ENGLISH MATHS

WELCOME TO ENGLISH MATHS

Dr. Nuria Serichol Augué Year 2023-2024

INDEX

1. Real Numbers

2. Algebra

3. Elementary Functions

4. Exponential, Logarithmic and Trigonometric Functions

5. Function limit. Continuity

6. Derivatives

7. Bidimensional Distributions and Probability

REAL NUMBERS

Classify the following numbers, small test:

INTERVALS

OPEN INTERVAL

CLOSED INTERVAL

SEMI-OPEN INTERVAL

OPEN INTERVAL

ON THE LEFT

ON THE RIGHT

REAL HALF-LINES

POSITIVE NUMBERS

NEGATIVE NUMBERS

to begin with

short summary with examples !!!

TEST - WHAT HAVE I LEARNED

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RADICALS

Properties:

EXERCISES

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RATIONALISATION

The process used to eliminate a radical or imaginary number from the denominator of an algebraic fraction. That is, remove the radicals in a fraction so that the denominator only contains a rational number.

EXERCISES

Simplify the following completely:

LOGARITHMS

Properties:

different ways of saying the same thing

EXERCISES

webgraphy

Google images youtube.com study.comhttps://thenounproject.com/search/icons/ https://www.apuntesmareaverde.org.es/grupos/mat/Bachillerato/BS1%2001%20Reales.pdf https://www.math10.com/tests/classify-numbers-natural-rational-real-irrational.html https://math.libretexts.org/Bookshelves/Algebra/Advanced_Algebra/05%3A_Radical_Functions_and_Equations/5.0E%3A_5.E%3A_Radical_Functions_and_Equations_(Exercises) https://www.shmoop.com/radicals-powers-roots/rational-denominator-exercises.html# http://www.wallace.ccfaculty.org/book/8.5%20Rationalize%20Practice.pdf https://math.libretexts.org/Courses/Monroe_Community_College/MTH_165_College_Algebra_MTH_175_Precalculus/04%3A_Exponential_and_Logarithmic_Functions/4.05%3A_Logarithmic_Properties/4.5e%3A_Exercises_-_Properties_of_Logarithms

ALGEBRA

Types of equations to solve

biquadratic equation

Ruffini method

used to solve third-degree or higher equations

how Ruffini's method works

Factorize the following polynomials:

watch the video

algebraic fractions

Factor and simplify:

* the easiest ones:

MORE EXERCISES

Simplify/reduce the algebraic fractions:

REMEMBER !!!

watch the video

radical equations

exponential equations

watch the video

logarithmic equations

watch the video

EXERCISES

EXERCISES

webgraphy

Google images youtube.com study.comhttps://www.mathsisfun.com/algebra/index.html https://www.siyavula.com/read/za/mathematics/grade-11/equations-and-inequalities https://ekuatio.com/ruffinis-rule/ http://ionma.org/share/maths/maths3c/ruffinis.pdf https://youtu.be/lGsw3cH0xl8 https://mathspace.co/textbooks/syllabuses/Syllabus-461/topics/Topic-8661/subtopics/Subtopic-114841/ http://maths.mq.edu.au/numeracy/web_mums/module2/Worksheet23/module2.pdf https://corbettmaths.files.wordpress.com/2013/02/algebraic-fractions.pdf https://www.math-only-math.com/solving-algebraic-fractions.html youtube.com https://mcckc.edu/tutoring/docs/blue-river/math/equat_inequ/Solving_Radical_Equations.pdf https://www.sd27j.org/cms/lib8/CO01900701/Centricity/Domain/572/3.1%20Practice%20Solving%20Exponential%20Equations%20no%20logs.pdf https://math.colorado.edu/math1300/resources/Exercises_LogarithmicFunction.pdf https://www.lmtsd.org/cms/lib/PA01000427/Centricity/Domain/621/Practice%20Test%201%20-%20Systems%20of%20Inequalities%20-%20Copy.pdf https://www.ranchorams.org/ourpages/auto/2017/1/23/63918681/01-10%20Intro%20to%20Systems%20of%20Inequalities.pdf

ELEMENTARY FUNCTIONS

"f(x) = ... " is the classic way of writing a function. And there are other ways, as you will see!

TEST - WHAT HAVE I LEARNED

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Types of functions

A linear function is of the form f(x) = mx + b where 'm' and 'b' are real numbers.

A quadratic function is a polynomial function with one or more variables in which the highest exponent of the variable is two.

The definition of inverse proportion states that "Two quantities are said to be in inverse proportion if an increase in one leads to a decrease in the other quantity and a decrease in one leads to an increase in the other quantity".

A radical function is any function that is defined in a root. This function also contains a square root, cubed roots, or any of the nth root.

An absolute value function is an important function in algebra that consists of the variable in the absolute value bars.

A piecewise function is a function with multiple pieces of curves in its graph. It means it has different definitions depending upon the value of the input. i.e., a piecewise function behaves differently for different inputs.

EXERCISES

Associate the graphs with the equations above

inverse proportion

linear

quadratic

radical

Associate the graphs with the equations above

Obtain the analytical expressions of the following piecewise functions

Function transformation

Domain

check your knowledge

Find the domain of the following functions:

and some other combinations that will be written on the blackboard . . .

with square roots, logarithms, like this one

EXTRA WORK

webgraphy

Google images https://www.mathsisfun.com/sets/function.html https://www.mathsisfun.com/sets/set-builder-notation.html https://www.highermathematics.co.uk/domain-restrictions/ https://www.math-exercises.com/analysis-of-functions/domain-of-a-function https://www.cuemath.com/calculus/linear-functions/ https://www.cuemath.com/calculus/quadratic-functions/ https://www.cuemath.com/commercial-math/inverse-proportion/ https://www.cuemath.com/algebra/absolute-value-function/ https://www.cuemath.com/calculus/piecewise-function/ https://www.cuemath.com/calculus/transformation-of-functions/ https://www.math-exercises.com/analysis-of-functions

EXPONENTIAL, LOGARITHMIC AND TRIGONOMETRIC FUNCTIONS

The composition of functions is the process of combining two or more functions into a single function.

Reciprocal function

Find the reciprocal function of y = x + 3 ; (f(x)=y)

follow the next steps: 1. Replace y with x and viceversa: x = y + 3 2. Solve for y: y = x - 3 3. Rename y as

= x - 3

EXTRA WORK

Exponential function, as its name suggests, involves exponents. But note that, an exponential function has a constant as its base and a variable as its exponent but not the other way round.

exponential growth

exponential decay

Logarithmic function

properties

Trigonometric functions are the basic six functions that have a domain input value as an angle of a right triangle, and a numeric answer as the range.

EXERCISES

hint

where

is the initial capital, k is the % and t is the years of investment

hint

webgraphy

Google images https://www.highermathematics.co.uk/wp-content/uploads/2023/06/Plotting-Exponential-Functions-Worksheet-A-A4.pdf https://www.cuemath.com/calculus/composite-funtions/ https://www.cuemath.com/calculus/reciprocal-function/ https://www.cuemath.com/calculus/exponential-functions/ https://www.cuemath.com/algebra/logarithmic-functions/ https://www.cuemath.com/trigonometry/trigonometric-functions/ https://www.humbleisd.net/cms/lib/TX01001414/Centricity/Domain/3205/Section%204.1%20Exercises.pdf https://www.brainkart.com/article/Exercise-1-5--Composition-of-Functions_39301/ https://math.libretexts.org/Courses/Monroe_Community_College/MTH_165_College_Algebra_MTH_175_Precalculus/04%3A_Exponential_and_Logarithmic_Functions/4.04%3A_Graphs_of_Logarithmic_Functions/4.4e%3A_Exercises_-_Graphs_of_Logarithmic_Functions https://rhay.weebly.com/uploads/8/7/3/2/8732474/pc12_sol_c07_7-9.pdf https://www.alamo.edu/contentassets/35e1aad11a064ee2ae161ba2ae3b2559/trigonometric/math2412-review-exercise-set-3.pdf https://www.math-exercises.com/analysis-of-functions

FUNCTION LIMIT. CONTINUITY

Limits are all about approaching. Sometimes you can't work something out directly, but you can see what it should be as you get closer and closer!

TEST - WHAT HAVE I LEARNED

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EXERCISES

A continuous function, as its name suggests, is a function whose graph is continuous without any breaks or jumps.

EXERCISES

An asymptote is a line that a function approaches but never reaches or crosses.

A vertical asymptote is a vertical line at the x value for which the denominator will equal to zero.

A hole or removable discontinuity is also an x value for which a function f(x) does not exist, but it is not a vertical line. It is a point (x, y) that has been 'removed' by canceling the same factor from the numerator and the denominator of a rational function! Sort of like a black hole!

watch the video

There are some x values that do not exist for rational functions. Think about it, when you divide by x, you already know that x cannot equal zero. What about possible y values that may or may not exist. Horizontal asymptotes are horizontal lines that discontinue the function on the y-values.

Oblique asymptotes exist if the degree of the polynomial in the numerator is one degree higher than the denominator. You can obtain the equation for the slant asymptote by performing polynomial division (synthetic or long division). If you divide the numerator by the denominator, the quotient is the linear equation of the slant asymptote.

TEST - WHAT HAVE I LEARNED

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EXERCISES

webgraphy

Google images youtube.com https://www.mathsisfun.com/calculus/index.html https://www.mathsisfun.com/algebra/asymptote.html https://www.cuemath.com/calculus/continuous-function/ https://www.math-exercises.com/limits-derivatives-integrals https://www.math-exercises.com/analysis-of-functions/continuity-of-a-function https://study.com/academy/lesson/graphing-functions-with-asymptotes.html https://www.math-exercises.com/analysis-of-functions/asymptotes-of-a-function

DERIVATIVES

The average rate of change of a function f(x) over an interval [a, b] is defined as the ratio of "change in the function values" to the "change in the endpoints of the interval".

Derivative rules

TEST - WHAT HAVE I LEARNED

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For practice:

Applications of derivatives

EXTRA WORK

Geogebra

TEST - WHAT HAVE I LEARNED

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EXTRA WORK

An Inflection Point is where a curve changes from Concave upward to Concave downward (or vice versa)

The tangent line of a curve at a given point is a line that just touches the curve (function) at that point.

equations:

The normal line is te line that goes through the point (x,y) and is perpendicular to the tangent line.

Practice Questions on Tangent Line:

EXTRA WORK

Mathematical optimization deals with the problem of finding numerically minimums (or maximums or zeros) of a function. In this context, the function is called cost function, or objective function, or energy.

Graphical representation of data is an attractive method of showcasing numerical data that help in analyzing and representing quantitative data visually. A graph is a kind of a chart where data are plotted as variables across the coordinate. It became easy to analyze the extent of change of one variable based on the change of other variables.

1. Domain
2. Parity, odd parity, periodicity
3. The intersection with the coordinate axes
4. The derivative and critical points
5. Intervals of increase, decrease, and extreme points (and the value of the function at these points)
6. The behavior of the functions at the ends of the scope and the asymptotes of the graph of a function (vertical, horizontal and inclined)
7. The second derivative and the study of the function
8. If you want to find control points specifying the behavior of the graph
9. On the basis of the study building sketch the graph of a function

Study of a function:

EXERCISES

webgraphy

Google images https://www.mathsisfun.com/calculus/index.html https://www.mathsisfun.com/calculus/derivatives-dy-dx.html https://www.mathsisfun.com/calculus/maxima-minima.html https://www.mathsisfun.com/calculus/concave-up-down-convex.html https://www.cuemath.com/average-rate-of-change-formula/ https://www.cuemath.com/calculus/tangent-line/ https://www.cuemath.com/geometry/tangents-and-normals/ https://www.cuemath.com/data/graphical-representation-of-data/ https://www.math-exercises.com/limits-derivatives-integrals/derivative-of-a-function https://www.superprof.co.uk/resources/academic/maths/calculus/derivatives/tangent-problems.html#chapter_exercise-1 https://www.superprof.co.uk/resources/academic/maths/calculus/functions/optimization-word-problems.html https://scipy-lectures.org/advanced/mathematical_optimization/ https://cubens.com/en/handbook/algebra-and-introduction-to-mathematical-analysis/investigation-of-a-function https://www.math-exercises.com/analysis-of-functions https://www.geogebra.org/classic?lang=es

BIDIMENSIONAL DISTRIBUTIONS AND PROBABILITY

Graphical representation of data is an attractive method of showcasing numerical data that help in analyzing and representing quantitative data visually. A graph is a kind of a chart where data are plotted as variables across the coordinate. It became easy to analyze the extent of change of one variable based on the change of other variables. Graphical representation of data is done through different mediums such as lines, plots, diagrams . . .Graphical representation helps to qualify, sort, and present data in a method that is simple to understand for a larger audience. Graphs enable in studying the cause and effect relationship between two variables through both time series and frequency distribution. Quantitative data is more structured, continuous, and discrete with statistical data whereas qualitative is unstructured where the data cannot be analyzed.

Rules of Graphical Representation of Data

Pros and Cons

• bar chart
• pie chart
• line graph - frequency polygon
• pictograph
• histogram

Graphical Representation

Mean, median, and mode are the three measures of central tendency in statistics. We identify the central position of any data set while describing a set of data. This is known as the measure of central tendency.

Mean: The mean is also known as the average, and it is calculated by adding up all the values in a data set and dividing by the total number of values. Median: The median is the middle value of a data set, which separates the highest and lowest values equally. It is calculated by arranging the data set in order from lowest to highest and finding the value in the exact middle. Mode: The mode is the value that appears most frequently in a data set.

Review the basics with the following activities:

Copy these data on a spreadsheet, sort them in a table, find out the main statistical parameters and make an appropiate chart: 1. Siblings: 3, 1, 2, 2, 1, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 1, 2, 3, 2, 2, 3, 1, 4, 2, 3, 2, 1, 1 2. Eating fish per week: 0, 2, 2, 0, 2, 2, 2, 1, 4, 1, 2, 2, 1, 2, 2, 1, 3, 2, 2, 1, 6, 1, 7, 2, 2, 2, 7, 2, 2 3. Gums under your desk: 3909, 4, 4, 15, 3, 4, 1, 10, 25, 6, 7, 5, 5, 12, 1, 1, 0, 6, 2, 4, 2, 2, 3, 6, 14, 4, 4, 4 4. Followers in instagram: 350, 284, 645, 607, 1450, 258, 1008, 152, 1220, 978, 294, 1680, 344, 953, 903, 537, 760, 137, 540, 43, 820, 611, 306, 379, 924, 674, 0, 284, 258

BIDIMENSIONAL DISTRIBUTIONS Two-way tables. Scatter plots. Correlation: the relation between two variables. The strength of that relation can be measured by some coefficients.

VOCABULARY:

• Population and samples
• Simple random sampling
• Statistical variable
• Collecting data
• Surveys
• Tallys
• Processing data
• Tables
• Frequencies
• Parameters: arithmetic mean (average), median, mode, standard deviation

Range in Statistics

The Interquartile range formula helps in finding the difference between the third quartile and the first quartile. The Interquartile range formula measures the variability, based on dividing an ordered set of data into quartiles. Quartiles are three values or cuts that divide each respective part as the first, second, and third quartiles, denoted by Q 1 , Q 2 , and Q 3 , respectively. Q 1 is the cut in the first half of the rank-ordered data set Q 2 is the median value of the set Q 3 is the cut in the second half of the rank-ordered data set.

The percentile formula determines the performance of a person over others. A percentile is a number that tells the percentage of scores that fall below the given number.

To calculate the percentile, here are a few steps to use the percentile formula:

Step 1: Collect the data set Step 2: Arrange the data set in ascending order Step 3: Determine the total number of observations Step 4: Identify the data value for which you are interested to find the percentile Step 5: Count the number of data values that are less than the above value Step 6: Divide the number from Step 5 by the number from Step 3 to find the percentile of the given data value

Standard deviation is the positive square root of the variance. It is one of the basic methods of statistical analysis. Standard deviation is the degree of dispersion or the scatter of the data points relative to its mean, in descriptive statistics. It tells how the values are spread across the data sample and it is the measure of the variation of the data points from the mean.

TEST - WHAT HAVE I LEARNED

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A regression line indicates a linear relationship between the dependent variables on the y-axis and the independent variables on the x-axis. The correlation is established by analyzing the data pattern formed by the variables.

watch the video

example:

Try to solve this example by yourself:

Probability

A tree diagram is a new management planning tool that depicts the hierarchy of tasks and subtasks needed to complete and objective. The tree diagram starts with one item that branches into two or more, each of which branch into two or more, and so on.

example:

Try to solve the examples by yourself:

https://www.mathsisfun.com/probability_line.html

TEST - WHAT HAVE I LEARNED

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Events can be:

• Independent (each event is not affected by other events) (coins ...)

• Dependent (also called "Conditional", where an event is affected by other events) (cards ...)

• Mutually Exclusive (events can't happen at the same time) (left/right ...)

TEST - WHAT HAVE I LEARNED

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https://www.mathsisfun.com/data/index.html#probability binomial and normal distribution

https://www.mathsisfun.com/activity/index.html examples interessants

webgraphy

Google images youtube.com https://www.cuemath.com/data/graphical-representation-of-data/ https://www.cuemath.com/data/range-in-statistics/ https://www.cuemath.com/interquartile-range-formula/ https://www.cuemath.com/percentile-formula/ https://www.cuemath.com/data/standard-deviation/ https://www.cuemath.com/data/probability-tree-diagram/ https://www.mathsisfun.com/data/standard-deviation.html https://www.mathsisfun.com/data/least-squares-regression.html https://www.mathsisfun.com/data/index.html#stats https://www.mathsisfun.com/data/probability.html https://www.mathsisfun.com/data/probability-events-types.html https://www.mathsisfun.com/data/probability-events-independent.html https://www.mathsisfun.com/data/index.html#probability binomial and normal distribution ??? https://sites.google.com/site/matesfrayluis/1%C2%BA-bach/u13-statistics-bidimensional-distribution ??? https://www.wallstreetmojo.com/regression-line/