## IB Mathematics HL SL-Maxima and Minima

In my previous post, we discussed how to find the equation of tangents and normal to a curve. There are a few more  Applications of Derivatives in IB Mathematics HL SL, ‘Maxima and Minima’ is one of them.

Maxima and Minima:-

1. A function f(x) is said to have a maximum at x = a if f(a) is greater than every other value assumed by f(x) in the immediate neighbourhood of x = a. Symbolically

gives maxima for a sufficiently small positive h.

Similarly, a function f(x) is said to have a minimum value at x = b if f(b) is least than every other value assumed by f(x) in the immediate neighbourhood at x = b. Symbolically

If x = b gives minima for a sufficiently small positive h.

## Mathematics-

In my previous post, we discussed how to find the derivative of different types of functions as well as the geometrical meaning of differentiation. Here we are discussing  Applications of Derivatives in IB Mathematics
There are many different fields for the Applications of Derivatives. We shall discuss a few of them-

Slope and Equation of tangents to a curve- If We draw a tangent to a curve y=f(x) at a given point   , then

The gradient of the curve at given point=the gradient of the tangent line  at given  point

and we already discussed that slope or gradient of the tangent at given point

m=

=()

Finally to find the equation of tangent we use the slope-point form of equation

The major part of this concept is also discussed in the previous post. We should also remember following points while solving these types of questions.

(i) If two lines are parallel to each other, their slopes are always equal
i.e
(ii) If two lines are perpendicular to each other, the product of their  slopes is always -1

(iii) If a line is passing through two points    and    then, slope of the line

# Continuity of functions-

The word continuous means without any break or gap. Continuity of functions exists when our function is without any break or gap or jump . If there is any gap in the graph, the function is said to be discontinuous.

Graph of functions like sinx,cosx, secx, 1/x etc are continuous (without any gap) while greatest integer function has a break at every point(discontinuous).

1. A function f(x) is said to be continuous at x = c,  if  .

symbolically f is continuous at x = c if .

It should be noted that continuity of a function at x = a is meaningful only if the function is defined in the immediate neighborhood of x = a, not necessarily at x = a.

## How To Solve Limit Problems

In my previous post on limits, We have discussed some basic as well as advanced concepts of limits. Here we shall discuss different methods to solve limit questions. Based on the type of function, we can divide all our work into sections-:

Algebraic Limits- Problems of limits that involve algebraic functions are called algebraic limits. They can be further divided into following sections:-

Direct Substitution Method –Suppose we have to find.  we can directly substitute the value of the limit of the variable (i.e replace x=a) in the expression.

► If f(a) is finite then L=f(a)

► If f(a) is undefined then L doesn’t exist

► If f(a) is indeterminate  then this method fails

Example-1:- Find value of  (x²-5x+6) Read more

# Limit of a function

Limit of a function f(x) is said to exist as,  when

finite quantity.

Fundamental Theorems On Limits :

Let    &     If l & m exists then :

(i) f (x) ± g (x) = l ± m

(ii) f(x). g(x) = l. m

(iii)   provided

(iv)    where k is a constant.

(v)   provided f is continuous at        g (x) = m

Standard Limits :

(a)  and Where x is measured in radians

(b)  both are equal to e

(c) then this will show that

(d)  and   (a finite quantity) then

where z=

(e)  where a>0. In particular

Indeterminant Forms:

etc are considered to be indeterminant values

We cannot plot  on the paper. Infinityis a symbol & not a number. It does not obey the laws of elementary algebra.

+=

×

(a/) = 0 if a is finite v is not defined

a b =0,if & only if a = 0 or b = 0  and  a & b are finite.

Expansion of function like Binomial expansion, exponential & logarithmic expansion, expansion of sinx , cosx , tanx should be remembered by heart & are given below:

(i)  ex =1+x/1!+x3/3!+x4/4!……

(ii)  ax=1+(xloga)/1!+ (xloga)2/2!+ (xloga)3/3!+ (xloga)4/4!+……….where a > 0

(iii)   ln(1-x)=x-x2/2+x3/3-x4/4……….    where -1 < x  1

(iv)  ln(1-x)=-x-x2/2-x3/3-x4/4……….     where  -1 x < 1

(v )

(vi)

(v)

In next post, I will discuss various types of limit problems, their solutions and L’ Hospital’s rule.In the meantime, you can solve these basic questions from this PDF. This PDF is for beginners only. I will post difficult and higher level questions in the next post on this topic

# Maths tutor-

Maths tutor give great importance to Trigonometry.

Trigonometry is one of the fascinating branches of Mathematics. It deals with the relationships among the sides and angles of a triangle.Word trigonometry was originated from the Greek word, where, ‘TRI‘ means Three‘GON‘ means sides and the ‘METRON’ means to measure. It’s an ancient and probably most widely used branch Mathematics. For basic learning,

## Maths Tutor divide trigonometry in two part:-

1. Trigonometry based on right triangles

2. Trigonometry based on non-right triangles.

Here, we are discussing trigonometry based on non-right triangles only.

In the third article of this series, we will discuss problems based on complementary angles

In the third article of this series, we will discuss problems based on complementary angles

In this right triangle Sin A=BC/AC & Cos C=BC/AC   clearly: Sin A=Cos C  In the given triangle A+C=90° so we can write C=(90°-A). This gives us freedom to write Sin A=Cos (90°-A) similarly we can write these relationships     Read more

# Mathematics tutor

Mathematics tutor give great importance to Trigonometry

Trigonometry is one of the fascinating branches of Mathematics. It deals with the relationships between the sides and angles of a triangle. Word trigonometry was originated from the Greek word, where, ‘TRI‘ means Three‘GON‘ means sides and the ‘METRON’ means to measure. It’s an ancient and probably most widely used branch Mathematics.

For basic learning, Mathematics tutor divide trigonometry in two part:-

1 Trigonometry based on right triangles

1 Trigonometry based on non-right triangles.

In this post, we will discuss problems based on trigonometric ratios of a few specific angles like 0°,30°, 45°, 60° and 90°. Mathematics tutor use different tricks to form this table. I will discuss my tricks in a separate post

## How to solve basic problems in trigonometry?(concept-1)

### Mathematics tutors give great importance to Trigonometry

Trigonometry is one of the fascinating branches of Mathematics. It deals with the relationships among the sides and angles of a triangle.Word trigonometry was originated from the Greek word, where, ‘TRI‘ means Three‘GON‘ means sides and the ‘METRON’ means to measure. It’s an ancient and probably most widely used branch Mathematics. For basic learning, I am dividing trigonometry in two part:-

1 Trigonometry based on right triangles

1 Trigonometry based on non-right triangles.

In this post, I will only discuss Trigonometry based on right triangles.

In a right triangle, there are three sides hypotenuse (the longest side), adjacent side(base) and the opposite side(perpendicular).

Many IB mathematics tutors consider quadratic equations as a very important topic of ib maths. There are following ways to solve a quadratic equation

► Factorization method

►complete square method

► graphical method

Given equation: ax²+bx+c=0

Step-1: transfer constant term to the right side

ax²+bx=-c

Step-2: divide both sides by coefficient of x²

x²+bx/a=-c/a
Step-3: write (coefficient of x/2)²     that is (b/2a)²=b²/4a²
Step-4: Add this value to both sides
x²+bx/a+b²/4a² =-c/a²+b²/4a²
(x+b/2a)²=b²-4ac/4a²
now, take square root on both sides

x+b/2=±√b²-4ac/a²

x=-(b/2a)±√b²-4ac/2a

This formula is known as quadratic formula, we can put values of a, b and c  from any equation and find the value of x (the variable) by directly using this formula.

IB Mathematics tutors can also explain the concept of conjugate roots with the help of quadratic formula. In a quadratic equation,
ax²+bx+c=0
if a, b and c are all rational numbers and one root of the quadratic equation is a+√b then the second root will automatically become a-√b. that can be understood easily as we use one +ve and one -ve sign in quadratic formula.
These types of roots are called Conjugate Roots.
If  a & b  are  the  roots  of  the  quadratic  equation  ax² + bx + c = 0,  then;
(i)    (ii)       (iii)
Nature  Of  Roots:
(a) Consider the quadratic equation ax² + bx + c = 0  where a, b, c  R &  then
(i) D > 0   roots  are  real & distinct  (unequal).
(ii) D = 0  roots  are  real & coincident  (equal).
(iii) D < 0 roots  are  imaginary
(B) Consider the quadratic equation ax2+ bx + c = 0 where a, b, c  Q &  then
If  D > 0  &  is a perfect  square , then  roots  are  rational & unequal.
A quadratic  equation  whose  roots  are  a & b  is  (x – a)(x – b) = 0  i.e.   x2 – (a + b) x + a b = 0 i.e.
x2 – (sum of  roots) x +  product  of  roots = 0
Consider  the  quadratic  expression , y = ax² + bx + c  , a, b, c  R &   then
(i) The graph between x, y  is always a  parabola.  If a > 0  then the shape of the parabola is concave upwards &  if a < 0  then the shape of the parabola is concave downwards.
Common  Roots  Of  2  Quadratic  Equations  [Only  One  Common  Root]-   Let
be  the  common  root  of  ax² + bx + c = 0  &  a’x2 + b’x + c’ = 0
Therefore
If we solve above pair by cramer’s rule we get

This will give us

Every pair of the quadratic equation whose coefficients fulfils the above condition will have one root in common.

The condition that a quadratic function-
f(x , y) = ax² + 2 hxy + by² + 2 gx + 2 fy + c  may be  resolved  into  two  linear  factors  is  that      or

Reducible Quadratic Equations-These are the equations which are not quadratic in their initial condition but after some calculations, we can reduce them into quadratic equations
(i) If the power of the second term is exactly half to the power of the first term and the third term is a constant, these types of equations can be reduced to quadratic equations.
i.e.,  ,  ,  all these equations can easily be reduced into quadratic equations by applying the method of substitution.

Example-Solve this equation and find x

Ans:

let y=   then given equation will become

it’s a simple quadratic equation we can be easily factorised it and solve  so y=–3,2

so  =-3

the final equation can be solved using Quadratic formula and the same process can be repeated for  y=2

(ii) If a variable is added with its own reciprocal, then we get a quadratic equation i.e, all these equations can be reduced into quadratic by replacing one term by any other variable.

Standard Form of a Quadratic Function-A quadratic function y=ax2+ bx + c can be

reduced into standard form      by the method of completing the square. If we

draw the graph of this function we shall get a parabola with vertex (h,k). The parabola will be upward for a>0 and downward for a<0

Maximum and Minimum value of a quadratic function- If the function is in the form

Then ‘h’ is the input value of the function while ‘k’ is its output.

(i) If a>0 (in case of upward parabola) the minimum value of f is f(h)=k

(ii) If a<0 (in case of downward parabola)the maximum value of f is f(h)=k
If our function is in the form of  y=ax² + bx + c then vertex of the parabola
The line passing through vertex and parallel to the y-axis is called the axis of symmetry.
The parabolic graph of a quadratic function is symmetrical about axis of symmetry.

f(x) has a minimum value at vertex if a>0 and   at

f(x) has a maximum value at vertex if a<0 and     at

In the next post about quadratics, I shall discuss discriminant, nature of roots, relationships between the roots. In the meantime, you can download the pdf and solve practice questions.