Inflection Points

Mohamed Salahuddin Blog Post 4

Definite integrals and the area under a curve

In this topic I want to explain the Definite integrals, and explain its relation with the area under a curve.

A Definite Integral has start and end values: in other words there is an interval from “a” to “b”.

We can find the Definite Integral by first calculating the Indefinite Integral at points x, or I(x);

The second step is to find substitute the “a” and “b”, to find I(a) and I(b);

The third step is to subtract the I(a) from I(b) to obtain the Definite integrals.

Now, that we know how to solve it, we need to explain what does it mean. Say if it is a Definite integrals of the f(x) from “a” to “b”, it is represented by the area of the curve from the x-axis to the f(x) graphically.

If however f(x) is crossing the x-axis between the interval [a, b], we need to find the crossing point c, and integrated it in two parts, from [a, c] and [c, b].

Now we will use an example to illustrate my point.

Say if we want to integrate f(x) =x-2 from [2, 3].

Graphically, it is exactly the area between the f(x) =x-2 and x-axis from x=2 to x=3.

Now if we want to find the area between [1, 3], we need to be careful, since f(x) is crossing the x-axis within the interval at x=2. So we need to integrate two parts: [1, 2], and [2, 3]

The total area is thus 1+3=4.

Now we can see the close relation between the Definite Integral and areas under a curve. It can also be used to find volumes, central points and many useful things.

Limits - Muneera Albahar

My name is Professor Muneera Albahar and today I am going to teach you the concept of limits.

Learning Objectives:

1.     Understand the meaning of limits and why we study it.

2.     Learn how to calculate the limits for a specified function. (Students should be able to explain how they to come to a particular solution.)

3.     Real World Application

Learning Materials:

Youtube Video Link: http://www.youtube.com/watch?v=skM37PcZmWE

Reference Book: “Fundamentals of University Mathematics” by C M McGregor, John Nimmo, Wilson Stothers.

Website: http://www.sparknotes.com/math/calcab/functionslimitscontinuity/section2.rhtml

       http://www.mathsisfun.com/calculus/limits.html

Lesson Outline

Ø  Why we need to learn limits?

Limits are the most fundamental concept in Higher Mathematics, especially Mathematical Analysis. Most concepts, such as continuity, differential, integration, are defined on the basis of limits. For example, the definition of function continuity at a certain point is always written as the following:

“In the function f, at the certain point x₀ of its domain, if the limits of f(x) from left hand side and right hand side as x₀ approaches f(x₀) through the domain of f exist and both are equal to f(c)”.

To simplify, a limit is our best prediction of a point that we are unable to observe, which means that we can learn a lot about the function using limits. For example, we can check what happens to the function at a discontinuity by taking a limit at this point. Graphically, it also tells us that the function approaches either positive or negative infinity.  

Ø  Definition

In the 19^th century, Augustin-Louis Cauchy, followed by Karl Weierstrass, formalized the definition of the limit, which became widely known as the (ε, δ)-definition of limit.

Suppose f is a real-valued function, and c is a certain number of its domain. The expression  means when the values of x whose distance from c is less than some positive number δ, that is, values of x within either (c − δ, c) or (c, c + δ), there is a certain positive number ε which represent any small positive number, that f(x) certainly lies in the interval (L-ε, L+ε).

Ø  Graph Explanation

Whenever a point x is within in δ units of c, f(x) is within ε units of L.

Ø  Rules For Computing Limits

Three fundamental Rules:

1.     The limit of a constant will simply be the constant

2.     If two sequences have limits that exist, then the limit of the product is the product of the limits.

3.     If two sequences have limits that exist, then the limit of the sum of sequences is the sum of the limits of the sequences.

If our sequence is described by a fraction with powers of n on the top and the bottom, we may follow the following rules:

1.     If the degree of n is higher on top, then the limit is infinity.

2.     If the degree of n is higher on the bottom, then the limit is zero.

3.     If the degree of n is the same in the numerator and denominator, then the limit is the ratio of the leading coeffecients.

Ø  Example:

Example1.

The definition of the derivative of a function at a certain point uses the concept of limits.

Suppose f is a real-valued function, and x₀ is a certain point of its domain in the f. The derivative f’(x₀) f’(x₀) = lim x

Example2.

Find f(x)=

Note: the point x=3 for the function  is meaningless since the denominator cannot be zero, so we should use the limit to predict the value at x=3.

Solution: f(x)==

Ø  Real World Application:

Suppose if I keep tossing a coin as long as it takes, how likely am I to never toss a head?

The possibility of never tossing a head is decreasing. Within the frequency increasing, the limit of the possibility is 0.

The mathematical answer to this is:

P(N)=( )^N

lim_N→∞p(N)=0




Class dismissed!

Hello Class, today we're going to be talking about limits. What are limits you ask?

The limit of f(x) is L as x approaches a. 



lim f(x) = L means that by taking a value of x very x->a close, but 

not equal, to a, the function value f(x) will be very close to L.  

Example:

From this graph we can see that as X comes in from the left side (positive infinity) it approaches 0. Likewise we also see that as X comes in from the right side (negtive infinity, it approaches 0). 

 

NOTE that you can also interpret this graph in different ways. Instead of x approaching infinity, maybe it approaches 0. How is the limit different then?

As x approaches 0 from the negative sector, we see that it approaches negative infinity but as x approaches 0 from the positive sector, it approaches positive infinity so in that circumstance THE LIMIT DOES NOT EXIST (https://www.youtube.com/watch?v=Lck5_YoxxGI) because they don't match. 

More examples:

What's the limit of F(x)= (x^2-9)/(x-3) as x approaches 3



Looking at the graph we see that as x approaches 3 it comes close to equaling 6

Average Cost

Average cost

Good morning students! Today we are going to learn about average cost. 

Lets assume we have a gelato company. When a company produces gelato there are two types of costs associated with its manufacturing:  fixed costs and variable costs. 

Fixed costs are what we are going to spend no matter how many buckets of gelato we are going to produce or how many employees we hire. Office space, electricity, equipment,  salary for managers are some example of fixed costs. It is important to stress that fixed cost are not related to the activity level of goods produced by the factory, in other words they are indirect costs. 

The fixed cost graph will always be a horizontal line at the point of the total expenses associated  with the production. The cost function will always start from the point of the fixed cost, unless they are zero. 

Variable costs are the full compensation for the employees wages and materials used to produces. They vary directly with the amount of buckets we make. For example: a lot more hours will be clocked & a lot of more materials such as eggs, milk, sugar etc  will be used  if we make 100 buckets of gelato compared with just 10 buckets. 

We need both fixed cost and variable cost to help us determine the average cost. To determine average cost we use the sum of fixed cost plus variable cost divided by number or gelato buckets produced. It is the formula we use to find out the unit cost at a certain production level. In other words it is the total cost to produce one unit divided by the number of good produced.  

Total Cost= Fixed Cost + Variable Cost

Average Cost= Total Cost/Quantity

We need to know the average cost in order to determine how to price our gelato buckets and how much money we need to produce them, among other reasons.

By visualizing the average total cost helps us understand more its importance. As the amount of units produced increase, the average total cost comes down for a while and then goes back up. At the lowest point of the cure is the area where the production is maximized to make the most profit for the company.

Now, lets put our new learned skills to practice.

Suppose our company’s fixed costs are $ 8,500 a month and variable costs of $15.

Please fill in the following table.

Recall

In order to find the average cost, we need to add fixed cost plus variable cost and divide it by the # of buckets produced. 

Check your work!

Let me know if you have any further questions on the subject or would like extra practice. 
Have a nice day!