Friday, May 12, 2017

Derivative computations

The formula lim f(x)-f(x+h)/h when  x→h that defines the derivative of a function f implies tedious calculations to calculate the derivative of some types of functions and combinations of functions..

Therefore some formulas have been established to determine the derivatives of a combination of functions and some specific types of functions.

The formulas for the constant function and the power functions are called respectively constant rule and power rule. The formulas for the sum, product and quotient of functions are called respectively addition rule, product rule and quotient rule. The derivative of a composition of 2 functions f and g is called the chain rule. It is an extension of the power rule The trigonometric, logarithmic and exponential functions have their specific formula.

The derivative of an implicit function is called implicit differentiation.

It is essential to memorize the formulas. Otherwise, it would be difficult to calculate the derivatives of these particular functions. Today we are going to limiting ourselves to the learning of the basic formulas: constant, power, sum, product and quotient rule.

Derivative of a constant

The derivative of the function constant is 0. If f(x) = c the derivative of f(x) is 0. We write:  f′(x) = 0.


The Power rule

The derivative of the function power defined by f(x) =  xn is equal to n multiplied by x to the power of n-1. The formula is .f’(x) = nxn-1

Derivative of the product of a constant by a function

The derivative of the product of a function by a constant is equal to the product of the constant by the derivative of the function power.

If  f(x) = axn its derivative is f’(x) = axn-1


Derivative of the function f(x) = x

The derivative of the function f(x) = x can be calculated using the formula for the derivative of the function power. In order to use this formula we have to write f(x) = x as the function power. We write f(x) = x as f(x) = x
By applying the formula for the function power we obtain f’(x) = x1-1 
 f’(x) = x0 ⇒ f’(x) = 1 

Derivative of a sum of functions

If f. g. h,;;; are differentiable for any value of x of their domain the derivative of the sum of these functions is f’+g’+h’+ ....

Derivative of the product of 2 functions

If f and g are differentiable for any value x of their domain the derivative of the product f.g is fg’+gf’

Derivative of the quotient of 2 functions

If f and g are differentiable for any value of their domain the derivative of the quotient f/g is (f∕g)’ = f’g-gf’∕g2

These formulas have to be demonstrated and the learners have to do some exercises to apply them. If anyone is interested in learning more subscribe to these courses via this link Free Introductory Calculus Course and Complete Calculus Course


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Tuesday, April 25, 2017

Introduction to the notion of derivative

In studying limit we observe what happens to the values of a function when the values of the independent variable become closer and closer to a certain value. If a function is defined for every value of its domain it is continuous there. Graphically it means that there is no hole, jump or infinite branch. Quantitatively the function has a value for every value of the independent variable that belongs to the domain of the function. In limit and continuity we have been observing some changes in the behavior of a function when the independent variable behaves in a certain way. A function might have a limit when the independent variable becomes closer and closer to a certain value. For other values of the independent variable the same function has no limit. The function is not continuous.

The notion of derivative allows us to study more systematically the notion of change in a function. It allows us to study the change at any value of a function. The slope of a function allows us to study the change in this function with respect to the change or the increase of the value of the independent variable. The slope of a line is the rate of change of the independent variable with respect to the change of the dependent variable. Since a non-linear function varies in different ways there is no precise method to define its slope. This leads to the study of the slope of a tangent line to a function.  In order to study the change of a function it is important to define the notion of rate of change or slope of a line. the slope of a secant line to a curve or average rate of change or speed and slope at a point of a curve or instantaneous rate of change.

Slope of a line

The notion of slope is familiar to the civil engineers when they build roads. They have to figure out what type of slope they have to give to a road especially when they build it on a hill or in mountains. They have to shape the road in the right slope because if the road is too steep the cars cannot climb it. The slope is calculated by taking the tangent of the angle opposed to the right angle in a right triangle where the hypotenuse is the side that is going to be inclined. The slope is the measure of the inclination.of the hypotenuse. Its measure is calculated by dividing the opposite side to the angle to the adjacent side :


The slope of a function is also called the rate of change of this function. The slope of a line is called the rate of change of this line. It is the rate of the increase of y to the increase of x. It is constant at any part of the graph. It can be positive, negative or equal to zero. The slope of a line is calculated by dividing the difference of the y-ordinates of two points of that line by the difference of the x-ordinates.




   Watch this video to get some understanding of the notion of slope:
     
Slope of a tangent line to a curve

Graph of the slope of a tangent line.png
We have a curve (C), a secant line (PQ) and a tangent line L to the curve at the point P. The problem is to find the slope of the tangent line at P. In order to do this we make the point Q become closer and closer to the point P. As the point Q becomes close to the point P the initial secant P occupies different positions. At each position the secant has a different slope, The slope of the tangent line is the limit of the slopes of the different positions of the secant (PQ). In order to come to this conclusion let's calculate the function that allows to find the slope of the secant line (PQ).



If you are interested in learning more about these concepts you can subscribe to this free Introductory Calculus course to start learning about limits and move on to this complete course Calculus AB

Monday, March 20, 2017

Properties.Methods and Procedures to calculate limits and continuity

Sometimes we seem lost through the details when studying a subject. However if we get the big picture it becomes easy to continue studying.  A math topic is structured in concepts, rules or properties and theorems. This is the theoretical part. Then come the applications. The theories are applied in the applications but the procedures and methods are mastered through practice. Knowing some key theories and procedures can help tremendously in the solutions of problems. In this post I will highlight the properties of limits and continuity, the methods and procedures to solve problems.

Properties of limits

The properties of limit show how to calculate the limits of a combination of functions like the sum, the difference, the multiplication and division of functions. It shows also how to calculate the square root of a function.


Properties of continuous functions


Methods for determining limits

There are three methods that can be used to determine a limit. These methods are: graph, table and algebra. The graph method consists in determining a limit from the graph. The table method consists in calculating the limit to the left and to the right by drawing a table for each one-sided limit. The table allows to see the behavior of the values of f(x) as x gets closer and closer to a fixed value. From there we can conclude if the limit to the right or to the left exists. If the limits from both sides exist and are equal then the limit of the function exists at the given value. The algebra method consists by substituting the value of the independent variable in the function.

Method for determining if a function is continuous

To determine if a function is continuous, we find out if it satisfies the three following conditions:
1) It is defined at a specified point "a"
2) The limit at the point "a" exists
3) The limit of the function at the point "a" is equal to f(a).

If you are interested in learning more about these concepts you can subscribe to this free Introductory Calculus course or this complete course Calculus AB

Saturday, March 11, 2017

Limits and Continuity vocabulary

These definitions can be best learned by watching some videos and observing the graphs of the functions. If you have learned the previous lessons there shouldn't be any problems mastering them

Limit

If the values of a function f approach a number L as the variable gets closer and closer to a number "a", then L is said to be the limit of the function f at the poin "a".

Two-sided limit

A two-sided limit is a limit where both the limit to the left and the limit to the right are equal

One-sided limit

A one-sided limit is a limit taken as independent variable approaches a specific value from one side (from the left or from the right).

Limit to the left

If the values of a function approach a number L as the independent variable gets closer and closer to a number "a"in the left direction. then the number L is said to be the limit of the function f to the left at the point "a"

Limit to the right

If the values of a function approach a number L as the independent variable gets closer and closer to a number in the right direction, then the number L is said to be the limit of the function f to the right at the point "a".  

Asymptote

An asymptote is a straight line to a curve such that as a point moves along an infinite branch of a curve the distance from the point to the line approaches zero as and the slope of the curve at the point approaches the slope of the line 

Vertical asymptote

A vertical asymptote is a vertical line to a curve such that as a point moves along an infinite branch of the curve the distance from the point to the line approaches zero

Horizontal asymptote

A horizontal asymptote is a horizontal line to a curve such that as a point moves along an infinite branch of the curve the distance from the point to the line approaches zero.  

End behavior

This is the behavior of the arm branches or infinite arm branches of a curve. In the case of a curve with a vertical asymptote the arm branch approaches the asymptote more and more.

Continuity of a function at a point

A function f is continuous at a point "a" if the limit of the function when x approaches "a" is equal to the value of the function at this point 

Continuity of a function on an interval

A function f is continuous on an interval if it is continuous at every point of the interval

Continuity of a function to the left at a point

A function f is continuous to the left at a point "a" if its limit to the left is equal to the value of the function at this point

Continuity to the right

A function f is continuous to the right at a point "a" if its limit to the right is equal to the value of the function at this function.

Continuous function

A continuous function is a function of which the graph can be drawn without lifting the pencil. Its graph has no hole, jump or asymptote. Algebraically a function f is continuous if for every value of its domain the limit exists.

Discontinuous function

A discontinuous function is a function of which the graph has hole, jump or asymptote. Algebraically a discontinuous function is either not defined at a point of its domain, doesn't have a limit at this point or the limit at this point is not equal to the value of the function at this point.

Removable discontinuity

Graphically a removable discontinuity is a hole in a graph or a point at which the graph is not connected there. The graph can be connected by filling in the single point.
Algebraically a removable discontinuity is one in which the limit of the function does not equal to the value of the function. This may be because the function does not exist at that point.

Non-removable discontinuity

A non-removable discontinuity is a point at which a function is not continuous or is undefined. and cannot be made continuous by giving a new value at the point. A vertical asymptote and a jump are examples of non-removable discontinuity.

Intermediate value theorem

If a function f is continuous over an interval [a b] and V any number between f(a) and f(b), then there is a number c between a and b such as f(c) = V (that is f is taking any number between f(a) and f(b)). We can deduce from this theorem that if f(a) and f(b) have opposite signs, there is a number c such as f(c) - 0. This can be used to find the roots of a function,

If you are interested in learning more about these concepts you can subscribe to this free Introductory Calculus course or this complete course Calculus AB

Saturday, January 28, 2017

Introduction to the notion of continuity of functions

In general, something that is continuous continues without interruption. If during a jogging you run from point A to B without stopping your running is continuous. However, if you stop even once the running is discontinuous. A line of cars in traffic that never stop is continuous. If the cars stop the line is discontinuous. If you draw a straight line without lifting your pencil the line is continuous. If you draw a straight line with dots you lift your pencil several times. The line is discontinuous at every dot.

The graph of the linear, parabolic, third-degree functions is an unbroken curve. It can be drawn without lifting the pencil from the paper. In general, the polynomial functions are continuous because their limit exists everywhere in the domain of the real numbers. A function of which the graph has holes, jumps or breaks is not continuous. Such functions are discontinuous.You have to lift your pencil to draw their graph.

Watch these videos to get an idea of what it means for a function to be continuous.




Interested about an Introduction to Calculus course take this one for free Introduction to Calculus or if you prefer take the complete Calculus course Calculus AB

Friday, December 2, 2016

Notions of limits (written lesson)

In the previous lesson on limits, I introduced the lesson by assigning some videos that you have to watch. Today we get to the written part of the lesson.

I am going to describe this lesson and give you the assignments that you should do.

Before I continue I have to tell you that your learning should not be limited to what your teacher gives you. There are plenty of resources that you can use to learn. You can learn from a teacher, from someone who knows a subject well and can teach it to others, from books, from electronic resources like CDs, from electronic communications like radio and televisions. More importantly today there is plenty of resources in the internet that you can use if you know how to access them. You should make yourself comfortable to all types of resources that you can use for learning. Videos are great to learn something but you can't limit yourself to this only. If you want to learn something deeply you have to get the written materials. The written materials allow you to get an overview of what you  are going to learn and give you also the content. You can choose which parts to learn first or which parts to drop depending on your interests. The most important thing also is you can review the materials as much as you can. If your reading skills are good you can learn a lot from written materials but for math there isn't a lot to read. You have to do the reading and memorize certain things. You have to practice a lot.

Here is the link for the lesson but before you start read the following;

Description of the lesson

This lesson starts by a definition of limits and shows you the three methods of limits using examples. The lesson ends by giving you some problems to do. Below I give you the readings that  you have to do under each sub-title and the tasks you have to do for each lesson.

Assignments

1. Objectives

You should start by reading the objectives again then read the definition of limits. The first video that you watched on the previous lesson with videos gave you verbally an idea of what a limit is. Now you are going to have a written idea of limit and three methods that allow you to calculate a limit.

2. The Idea.

You read the paragraph giving you an idea of what a limit is. You already have a video demonstration giving you the idea of a limit. Now you have a conceptual definition of a limit. You should try to state this definition either in your own words without compromising the concept or verbatim for more accuracy. Now that you have a definition of limit you are going to learn in written words how to find the limit of a function using three methods: graphtable, algebra.

3. Methods for determining limits

a) The graph method

Under this title you should see a problem named "Example 1". This problem asks you to find three limits using the graph on the right. This problem is already solved for you. You are going to do two things with this problem.

Task I 

Read the example and its solution. Read the explanations provided for the solution of the problem in case you don't understand it. Below is a guide to the explanations.

Explanation of the solution a) 

You should be able to understand the solution easily. I provide the explanations of the solution in case you don't understand it. I give you a method to understand the solution. It's graphic. You should read and do it.

Explanation of the solution of b) and c)

The same method is used for the solution of a) and b)

Explanation of the  solution of d)

You can use the same method for the solution of d) but this time notice that the function doesn't have a limit.

Task 2 

Do a pencil and paper to do the example yourself without referring to the solution. After you finish verify that your answers are correct. 

Task 3

Do Practice I.

b) Table method

In this method you are going to use two tables to find a limit. You start by giving x some values to the left of the given value of x and you group the values of x and f(x) in a table. You do the same thing to the right of the given value of x to have a second table. Even though I don't mention the tasks in the lesson you are going to do them in the same way you do for the previous problem.

Task I

Start by reading the problem they ask you to find the solution. Then read the solution. I didn't provided any explanation of the solution because the solution is explanatory by itself. Below is a guided explanation

Explanation

You start by giving x a value less than 0 and closer to 0. This value has to be to the left of 0. Then you calculate the value of f(x). You give to x a second value and closer to zero than the previous one You calculate the second value of f(x). You continue to give some values to x closer and closer to x and calculate the corresponding values of x. You do a table grouping all the values of x and f(x) in a table. When you look at the table you notice that f(x) gets close to 1 to the left as x gets closer and closer to 0 to the left.

Now you give x values to the right of 0 but closer to 0 and you calculate the corresponding values of f(x). You do a table grouping the values of x and f(x). When you look at the table you notice that as x gets closer and closer to to 0 to the right f(x) gets close to 1 to the right.

Since f(x) gets close to 1 as x gets closer and closer to 0 both to the left and right to 0 we conclude the limit of f(x) is 1 when x gets close to 0.

Task 2

Take a pencil and a piece of paper to do the problem by yourself.

Task 3

Do the practice problem

Algebra method

This method is very simple but it involves some calculations to do. In this method you substitute x in the function

Task 1

Read the problem first. Then write its solution. Below is a guided explanation.

Explanation

You substitute x in the function and you do the calculations to find f(x). The value of f(x) is the limit of the function

Task 2

Do the problem by yourself

Task 3

Do the practice problems

Review problems

Do the review problems involving the three methods.

Interested in learning more about limits get this free course Introduction to Calculus
You can also be enrolled in the complete Calculus course



Friday, November 4, 2016

How to learn a subject deeply

Learning a subject deeply means you know its theories and are able to apply it. Very often people learn a subject because they are required to without knowing its applications or if they would ever apply it. People learn practical subjects and are not able to apply them. These subjects require practice. But when you learn a subject deeply its practice becomes easy.

To learn a subject deeply requires to know "how to learn". You start by learning the concepts or key words in the subject. Sometimes there are words that are not known or are not well understood. Having a clear definition of these words helps to learn the subject deeply. Besides knowing key vocabulary it is necessary to master the theories. It is also important to have a clear understanding of the concepts of the subject. This can be done by having a clear mental picture of these concepts in one's mind. If it's not possible to imagine the concepts one can try to represent them by a visual representation. In order to learn a subject deeply it has to be absorbed gradually. so that the previous concepts can be applied to the following ones.

Deeper learning is the ability to apply knowledge to new situations. Deeper learning is associated  with better life and work outcomes according to a 2012 report.

Superficial  learning is associated with poor performance. On the 2012 Program for International Student Assessment (PISA), a test that measures students' abilities to apply their knowledge to real-world problems U.S fifteen years old scored 26th of the 34 industrialized nations in mathematics.

Schools that practice deeper learning have their students graduated and attended college at higher rates than schools that don't use deeper learning.

Students who practice "deeper learning" take responsibility for their learning. In "deeper learning" students master their subjects deeply. They know the concepts, can apply them and reflect deeply on the subject.

Deeper learning is defined by 6 competencies: mastering content, critical thinking, effective written and oral communication, collaboration, learning how to learn and developing academic mindsets.

Deeper learning is associated with practice and reflection. In practical subjects learners can build things. Imagination, intuition and inspiration are some of the characteristics of deeper learning.

Deeper learners cultivate academic mindsets. They make the most out of their learning experiences. They hold the following key beliefs:
"I can change my intelligence and abilities through effort"
"I can succeed"
"This work has value and purpose for me"

Beliefs and learning skills bring success for learners.

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