Problems with parameters

Name: Problems with parameters
Rating: 5.0 (3 reviews)

Analytical, graphical and functional methods of solving problems with parameter

Created byAibulat Abdullin

Last updated 5/2023

English

What you'll learn

Understand basic mathematics deeper
Master solving all kinds of problems with parameters
Find interesting and complex problems to make your math lessons better
Challenge your mind with some of the most tricky and neat problems of basic mathematics

Course content

2 sections • 17 lectures • 59m total length

Introduction1:04
Meet a Russian mathematics teacher from Mipt who explains the unified state exam and its difficulty. Learn about his 100-point achievements and his math and technology passions.
What's this course about?0:30
Explore problems with parameters across equations, inequalities, systems, and function properties, and learn graphical, analytical, and functional methods to solve them.
What is a problem with parameters?7:57
Explore linear equations with a parameter, cases with one, no, or infinitely many solutions. For a ≠ −2, x = a + 2; for a = −2, any solution.
What will you learn?1:06
Solve problems with parameters to train your brain through three main approaches, choosing the best strategy from a toolbox of tools like mathematicians.

Linear equations with parameters3:25
Explore solving linear equations with parameters, including substitution and division caveats, critical values like a=0 and a=-2, and cases with no, infinite, or unique solutions.
Systems of linear equations4:48
Analyze a system of linear equations to find a values that yield no solutions; a = -4 gives no solution, a = -2 yields infinitely many, otherwise a unique solution.
Linear inequalities4:15
Analyze linear inequalities by three cases for a: positive, negative, and zero; note sign flips when dividing by negatives and the empty set for a=0; deduce a<2.
Quadratic equations5:58
Learn how the discriminant decides real solutions of quadratic equations with a parameter a, yielding two, one, or no solutions, and note the linear case when a equals zero.
Using Vieta's formulas4:09
Apply Vieta's formulas to a quadratic with parameters, form a root pair, and require positive discriminant to find parameter values, such as a = 4 and a = -3/8.
Signs of the roots2:40
Apply Vieta's formulas to determine when both roots are positive by ensuring the sum and product of the roots are positive, avoiding cumbersome discriminant inequalities.
Quadratic inequalities3:50
Learn to solve quadratic inequalities with a parameter by analyzing the parabola, using discriminant cases and a number line to determine the solution sets for a and x.
Conditions on a segment3:39
Explain quadratic inequalities with a parameter, requiring a solution segment longer than six; use discriminant and root difference to derive a less than minus one or a greater than three.
Nailing theorem2:13
Consider a quadratic ax^2+bx+c with a>0; the parabola opens upward, its vertex is at x=-b/(2a) and y=-D/(4a), and if f(t)<0 for some t, the nailing theorem guarantees two distinct roots.
Position of roots of quadratic equations 12:28
The quadratic in x opens upward since a^2 - a + 1 is positive for all a, and x = 1 yields f(1) < 0, implying two distinct roots.
Position of roots of quadratic equations 23:32
Use the nailing theorem to locate roots of a quadratic relative to 2, deriving a < -4, then for the threshold 1 obtain -1 < a < 0.
Position of roots of quadratic equations 35:53
Determine parameter values for which the quadratic’s roots exceed -1 or 1/2, using discriminant, vertex position, and unified inequalities; consider distinct and equal-root cases to obtain the range of a.
Roots in the interval1:59
Determine a for which x^2 + a x + 4 has roots in [1,3], using discriminant nonnegative and a vertex between 1 and 3, yielding a in -13/3 to -4.

Requirements

Anyone who knows basic mathematics can take this course. It doesn't require any special knowledge.

Description

This course will challenge your knowledge in high-school mathematics. You will learn new ways of solving problems with parameter and comparing ways of solving these problems and choosing the best one.

This course is first of all made for school and university teachers and technical university students, but anyone who wants to put their knowledge of basic mathematics in systematic order can take it.
The course covers three main ways of solving problems with parameter:

Analytical method.
You will learn about:
- linear equations and inequalities with parameter
- systems of linear equations and inequalities with parameter
- rational equations and inequalities with parameter
- quadratic equations and inequalities with parameter
- theorems about positions of the roots of quadratic equations and inequalities with parameter
- problems reducible to quadratic equations and inequalities with parameter
Graphical method.
You will learn about:
- graphs of linear, quadratic, rational, modular, irrational, exponential, logarithmic and trigonometric functions
- transformations of graphs of functions
- graphs of equations and inequalities
- using graphs as a visualisation tool to solve problems with parameter
Functional method.
You will learn about:
- monotonous functions
- limited functions
- even\odd functions
- using properties of functions to solve problems with parameter

In conclusion you will face some problems solvable in every mentioned way and learn to compare different ideas and choose the one that fits the best for every unique situation.

Who this course is for:

High school students, mathematics teachers, anyone curious about mathematics

Problems with parameters

What you'll learn

Explore related topics

Course content

Introduction4 lectures • 11min

Analytical solutions13 lectures • 49min

Requirements

Description

Who this course is for: