Solving Monotonicity and Concavity

A function is said to be rising on an interval [a, b] = I if f (z1) < f (z2) whenever z1 < z2 for z1, z2 `in` I. A function is said to be decreasing on an interval [a, b] = I if f (z1) > f (z2) whenever z1 < z2 for z1, z2 `in` I. A function which is strictly increasing or strictly decreasing on an interval is said to be monotonic on that interval. A function is said to be curved in up on an interval [a, b] if f ’’ (z) > 0 on [a, b]. A function is said to be curved in down on an interval [a, b] if f ’’ (z) < 0 on [a, b]. A point (c, f(c)) on the graph of y = f (z) is called an inflection point if the concavity changes at the point. (I.E., it is curved in up on some interval (a, c) and curved in down on some interval (c, b) or vice versa.)

Solving monotonicity and concavity - Examples:

Solving monotonicity and concavity - Example 1:

Find inflection points & determine concavity for f (x)

f ‘(x) = `(x^5)/20 + (x^4)/12 - (3x^3)/3 - 10`

f ’(x) = `(x^4)/4 + (x^3)/3 - 3(x^2)`

f ‘’ (x) = x3 + x2 – 3x

= x(x2 + x - 3)

f ‘‘(x) = 0

x = 0, -2.3, 1.3

Inflection pts: x= -2.3, 0, 1.3

Curved in up: (-2.3,0), (1.3,infinity)

Curved in down: (-oo,-2.3), (0,1.3)

Solving monotonicity and concavity

Solving monotonicity and concavity - More Examples:

Solving monotonicity and concavity - Example 1:

Find inflection points & determine concavity for f (x)

f (x) = `(x^5)/20 + (x^4)/12 - (x^3)/3 +10`

f ‘ (x) = `(x^4)/4 + (x^3)/3 - x^2`

f ‘’ (x) = x3 + x2 – 2x = x(x2 + x - 2)

= x (x+2) (x-1)

f ‘‘ (x) = 0

x = 0, -2, 1

f ’’(-5) < 0, f’’(-1) > 0, f ’’(.5) < 0, f ’’(10) > 0

Inflection pts: x=-2,0,1

Curved in up: (-2,0), (1,`oo` )

Curved in down: (`-oo` .,-2), (0,1)

Solving monotonicity and concavity

Trinomial Multiplication

In algebraic expression consisting of only one term is called a monomial, two terms is called a binomial and when there are three terms separated by an addition or a subtraction operation is called a trinomial. ‘Tri’ in the word trinomial means three and hence the name. It is also referred to as a polynomial as ‘poly’ means more than two terms. For example, a+3b-c consists of three terms a, b and c and hence a trinomial. In 3b, 3 is called the coefficient of b. In a given polynomial we come across like terms and unlike terms.

Like terms are the terms which have the same variable or literal but a different coefficient. For example, (3b, -7b);(5x2, 12x2);(-4xy, 7xy) are some of the like terms. Unlike terms as the name suggests are the terms which have different variables. For example, (7x, 8y); (4ab, -4ac); (2y2, -2x2) etc are some of the unlike terms.

It is not necessary that there would be only two like or unlike terms, it depends on the number of terms in the given polynomial. Now let us learn as to How do you multiply trinomials. There are two methods in which we can multiply trinomials, one is the horizontal method the other is the vertical method.

The steps to be followed in a Horizontal method of multiplication are as follows:
Example: Multiply (4x2-3x+5)(x2+5x-3)
Solution: First and second terms are chosen irrespective of the order
Let (4x2-3x+5) be the first term and (x2+5x-3) be the second term

Arrange the two polynomials horizontally
(4x2-3x+5) X (x2+5x-3)
Now distribute each of the terms of the first trinomial with each of the terms of the second trinomial
=4x2(x2+5x-3)- 3x(x2+5x-3) +5(x2+5x-3)
=4x4+20x3-12x2 - 3x3-15x2+9x +5x2+25x-15

Combine the like terms and simplify
=4x4+x3(20-3)+x2(-12-15+5)+ x(9+25) – 15
= 4x4+17x3-22x2+34x-15

The steps to be followed in a Vertical method of multiplication are as follows:
Example: Multiply (4x2-3x+5)(x2+5x-3)
The first and the second terms are chosen irrespective of the order
Let (4x2-3x+5) be the first term and (x2+5x-3) be the second term
Arrange the two polynomials in a vertical form
4x2-3x+5
X  x2+5x-3

Working from right towards left each term of the lower trinomial is multiplied with each of the terms of the upper trinomials. Then the products are written underneath the second trinomial in three rows in the order of the degree of each term and then the like terms simplified as shown below
4x2-3x+5
X  x2+5x-3
-12x2+ 9x – 15
+20x3-15x2 + 25x
4x4 - 3x3  + 5x2                      .
4x4+17x3-22x2+ 34x-15 is the final product!

Solve Explicit Differentiation

In calculus,Explicit is a function which the independent variable. The function f explicitly is to provide a preparation for determining the output of the given function y in terms of the input value x: y = f(x). Derivative of an explicit function is called as explicit differentiation. for example  y = x3 + 5. The process of finding the differentiation of the independent variable in an explicit function by differentiating each term separately, by expressing the derivative of the independent variable as a symbol, and by solving the resulting expression for the symbol.

Solve explicit differentiation problems:

Let us see some problems and its helps to solve an explicit differentiation.

Solve explicit differentiation problem 1:

Find the differentiation of given explicit function   y = x2 - 15x + 3.

Solution:

Given explicit function is  y = x2 - 15x + 3.

Differentiation of explicit function is   dy/dx  = d/dx ( x2 - 15x + 3)

Separate the each term, so, we get

= d/dx (x2) - d/dx (15x) + d/dx (3)

= d/dx (x2) - 15d/dx(x) + d/dx (3).

= 2 x(2-1) - 15 + 0

= 2x - 15

The differentiation of an explicit function is  2x - 15

Solve explicit differentiation problem 2:

Find the differentiation of an explicit function  x2 +  cot x = - y

Solution:

Given explicit function is  x2 +  cot x = - y

Multiply by (-1) on both sides,  y = - x2 -  cot x

Differentiation of an explicit function,

y =   - x2 -  cot x

dy/dx = d/dx -x2   - d/dx (cot x) .

=  - 2x - (-cosec2 x) .

= - 2x + cosec2x

The Differentiation of an explicit function is  - 2x + cosec2x

Solve explicit differentiation problem 3:

Find the differentiation of given explicit function  2x2 + y2 = 1

Solution:

Given explicit function is  2x2 + y2 = 1

Subtract 2x2 on both sides.we get,

2x2 + y2 - 2x2 = 1 - 2x2

y 2 = 1 - 2x2

Take square root on both sides, we get

 sqrt (y^2) =  +- sqrt (1 - 2x^2)

y = +- sqrt (1 - 2x^2) .

Differentiate the function,  Let u = 1 - 2x2            and           y = sqrt u

(du)/(dx) = - 4x                              dy/(du) = 1/(2sqrtu)

So,   dy/dx =  ((dy)/(du)) ((du)/(dx)) .

=  1/(2sqrtu) . (-4x )

=  (- 2x) / sqrtu .

Substitute u = y, So we get

= - (2x)/y   .

The differentiation of an explicit function is  - (2x)/y    .

Substitution Geometry

Geometry is a module of mathematics, which involves the study of shapes, line equation, angles problem, dimensions, relative position of figures etc.  The term ‘Geometry’ means study of properties. A point is used to represent a position in space. A plane to be a surface extending infinitely in all directions such that all points lying on the line joining any two points on the surface. Substitution geometry problems are given below.

Example problems for substitution geometry :

1. Find out the geometry equation of straight line passing through the points 2x + y = 8 and 3x - 2y + 7 = 0 and parallel to 4x+ y - 11 = 0
Solution:
Let (x1, y1) be the intersection lines
2x1 +  y1 =  8       …  (1)
3x1 - 2y1 = - 7   …   (2)


(1) × 2 ? 4x1 + 2y1 = 16      …    (3)
(2) + (3) ? x1 =9/7 `=>` y1 =38/7   (x1, y1) =( 9/7 ,38/7)


The straight line parallel to 4x + y - 11 = 0 is of the form 4x + y + k = 0
But it passes through (9/7 ,38/7)


36/7 +38/7 + k = 0 ? k = -74/7
4x + y -74/7 = 0
28x + 7y - 74 = 0 this is the equation of straight line.


2. For what values of ‘a’, the three straight lines 3x + y + 2 = 0, 2x - y + 3 = 0and x + a y - 3 = 0 are concurrent?

Solution:

Let (x1, y1) be the point of concurrency. This point satisfies the first two equations.
3x1 + y1 + 2 = 0 … (1)
2x1 - y1 + 3 = 0 … (2)

Solving (1) and (2) By using substitution method, we get (- 1, 1) as the point of intersection. Since it is a point of concurrency, it lies on x + a y - 3 =0
- 1 + a - 3 = 0
a -4 = 0

a = 4

Practice problems for substitution geometry:

1. Find the point of intersection of the straight lines 5x + 4y - 13 = 0 and 3x + y - 5 = 0

Ans: The point of intersection is (1, 2)

2. Find the geometry equation of straight- line perpendicular to the straight line 3x + 4y + 28 = 0 and passing through the  point (- 1, 4).

Ans: 4x - 3y + 16 = 0

3. Find the equation of straight line passing through the intersection of straight lines 2x + y = 8 and 3x - y = 2 and through the point (2, - 3).

Ans: x = 2

Proportionality Problems

In mathematics, proportionality indicates that two variables are related in a linear manner. If one number doubles in size, so does the other; if one of the variables diminishes to 1/10 of its former value, so does the other.The symbol for proportionality resemble a stretched-out, lowercase Greek letter alpha . When this symbol appear that two quantities or variables, it is read "is proportional to" or "varies in direct proportion with." Thus, the expression x alpha y is read " x is proportional to y " or " x varies in direct proportion with y ." In this condition, as long as x and y do not attain values of zero, the quotient x / y is always equal to the same value k , which is called the proportionality constant. (source: wiki)

Example proportionality problems:

Proportionality problem 1:

Find a if a/4= 3/2.

Proportionality  solution:

By using property 1:

a/4=3/2

(a) (2) = (4) (3)

2a = 12

Dividing both side by 2

a=6

Therefore the value of a=6

Proportionality problem 2:

Is 6: 4 = 3: 2 a proportion?

No. If this were a proportion, Property 1 would produce

(6) (2) = (4) (3)

12   = 12, this is true. It is a proportion.

Practice proportionality problems:

Practice proportionality problem 1:

Janette's car uses 9 gallons to go 200 miles.

a) How many gallons will she use to go 400 miles?

B) How many gallons will she use to go 600 miles?

c) How far can she drive with 36 gallons?

(a)18

(b) 27

(c)800 miles

Practice proportionality problem 2:

Carlos makes $25 in 4 hours.

a) How much will he make in 2 hours?

b) How much will he make in 20 hours?

c) How much will he make in 22 hours?

(a)$12.50

(b)$125

(c)$137.50


practice proportionality problem 3:

Which is a better value?

a) 15 ounces for $ 9.25 or 30 ounces for $18.00?

b) 15 ounces for $ 9.25 or 5 ounces for $3.29?

Answer:  ( a )30 ounces for $18.

( b )15 ounces for $9.25.

practice proportionality problem 4:

6 bottles cost $7.00.

a) How much will 18 bottles cost?

b) How much will 15 bottles cost?

c) How much will 27 bottles cost?

d) How many bottles can you buy with $10.50?

Answer:   ( a ) $21

( b ) $17.50

( c ) $31.50

( d ) 9

Geometry Area and Volume

Geometry” Earth-measuring" is an part of the mathematics concerned with questions of size, shape, relative position of figures, and the properties of space. Geometry is one of the oldest sciences. Initially a body of the  practical knowledge concerning lengths, areas, and volumes. And now we can see about the problems in geometry area and volume.

Geometry area and volume problem 1:

Pro 1 :Find the volume of cylinder with the radius 8 cm and the height 12 cm.

Solution:We can find the volume of  an cylinder by using the following formula:

Volume of cylinder V=πr2h

Substitute the values of r and h into the above formula. Than, we get

V=π*82*12

=3.14*64*12

=2411.52 cm3

Pro 2 :Find the volume of sphere with the radius is 12 cm.

Solution:We can find the volume of  the sphere by using the following formula

Volume of sphere V= (4/3) πr3

Substitute the value of radius into the above formula. Then we get,

V= (4/3) *3.14*123

= 1.333*3.14*144

= 602.72 cm3

Ans: 602.72 cm3

Problem 3:Find the amount of pyramid with the base 8.2 mt and height 10.2 mt.

Solution:We can find the volume of  the pyramid by using the following formula

Volume of pyramid V= (1/3) b h

Substitute the values of  the base and height into the above formula. Then we get,

V= (1/3)*8.2*10.2

=0.333*8.2*10.2

=27.8521 mt3

Ans: 27.8521 mt3

Geometry - Find the volume of shapes when area is given

Find the volume of the right prism whose area of the base is 550 cm2 and height is 38cm



Solution:Given that area of the base, A = 550 cm2 and height (h) of the prism = 38 cm

Volume of the right prism = area of the base * height cu.units

= A * h

= 20900

Volume of right prism = 20900 cm3

Find the volume of the right prism whose area of the base is 450 cm2 and height is 34cm

Solution:Given that area of the base, A = 450 cm2 and height (h) of the prism = 34 cm

Volume of the right prism = area of the base * height cu.units

= A * h

= 15300

Tutoring About Correlation Coefficient

Concept of correlation:

Correlation is a method of studying the relationship between the two variables. In statistical analysis we come across the study of two variables wherein the change in the value of one variable produces a change in the value of other variable. In that case we say that the variables are correlated or there is a correlation between the two variables.

The formula for the correlation coefficient r can be expressed in the form,

r = `(sum (X - barX) ( Y - barY))/(sqrt(sum (X - barX)^2) sqrt(sum(Y - barY)^2))`

It is conventionally taken as x = X - X and y = Y - Y and hence we write

r = `(sum xy)/(sqrt(sum x^2) sqrt(sum y^2))`

The above formula is expressed in terms of deviations of the variables from their means. Instead, if the actual values of the observations are taken then the formula can be written as,

r= `(N sum XY - sum X sum Y)/(sqrt(N sum X^2 - (sumX)^2) sqrt(NsumY^2 - (sum Y)^2))`

Instead of the deviations from their means, the deviations are measured from the value A and B for X and Y variables by taking dx = X - A, dy = Y - B, the correlation coefficient r is given by,

r = `(N sumdxdy - sumdx.sumdy)/(sqrt(Nsumdx^2- (sumdx)^2) sqrt(Nsumdy^2 - (sumdy)^2))`

Tutoring about formulas for calculating correlation coefficient:

r = `(sum xy)/(sqrt(sum x^2) sqrt(sum y^2))`

This formula is used when deviations are measured from their mean.
r= `"(N sum XY - sum X sum Y)/(sqrt(N sum X^2 - (sumX)^2)(sqrt(N sum Y^2 - (sumY)^2))) `

This formula is used if no assumed average is taken for x and y series
r = `"(N sumdxdy - sumdx.sumdy)/(sqrt(Nsumdx^2- (sumdx)^2)(sqrtNsumdy^2 - (sumdy)^2)) `

This formula is applied when deviations for x and y series are taken from some assumed values.


Tutoring on problems on correlation coefficient::

Calculate the correlation coefficient between x and y from the following data:
x 1 3 5 8 9 10
y 3 4 8 10 12 11


Solution:
x y x - `barx` y - `bary` (x - `barx` )2 (y - `bary` )2 (x-`barx` )(y - `bary` )
1 3 -5 -5 25 25 25
3 4 -3 -4 9 16 12
5 8 -1 0 1 0 0
8 10 2 2 4 4 4
9 12 3 4 9 16 12
10 11 4 3 16 9 12
36 48 0 0 64 70 65


`barx` = `(sum x)/(n)` = 36/6 = 6

`bary` = `(sum y)/(n)` = 48/6 = 8

r = `(sum(x- barx)(y - bary))/(sqrt(sum(x-barx)^2)sqrt(sum(y-bary)^2))`

= `(sum xy)/(sqrt(sumx^2)sqrt(sumy^2))`

= `(65)/(sqrt(64)sqrt(70))` = 0.97