Extraneous solutions to radical equations

Extraneous solutions - a solution of a simplified  version of an equation that does not satisfy the original equation. Referred from  http://www.mathwords.com/e/extraneous_solution.htm


    Extraneous solutions to radical equations: Extraneous Solutions to Radical Equations

Dependent and independent variables

For p=5q

p is dependent on what happens to q so p is the dependent variable.


    Dependent and independent variables exercise: the basics: Here we have a problem that asks us to identify which variables are dependent and independent. Hint: independent variables are not influenced and remain unchanged by the other variable.

How many people can a blimp carry

Density = Mass /Volume so volume is proportional to mass.




    How many people can a blimp carry:

Surface Area

The problems for this topic may be quite hard to get through.

You may need to draw the object or the net of it.

 Write down the dimensions one by one if you are having difficulty, calculating the area first, then the number of sides.

Read the question carefully!

Here is one of the odd objects you may encounter
-tetrahedron

 


 

Triangle inequality theorem

The triangle inequality theorem states that any side of a triangle is always shorter than the sum of the other two sides.

By the same theorem, the third side must be also larger than the difference between the other two sides.

Recognizing functions (example 1)

If y has more than one value, then y is not a function of x.


    Recognizing functions (example 1):

Limits

From Khan Academy 

Evaluate limx→11−x√1−x using algebraic methods.

If we try substitution, we obtain the indeterminate form 

. 

The first solution requires rationalizing the numerator  

limx→11−x√1−x=limx→11−x√1−x⋅1+x√1+x√=

 The rest is not too difficult. Sometimes, have to factorise expressions like 25-x.
This becomes (5-x^0.5)(5+x^0.5). The above case can be done with this method.
1-x becomes (1-x^0.5)(1+x^0.5)