12.3 The Tangent Line Problem

The slope of a curve at a point is defined to be the slope of the tangent line. Thus the slope of a curve is found using the derivative.

Everyone knows slope is the change of y over the change of x

You use that to find the AVERAGE slope between two points. But some may ask how to find the slope at a GIVEN POINT.

They may think it is impossible by combining what they have learned and common knowledge. But they are wrong.

With derivatives you shrink the point (two points) to the smallest possible distance... but then you eventually shrink it down to zero.


How to Find a Derivative!

You start off with the original slope formula like the one above, but with a little twist.

f(x+h)-f(x)= y
              h=x

Once you simply that formula, that gives you the average slope of the graph. Then from that you plug in a point and voila! the slope at that given point.

Like I said earlier, eventually you turn "h" into 0
So the final average slope of the formula above is "2x"

For more help visit: http://www.coolmath.com/precalculus-review-calculus-intro/precalculus-algebra/24-the-difference-quotient-01.htm

Thank You,

Peter Kessel

12.2- Techniques for Evaluating Limits

Limits of Polynomial and Rational Functions

--1. If p is a polynomial function and c is a real number, then  



--2. If r  is a rational function given by r(x)= p(x)/q(x), and c is a real number such that q(c) does not equal 0, then:
     as x approaches c,   lim r(x) = r(c) = p(c)/q(c)

Evaluating Limits by Direct Substitution

      - when evaluating a limit by direct substitution, you insert the c value in for the x value of the function and the resulting number is the limit as x approaches c.

     - This method only works if the function is continuous, ehich means there are no holes in the graph

Evaluating a limit by Dividing Out

  

     - To evaluate a limit by dividing out, you can factor the numerator and divide out the common factors.

     - This particular example has a limit of 7, and has a hole at the point (1,7) because the factor of (x-1) is divided out, and at that point, y=7.

Evaluating a Limit by Rationalizing

    - To evaluate a limit by rationalizing, you have to first rationalize the numerator of the function.

   - In this example, once you get the numerator rationalized, you can plug in the c value for x and that gives the limit as x approaches c, where at c there will be a hole in the graph.

Evaluating One-Sided Limits

To evaluate a one sided limit, you find what y value is approached from either side as x approaches c.  In this graph, as x approaches c (2) from the left, y also approaches 2.  As x approaches c from the right, however, the y value approaches 1.  These can be solved by simply looking at the graph, or applying direct substitution to the 2 functions for their correct sides of the graph.  

Some helpful videos:

http://www.youtube.com/watch?v=0u7NtyGGlv4

http://www.youtube.com/watch?v=mGPyMDi1Mdc

Thanks,

Michael 

Chapter 12.1: Introduction to Limits

This next chapter is all about limits. So the first question was, what is a limit?
The book defines a limit as:

If f(x) becomes arbitrarily close to a unique number L, as x approaches c from either side, the limit of f(x) as x approaches c is L.

More simply, a limit is the value y approaches as x approaches a specified value.

 

 

Today we learned two different ways of finding the limit:

1. Tables

2. Graphs



 

First, insert the equation into your calculator.

Then insert values into the table that are close to 2.

From this table, you can tell that as x approaches 2, y approaches 5. So the limit is 5.

From the graph, we can see that as x approaches 0, y approaches 1. All these types of problems are solved by just looking at the graph.

Limits That Do Not Exist:
There are also limits that don't exist. There are 3 scenarios where this happens.

1. When y approaches two different numbers
-In the example above, if you try to solve for the limit when x approaches 1, you get two values that y approaches. Therefore, there is not a limit at all.

2. When y approaches infinity or negative infinity
-In the graph below, as x approaches 1, y approaches both infinity and negative infinity. In this example, the limit DNE.
 

 3. In oscillating graphs
-In the graph of sin(1/x), the wave only goes between 1 and -1 so there is no limit.

That's mostly everything we learned today. Hope it helped!
-Natalie

 

10.6 Polar Coordinates

Hey everyone, so today we learned a new way to graph with the polar coordinate system.


We are all very familiar with the Cartesian Plane or the rectangular coordinate system.

Points are labeled (x,y)
Each set of points represent the directed distances from the coordinate axes to the points.  Each set of points is unique.














Now we are learning about a new coordinate system.
The polar coordinate system consists of concentric circles.  The polar axis is similar to the x axis.

Each point P can be assigned polar coordinates (r,θ)
r represents the distance from the origin (it can be negative)
θ represents the rotation from the polar axis (also can be negative)

Unlike the rectangular coordinate system, points on the polar plane are not unique. The same point can be represented by infinite sets of polar coordinates.  









Now that we know about the polar coordinate system, we can now relate both of the graphs.  Points can be converted from (x,y) to (r,θ) and vice versa.  

Coordinate conversions 

x=rcosθ

y=rsinθ

tanθ=(y/x)

r^2=x^2+y^2

Thats about it!!!

 Helpful video

-Jen Kendall

10.5 Parametric Equations

This section was all about parametric equations

Most equations just use x and y variables, but parametric equations use a third variable, which is often t for time and is called a parameter.  You can then write x and y as functions of the parameter to get two parametric equations.

Graphing Parametric Equations:

Here's an example of a parametric equation and its graph:

x = t - 2

y = t + 1

To graph this, you can pick values of t and plug them into the equations to calculate the coordinate points of the graph.  For example, when t = -2, the x value is -4 and the y value is -1 so the coordinate is (-4, -1).  This is what the graph should look like:

Eliminating the Parameter:

Once you've graphed the parametric equations, you can rewrite it as an equation using only x and y.  The equation for the line above is y = x + 3.

You can also eliminate the parameter algebraically without graphing to get, as the book says, a rectangular equation, which is just a normal equation with two variables.  You do this by solving for the parameter in one of the equations, then plugging that into the other equation.

x = t -2
t = x +2

y = t + 1
y = x + 2 + 1
y = x + 3


Finding Parametric Equations:

 

You can  find parametric equations from a rectangular equation.

ex.  x² + y² = 1

There are many different ways to rewrite this equation parametrically.  One of them is...

x = sinθ

y = cosθ

Calculator:

You can graph parametric equations on your calculator by going to MODE and selecting PAR, then you just type the equations in and graph.

That's basically it.
-Olivia R