Calculus Formula Sheet

Calculus is a branch in mathematics which deals with the study of limits, functions, derivatives, integrals, and infinite series. It has two main branches

• Differential calculus
• Integral calculus

It is a formula sheet which contains differentiation and integration formulas which are needed to solve the calculus problems. We should know all the calculus formulas before get into the problems. Calculus formula sheet helps you to learn all those formulas. It is very much helpful for the students at the time of examination for reviewing calculus formulas.

Integration is a limiting process which is used to find the area of a region under a curve.  We can also say that integration is an anti derivative of differentiation. Integration of a function is shown as,
int` f(x) dx = F(x) + C
int `  = Sign of integration
The variable x in dx is called variable of integration or integrator.
C= constant          

Differential calculus formula sheet:

Derivatives of polynomial functions:
`d/dx` (c) = 0
`d/dx` (x) = 1
`d/dx ` (cx) = c
`d/dx` (xⁿ) = nx^n-1
`d/dx` (cxⁿ) = ncx^n-1

Derivatives of trigonometric functions:
`d/dx` (sin x) = cos x
`d/dx` (cos x) =  - sin x
`d/dx ` (tan x) = sec² x
`d/dx` (cot x) = - cosec²x
`d/dx` (sec x) = sec x tan x
`d/dx` (cosec x) = - cosec x cot x

Derivatives of inverse trigonometric functions:
`d/dx` (sin^-1 x) = `1/sqrt(1 - x^2)`
`d/dx` (cos^-1 x) = - `1/sqrt(1 - x^2)`
`d/dx` (tan^-1 x) = `1/(1 + x^2)`
`d/dx` (sec^-1 x) = `1/(|x|sqrt(x^2 - 1))`
`d/dx` (cosec^-1 x) = - `1/(|x|sqrt(x^2 - 1))`
`d/dx` (cot^-1 x) = - `1/(1 + x^2)`

Derivatives of hyperbolic functions:
`d/dx` (sinh x) = cos hx
`d/dx ` (cosh x) = sin hx
`d/dx ` (tanh x) = sec h²x
`d/dx` (sech x) = - tanh x sech x
`d/dx` (cosech x) = - coth x cosech x
`d/dx` (coth x) = - cosech²x

Integral calculus formula sheet:

List of integrals of rational functions:
`int` k dx = kx + C
`int` x^a dx = `(x^(a+1))/(a+1)` + C
`int` 1/x dx = ln|x| + C

List of integrals of logarithmic functions:
`int` ln x dx = x ln x - x + C
`int ` log_ax dx = xlog_ax  - x(ln a) + C

List of integrals of exponential functions:
`int ` e^x dx = e^x + C
`int ` a^x dx = `(a^x)/(ln a)` + C

List of integrals of trigonometric functions:
`int` sin x dx = - cos x + C
`int` cos x dx  = sin x + C
`int` tan x dx  = - ln |cos x| + C
`int` cot x dx = ln |sin x| + C

Computing Ratio

A relation obtained by comparing two quantities similar in some sense is called a ratio. Ratio can be expressed in terms of fraction, that is,  a:b is equal to a/b For example 25 is 1/4^th of the hundred, Therefore, the ratio of 25 to 100is ¼. We write it in ratio as 1:4.

Facts for computing ratio:

The numbers a and b in a ratio a:b are called the term of the ratio. Here, a is called the first terms and b is called the second term of the ratio

Computing Nature of ratio:
A ratio is purely a number: it has no unit attached to it.

Computing Comparison of ratio:
Since, the ratio a:b represents what multiple are part a is of b, it is also equal to a/b.
Accordingly ratio a : b is said greater than a ratio c : d if, a/b > c/d ( b and d >0)
Or  ad – bc > 0

Computing simplest form of ratio:
Since, a/b=ma/mb for every non-zero number m, therefore the ratios a:b and ma:mb are the same.
If,a/b is in its lowest terms, the ratio a:b is said to be in its simplest form.

Computing Composition of ratios:
Two or more ratios can be compounded.
For example: the ratio compounded of the ratios.

1. 2:3 and 4:5 is 2 * 4 : 3 * 5
2. a : b  and c : d is ac : bd

Note: The ratio a : B and B : a are generally different.
For example: the ratio 3:4 is ¾ where as, the ratio, 4:3 is 4/3 and surely ¾ is not equal to 4/3

Example problem for computing ratio:

Example:

In red and green tiles, the ratio of red tiles to green tiles is 2:3. If there is 90 green tiles, how many red tiles are there?

Solution:

Step 1: Assign variables:

Let x = red tiles

Write the items in the ratio as a fraction.

Red / green = 2/3 = x / 90

Step2: Do the cross multiplication

2 * 90 = 3 * x

3 * x = 2 * 90

3x = 180

3x / 3 = 180 / 3

X = 60

Example 2:

The length and width of the rectangle is 4:3 whose perimeter is 350.Find the are of the rectangle.
Solution:
Take the ratio 4:3
Consider 4x be the length and 3x be the width.
Perimeter of the rectangle is =350
That is, 2(l+w)= 350
2(4x+3x)= 350
2*7x=350
14x=350
X=25
Therefore,
The length of the rectangle is 4x= 4*25=100.
The width of the rectangle is 3x=3*25=75

Area of the rectangle = length * width = 100 * 75 = 7500

Ellipse Line Intersection

In this article we discuss ellipse line intersection. Ellipse is a conic section of geometry function. Ellipse can be defined in mathematically a plane passing through the right circular at angle between `0^o` to `90^o` . Ellipse can be included in two points like A1, A2. These two points called  focus.                                                     

                     

We are taking any two points on an above ellipse, the sum of the distance from the focus points is constant.

Shape of the ellipse line intersection:

The intersection of line in ellipse at centered origin. The line can passing through the points is `(x_o, y_o)` or origin. 

                           

The formula for ellipse is given below,
x^2/a^2+y^2/b^2=1`
where,
a=horizontal semi- axis
b=vertical semi-axis
 The trigonometry formula for ellipse is,
x=acos(t) and
y=asin(t)
where,
t =ellipse parameter
a=horizontal axis
b=vertical axis

Some important properties of ellipse line intersection:

• Center of ellipse :                 

It can be defined as connecting the focus points by using inside the single midpoint. It is also called as intersection of major and minor axes.                                             

• Major and Minor axes:

The longest diameter of the ellipse is called as major axes and shortest diameter of the ellipse is called as minor axes.                               

                    

• Tangent:

A line just touching one point on a ellipse is called as tangent.
                                             

                

• Secant :

The intersection of two points on an ellipse is called as secant.                                                            

                    

• Chord :

It can be defined as a line just linking any 2 points on ellipse. 

Triangle Online

A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ABC. In Euclidean geometry any three non-collinear points determine a unique triangle and a unique plane. (Source: Wikipedia)

Once you've gone through these, take a look at our Equilateral Triangle Area for more refernce.

Example problems for triangle solver online

Triangle solver online example 1:
                  A triangle has the base length of 20 cm and height is 23 cm. Find the area of the triangle in cm².
Solution:
   Given, base length (b) = 20 cm and height (h) = 23 cm.
Formula:
         Area of the triangle = `(1 / 2)` * (b * h)
Substitute the given base length and height values in the above formula, we get
                                            = `(1 / 2)` * (20 cm * 23 cm)
                                            = 230 cm²
Answer:
  Area of the triangle is 230 cm²

Triangle solver online example 2:
    Find the area of the given triangle.
        
Solution:
   From the given triangle ABC,
             Base length (b) = 20 cm and height (h) = 23 cm.

Formula:
         Area of the triangle = `(1 / 2)` * (b * h)
Substitute the given base length and height values in the above formula, we get
                                            = `(1 / 2)` * (20 cm * 23 cm)
                                            = 230 cm²
Answer:
  Area of the triangle is 230 cm²

Triangle solver online example 3:
    Find the x value of the given triangle.
       
Solution:
   From the given triangle ABC,
            Angle (B) = 60° and Angle (C) = 55° and Angle (A) = x
We know,
    Sum of the angle in triangle is 180°
 Therefore,
   Angle (A) + Angle (B) + Angle (C) = 180°
Substitute the given values in the above formula, we get
              x + 60° + 55° = 180°
                      x  + 115 = 180°
Subtract 115 on both the sides, we get
                                  x = 65°
Answer:
  The final answer is Angle (A) = 65°

Triangle solver online example 4:
    Find the x value of the given triangle.
    
Solution:
       Given triangle is right triangle ABC.
From the triangle ABC,
AC = x, AB = 48, BC = 36
Using Pythagoras theorem,
           AC² = AB² + BC²
Substitute the given values in the above formula, we get
              x² = 48² + 36²
               x² = 3600
Take square root on both the sides, we get
                 x = 60
Answer:
The final answer is 60

Practice problem for triangle solver online

Triangle solver online practice problem 1:
    A triangle has the base length of 32 cm and height is 53 cm. Find the area of the triangle.
Answer:
 The area of the triangle is 848 cm²
Triangle solver online practice problem 2:
    Find the area of the given triangle.
    
Answer:
 The area of the triangle is 210cm²

Triangle solver online problem 3:
    Find the area of the given triangle.
        
Answer:

 The area of the triangle is 80 cm²

Four Sided Polygons

If polygon has 4 sides then its represented as quadilateral.A quadrilaterals is generally mentioned as four sided shape.the following are the examples for four sided polygons,they are
1.square
2. rectangle.
3.Rhombus.
4.paralellogram.
Now in this article we are going to discus about the four sided polygons, but before that take a look at some example for four sided polygons.



Please express your views of this topic Similar Polygons by commenting on blog

Description to four sided polygon:

Let us see the brief discussing about the four sided polygons:
Square:
square has four equal sides. the following properties are used in square.



Area of the square :
A = a²
where a represents the length of one side.

Perimeter of the square :
P = 4a
where a is one side length value.
angle: all the angles are right angle (90^o)
Rectangle :
The opposite sides of the rectangle is equal.


Perimeter of the rectangle:
P = l+w+l+w
where l = length
and w = width
P= 2l+2w
    = 2(l+w)

Area of the rectangle :
Area A = length and width
         = lw.
opposite angles are congruent.
opposite sides are parallel.
Length of diagonal is calculated by
d= `sqrt (l^2+w^2)`

parallelogram:
Parallelogram has four sides, and the properties are listed below



Perimeter of the parallelogram
P = a+b+a+b
 = 2a+2b.

Area of parallelogram:
The are of the parallelogram is represented as
A = base * height
 = b*h
opposites sides are same length and also parallel.
opposite angles are congruent.

Rhombus:
rhombus is a quadrilateral which has four sides, the properties are listed below.


Perimeter of Rhombus:
P = a+a+a+a
 = 4a
where a is side length of the rhombus.
Area of Rhombus:
The are of the rhombus is calculated by
A = 1/2 ab
where a and b are diagonals of the rhombus.
all the sides of the rhombus are congruent.
opposite angles are congruent.

Example problems for four sided polygons:

Let we see some practice problems for four sided polygons.
Example 1)
Find the area and perimeter of the square whose side is 5 cm
Solution:

Area of the square :

A= a²

where a = 5.

A= 5²

=5*5

=25cm²

Perimeter of square:

p = 4a

=4*5

=20 cm.

Example 2)
Find the area and the perimeter of the rectangle whose height = 6 cm and base =8 cm
Solution:

Area of the rectangle :

A = l*b

where l=6 cm and

b= 8 cm

A = 6*8

=48cm²

Perimeter of rectangle:

P = 2(l+w)

=2(6+8)

=2(14)

=28cm.

Triangular Prism Net

A triangular prism is a solid has two parallel faces which are congruent triangle at both ends. These faces form the bases of the prism

The others faces are in the shape of parallellogram.They are called lateral faces.
A right triangular prism that has its bases perpendicular to its lateral surfaces.

Once you've gone through these, take a look at our Volume of Rectangular Prism for more refernce.

Surface area of triangular prism=2base area+peripeter of base x height prism
Volume of triangular prism = Base area of prism x height of prism.

Under this concept of triangular prism we draw the triangular prism net.
 Prepare the model of triangular prism net ,we need a paper , a pencil or pen, a scissor and gum.

Method of preparing triangular prism net

 First we draw the rectangle and two triangles to two ends of the rectangle on the paper. Then cut out along solid sides.
 Fold along dotted lines.
 Use clear tape to fasten .
 If you want to draw or colour the net,do it before you tape it together.
 If you want to decorate it by gluing on decorations, tape it together first.

Quadrilateral Prism Shape

In geometry, an n-sided prism is a polyhedron made of an n-sided polygonal base, a translated copy, and n faces joining corresponding sides. Thus these joining faces are parallelograms. All cross-sections parallel to the base faces are the same (source-Wikipedia). If the prism has four sides then it is called quadrilateral prism. in this article we shall discus about some quadrilateral prism.

Quadrilateral prim shapes:

Cube:

It is one type of quadrilateral prism. Its all dimensions are equal in measure (all side are equal).it has eight vertices. The shape of the cube is shown in below.
Formula for Volume of the cube V= a3 cubic units
Where,
            V = volume of the cube
 a =side length (or edge) of the cube.
Formula for surface area of the cube (A) = 6 a² square units

Rectangle prism:

It is one type of quadrilateral prism. The shape of the cube is shown in below.
Volume of rectangular solid (V) = length x width x height Cubic units
     = l x w x h cubic unit.
Total surface area of rectangular prism
                                    = 2 (length x breadth + breadth x width + length x height)
                                    =2(lb + bh+ lh) square units

Trapezoidal prism:

It is one type of quadrilateral prism.it has one set of parallel sides. The shape of the trapezoidal prism is shown in the figure.
Volume of trapezoidal prism = l x area of the base cubic units
                                                       l – Length of trapezoidal prism
                              Area of the base= 1/2 x (a + b) x h square units
                                                       a and b are length of parallel sides.
                                                       h – Height

Quadrilateral prism shape - Example problems:

1. Find the volume and surface area of the cube whose side length is 17 cm.
Solution:
            Given:
                        Side length (a) = 17 cm
Formula:
            Volume of the cube V= a³ cubic units
                                                = 17³
                                                = 17 x 17 x 17
                                                = 4913 cm³
            Surface area of the cube (A) = 6 a² square units
                                                          = 6 x 17²
                                                          = 6 x 289
                                                          = 1734 cm²

2. Find the lateral surface area and total surface area of a rectangular whose length is 6 cm, width 3cm and height 6cm.
Solution:
Given:
                              Length (l) = 6 cm
                         Width (w) = 3 cm
                           Height (h) = 6 cm
Total surface area of rectangular prism =2(wh + lw+ lh) square units
                                                               = 2(3 x 6+ 6 x 3 +6 x 6)
                                                               = 2 (18 + 18 + 36)
                                                               = 2 (72)
                                                               = 144
Total surface area of rectangular prism = 144 cm²
            Volume of the rectangular prism (V) = l x w x h cubic units
                                                                           = 6 x 3 x 6
                                                                           = 108
                                                                           = 108 cm³
3.figure out the volume trapezoidal prism whose length 14 cm, height 5cm, length of parallel sides a=7 cm and b=4cm.
Solution:
Given:
                  Length (l) = 20 cm
                  Height (h) = 5 cm
                  Parallel sides a=7 cm and b=4 cm
Formula:
Volume of trapezoidal prism = l x area of the base cubic units

Area of the base:
Area of the base= 1/2 x (a + b) x h
                                                   = 1/2x (7 + 4) x 5
                                                   =1/2 x 11 x 5
                                                     = 27.5 cm²
Volume of trapezoidal prism = 20 x 27.5
                                                  = 550

Volume of trapezoidal prism = 550 cm³