part helping to find derivatives. Now, under the guidance of l'Hôpital's rule, derivatives are looking to show their gratitude by helping to find limits. Ever try to evaluate a function at a point and get 0/0 or infinity/infinity? Well, that's a big clue that l'Hopital's rule can help you find the limit of the function at that point.

8:51

Introduction to l'Hôpital's rule

When you are solving a limit, and get 0/0 or infinty/infinity, L'Hopistal's rule is the tool you need.

7:43

L'Hôpital's rule example 1

L'Hôpital's Rule Example 1

5:15

L'Hôpital's rule example 2

L'Hôpital's Rule Example 2

7:50

L'Hôpital's rule example 3

L'Hôpital's Rule Example 3

4:10

L'Hopital's Rule to solve for variable

Unlike previous example, we now have a variable inside a limit which we can tweak, and we are trying to make the limit equal a specific value.

13:11

Tricky L'Hopital's Rule problem

L'Hopital's rule can apply to situation which don't look like 0/0 or infinity/infinity, if you are clever enough.

Exercise

L'Hôpital's rule

Practice using L'Hopital's rule.

5:21

Proof of special case of l'Hôpital's rule

This isn't a full proof of L'Hopital's rule, but it should give some intuition for why it works.

## Local linearization

Let's see how we can local linearization can be used to approximate values of functions near values that we know.

9:38

Local linearization

Approximating square roots using tangent lines.

6:31

Local linearization example

Approximating ln(x^2) using local linearization.

Exercise

Local linearization

Practice approximating functions with local linearizations.

## Tangents to polar curves

Here you have the chance to practice thinking about tangent lines when curves are defined in polar coordinates.

Exercise

Tangents to polar curves

Practice tangent lines of functions defined with polar coordinates.