Why do we graph functions?

**Calculus in the Real World**

Lets start with the things we already know!

More things you should already know:

How does knowing these things help you graph something like

x^3 + 4x^2 + x - 6 ?

The answer is that knowing about different types of graphs and differentiation techniques is most of what you need to know to graph a complex function. What you still need to learn is how to use those techniques to find where the function is increasing/decreasing, where it has minimum/maximum values, and where it switches concavity.

Lets start with a video from the Khan Academy that will tell us how to use the derivative of a function to tell us about its critical points and its graph.

f '(x) = slope of the function

A function is increasing when f '(x) is positive

A function is decreasing when f '(x) is negative

When f '(x) = 0, or undefined, there is a critical point

A critical point is a minimum when the function switches from decreasing to increasing at that point.

A critical point is a maximum when the function switches from increasing to decreasing at that point.

What does the first derivative tell us?

In order to graph more complex functions we also need to use the second derivative, f ''(x).

No.

We are going to play a matching game. Please match the graph of the function with the correct type of mathematical equation on the guided notes.

A

B

C

D

E

Try to answer this question on your own before continuing.

Is this enough for us to be able to graph a complex function?

When f ''(x) is positive the graph is concave up, or curved upwards

When f ''(x) is negative the graph is concave down, or curved downwards

When f ''(x) = 0 we have a possible point of inflection

A point of inflection is where the graph switches concavity (from up to down or down to up)

The image below has a sign chart for some f '(x) and f ''(x). Many people use sign charts to help the visualize the intervals on which a function is increasing/decreasing and concave up/down. The numbers are the critical points for f '(x) and points of inflection for f ''(x).

inc = increasing

dec = decreasing

conc = concave

Find the critical points, maximum/minimum points, and increasing/decreasing intervals of the function x^3 + 4x^2 + x - 6.

Find any points of inflection and concave up/down intervals for x^3 + 4x^2 + x - 6. Draw sign charts for f '(x) and f ''(x) to help you visualize what the graph will look like.

At the start of this lecture, x^3 + 4x^2 + x - 6 was something that you were not able to graph.

However, now you have everything you need to graph the function. Give it a try on your own.

Your graph should look something like this:

(This

is

my

music)

Graphing complex functions, like the one we just did, is an important skill for engineers, mathematicians, physicists, and many other professions.

Write down 3 real things, situations, or relationships that you think can be graphed. Why is it important to graph the 3 things you chose?

Here is one example of how to obtain the mathematical equation for a real life situation:

Lets say we want to graph this:

How high will a ball go that is thrown straight up in the air?

(This is a question of physics)

Unfortunately for us this relationship uses more variables than then math we have learned so far, so we will restrict some dimensions.

Say there is no wind, the ball is going straight up with an initial velocity of 10 meters per second, a starting height of 2 meters, and the force of gravity is 9.8 m/(s^2) downwards.

2

Starting height (in meters)

y (height) = 2

Upwards velocity of 10 (in meters per second)

y = 2 + 10t

where t is time in seconds

Downward acceleration of 9.8

(in meters per second squared)

y = 2 + 10t - 9.8(t^2)

We can use our new mathematical tools and graph to see the relationship between the time in the air and height of the ball.