Taylor Holiday blog #1

CHAPTER 1 REVIEW
• Remember that estimating limits numerically means to set up and complete the table
The first step to setting up the table is to step up the left side of the graph. To set up the left side of the graph you subtract .1 .01 .001 from the number that X is approaching in the problem and set the subtracted answers to each number into three boxes.
The next box will simply have the number x is approaching in it
The final three boxes will have the results of adding .001 .01 .1 to the number x is approaching respectively.
The first step to estimating the limit is complete

• To complete the table you must enter the given equation into your y=
• After you will hit “2nd” and table and enter in the numbers that you solved for in the seven boxes of the first step.

• now you will fill in the bottom seven boxes of the chart with the results from the table

• (( a quick way to know if you’ve done the steps correctly is to make sure that the box under the box with the number x is approaching should have an error on the table.

Finally to estimate the limit you will read each side toward the number x is approaching and decipher what number each side is headed toward as it approaches the given number x is approaching.
• Remember the three boxes on the left side will read toward the right and the right three boxes will read toward the left

You have the possibility to have two different out comes

• If the numbers on each side of the table match then the number they are headed toward is the estimated limit
• If the numbers on each side of the table do not match the limit does not exist.

I will now give an example of each outcome

EX:1
Lim X-> 0
F(x)= x/ squareroot of (x + 1) -1
The top half of the table will read
[-.1] [-.01] [-.001] [0] [.001] [.01] [.1]
The bottom half of the table will read
[1.9487] [1.995] [1.995] [ERROR] [2.0005] [2.005] [2.0488]
From the left the numbers are approaching 2
From the right the numbers are also approaching 2
Therefore the estimated limit is 2

Ex:2
Lim X-> 0
F(x)= sin 1/x
The top half of the table will read
[2/pi] [2/3pi] [2/5pi] [0] [2/7pi] [2/9pi] [2/11pi]
The bottom half of the table will read
[1] [-1] [1] [ERROR] [-1] [1] [-1]
From the left the numbers are not approaching anything
From the right the numbers are also not approaching anything
Therefore the estimated limit DNE



CHAPTER 2 LESSON 1

SLOPE OF A TANGENT LINE
There are two formulas which need to be memorized
(((& means delta)
• The first formula is
f(x+&x)- f(x)/&x
This is the formula for a derivative. This formula is known as the secant line formula.

• The second formula is only a tiny bit different from the first

Lim f(x+&x)- f(x)/&x
&x -> 0

This formula is known as the slope of a tangent line

Solving for the problems we’ve had thus far in chapter 2 have consisted of plugging into these formulas and solving.

When given an equation you must plug x+&x into all x’s of the given equation and fill in the rest of the formula by placing – f(given equation exactly how its given)/ &x

Ex:
Find the slope of the graph of f(x)= 2x-3 at (2,1)

First you would plug in:

f(x) 2(x+&x)-3-(2x-3)/ &x
&x-> 0

Expand:

2x+2&x – 3 – 2x + 3 / &x

Take out what cancels:
2x + 2&x -3 -2x +3/&x

And you’re left with:
2&x/&x

Simplify:

2 &x/ &x
Therefore the answer is 2

((because there are no x’s left in the equation you can ignore the point (2,1) however if there had been any x’s you would have also plugged in 2 for the x’s before you simplified then solved as normal.))

There are a few helpful hints that need to be remembered
**remember that for each of these formulas any variable can be used for delta x
** remember that slope of the tangent line means to use the derivative formula
** there are many was to ask for a derivative these ways are:
Dy/dx
Y^1
F’(x)
d/dx {f(x)}
Dx[y]
D/dx

CHAPTER 2 LESSON 1

There are two rules to follow when using the derivative shortcut
These rules are : * The Constant Rule
• The Power Rule
The constant rule states that the derivative of a constant is 0
Therefore when asked to find a derivative of an equation with a constant the constant will automatically become 0.

The Power rule states that every time you take a derivative you lose a power.
Therefore if you are not taking the derivative of a constant you bring the exponent to the front of the variable and subtract one from the exponent.
The formula that displays this is
d/dx [x^n ] = nx^n-1
Example: d/dx[x^3] = 3x^2

There are many variations of equations that will call for simplification before you can take the derivative

When taking the derivative of a fraction you must simplify the equation by taking the numerator and placing it in front of the x and the exponent behind the denominator will become a negative
Example: d/dx [1/x^2] = x^-2
Then you would proceed to use the shortcut method to find the derivative.

When taking the derivative of a root you will turn the value of the root into a fraction with the exponent of x over the value of the root.
Example: d/dx [cuberoot x] = X^1/3
Then you would proceed to use the shortcut method to find the derivative.
**do not forget that after finding the derivative of a simplified equation you must convert the derivative back to an unsimplified form
Example: d/dx [ x^1/3] = 1/3x^-2/3 = 1/3cuberoot x^2