15-5: Instantaneous Rate of Change of a Function: The Derivative

Chapter 15 Section 5:

Review:

So, this is the last section of the last chapter of the year! I think that it is appropriate that I am the last person to post a blog seeing as I was the first person as well.

This section is our introduction to Calculus. Calculus is all about the rate of change and through this section we are figuring out how to predict it.

Now something to remember:

Instantaneous rate of change = Derivative = Slope of a tangent line

for the point x sub 0

Definitions:

Derivative= instantaneous rate, velocity

--derivative of the time-height function

Average rate of change of r(x) of function f(x) on an interval starting at x=c is the change in the y-value of the function divided by the corresponding change in the y-value of the function divided by the corresponding change in the x-value.

Instantaneous rate of change of f(x) at x=c is called the derivative and is denoted f '(x) "f prime of x." It is equal to the limit of the average rate as x approaches c.

The value of the derivative of f(x) at x=c equals the slope of the tangent line to the graph of f at x=c.

Example Problem:

Example 1:

Given: f(x)= x^3 + 2x + 3 what is the instantaneous rate of change of f(x) at x=1?

Remember, first you plug in 1 to f(x), so you have 1^3 + 2(1)+3 = 6

f(x)= f(x)-6/(x-1) = ((x^3 + 2x + 3)-6)/(x-1)

Now, use synthetic substitution

1 (pretend it is in the box thing) ___ 1 __ 0 __ 2__-3

______________________________________1 __ 1 __ 3

________________________________------------------------

________________________________1___1___ 3 __0

(sorry if all of the marks above confused you, but when I tried to publish the post, the computer automatically pushed all of the numbers to the left and it just looked like a jumble of numbers)

Yay! We got zero! which means that it is removable~

so:

((x-1)(x^2 + x + 3))/(x-1) the x-1 cancel out so you are left with

x^2 + x + 3. If you plug in 1 for x you get 1^2 + 1 + 3 = 5

So 5 = the Instantaneous rate of change = derivative = slope!

Now, use this information to create the equation for the line. From the information in the problem, you know the slope (= the Instantaneous rate of change = derivative) which is 5 and you also have a point, (1, 6) and as Mr. French said, it is always better to use point slope form because it is so easy!

So, y - 6 = 5 (x - 2).

Example 2:

You are at Six Flags and you are riding a roller coaster. On your favorite ride, there is a dip that has the equation y= x^4 + 5 where the x-axis is the ground and the y-axis cuts the dip in half. You are such a good Precalculus student that you want to use the knowledge that you learned in class to calculate your average velocity between 1 and 1.5 seconds after you reach the minimum of the dip.

Given this information, you know only your x values, 1 and 1.5. To find the y-values, plug the information into the equation. When you plug in 1, you get 6 and when you plug in 1.5, you get 10.0625. Using this information, you can calculate the average velocity. Your points are (1, 6) and (1.5, 10.0625). (10.0625 - 6)/ (1.5-1) = (4.0625/.5) = 8.125 = the average velocity!

For some extra help visit:

http://people.hofstra.edu/faculty/Stefan_Waner/RealWorld/tutorials/frames2_1.html
It is actually pretty cool! If you click on "previous tutorial" and "next tutorial" in the left hand corner there is more information about what this section covers.

This is a picture of me, Juan, and Alex. Juan Alderete de la Pena is the base player from my favorite band, The Mars Volta. I think Henry may have told this story on a previous blog, but I will tell you again anyway. So, ok, we were going to the Detour Music Festival in LA and we took the metro there and when were were in Union Station, we saw him and were like "Juan" and he turned around. It was amazing. We took the metro with Juan and walked down the streets of LA with him looking for the entrance. I couldn't believe that night; it was awesome! Anyway, we became buds and it was pretty amazing.

Reminder:

Oh yeah!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! No more blogs!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!