One-sided limits from graphs

When approaching from the -ve side, there is a negative superscript for the limit number


    One-sided limits from graphs:

Function as geometric series

Find the common ratio r.

convergence occurs if  -1 < r < 1

For such an interval of convergence, S=a/(1-r)

  Function as geometric series:

Tangent slope as limiting value of secant slope example 1

Draw out the graph to see what is needed.
It may be possible to substitute algebraically to get the solution.


    Tangent slope as limiting value of secant slope example 1:

Relatively prime numbers

In number theory, two integers a and b are said to be relatively prime, mutually prime, or coprime (also spelled co-prime)^[1] if the only positive integer that evenly divides both of them is 1. That is, the only common positive factor of the two numbers is 1. 

For example, 14 and 15 are coprime, being commonly divisible by only 1, but 14 and 21 are not, because they are both divisible by 7. 

Trapezoidal Rule to find the average value of f(x)

From http://www.intmath.com/integration/5-trapezoidal-rule.php

Area≈​2​​1​​(y​0​​+y​1​​)Δx+​2​​1​​(y​1​​+y​2​​)Δx+​2​​1​​(y​2​​+y​3​​)Δx+

Area≈Δx(​2​​y​0​​​​+y​1​​+y​2​​+y​3​​+…+​2​​y​n​​​​) for uniform grid


The trapezoidal rule finds the area. To get the average 
value, you will need to divide the area by b-a which is the
 the difference between the final and initial x values.

Inverse of a straight line

The straight line graphs shown below are inverses of one another.

 

Given the invertible function f(x), we determine the inverse by:

• replacing every x with y and y with x;

Moving left and right, speeding up or slowing down?

From Khan Academy

Let's deal with moving left and right to start.

The particle is moving to the right when its velocity is positively valued; that is, when v(t)>0.

The particle is moving to the left when its velocity is negatively valued; that is, when v(t)<0.

Now we consider when the particle is speeding up and slowing down.

The particle is speeding up for those times when the product of its velocity and its acceleration is greater than zero. Therefore we want

v(t)⋅a(t)>0.

For this condition to be satisfied, we want those times when v(t) and a(t) are both positively valued or both negatively valued.

In contrast, he particle is slowing down for those times when the product of the velocity and the acceleration is less than zero. Therefore we want

v(t)⋅a(t)<0.