Introduction to solve advanced math problems:
There are many topic involved in advanced math. Here we consider only three topics. They are quadratic functions, matrices and Integration. Here in quadratic functions, we need to write the equation in the form of y = A [x – h]2 + k. Here the vertex of the parabola or quadratic function is [h, k]. Based on the value of A that is, based on its sign, the parabola will open up or down. This will help us to find either minimum value or maximum value of the function.
To form that standard form we need to be aware of completing the square method. In matrices, we can learn how to find the solution of the variable using row reducing echelon method. So a set of simultaneous equations can be solved using that method in matrices Integration will help us to find the area under the curve.
Example Problems on Advanced Math Problems.
Ex 1: Find the vertex of the quadratic function y = x2 + 6x + 7.
Solution: Given: y = x2 + 6x + 7.
`=> ` y = [x + 3]2 – 32 + 7 = [x +3]2 – 9 + 7.
`=>` y = [x + 3]2 – 2.
Therefore The vertex is [-3, -2]
Ex 2: Solve using row reduced echelon method: 2x + y – z = 3; x + y + z = 1; x – 2y – 3z = 4.
Solution: 2x + y – z =3
x + y + z = 1
x -2y -3z = 4
`=>` ` [[2,1,-1],[1,1,1],[1,-2,-3]] [[x],[y],[z]] = [[3],[1],[4]]`
therefore By row reduced echelon form,
let us find the values of x, y, z.
`[[2,1,-1, 3],[1,1,1, 1],[1,-2,-3,4]]`
= `[[2, 1,-1, 3], [0, 1, 3, -1], [0,-5,-5, 5]] R2 -> 2R_2 ** R_1`, R3 `-> ` 2R2 – R1
= `[[2, 1,-1, 3], [0, 1, 3, -1], [0, 1, 1, -1]]` `R_2 -> R_2 / -5`
= `[[2, 0,-4, 4], [0, 1, 3 , -1], [0, 0,-2 , 0] ]` R1 `-> ` R1 – R2, R3 -> R3 – R2
= `[[2,0,-4 , 4],[0,1,3 , -1],[0,0,1, 0]]` `R_3 -> R_3 / -2` .
`=>` 2x – 4x = 4
y + 3x = -1
z = 0
` =>` y = -1, x = 2
Therefore x = 2, y = -1, z = 0.
Ex 3: Find the area under the curve y = x + 3 between 0 to 5.
Solution:
Area = `int_ 0^5` (x +3) dx
= 0 [`x^2 /2` + 3x`] _ 0^5` = [`25/2` + 15 – 0]
=` 55/2` sq.units.Having problem with trig help online keep reading my upcoming posts, i will try to help you.
Practice Problems on Advanced Math Problems.
1. Find the area under the curve y = x2 + 6x + 5. between 0 to 4.
[Ans: Area = `49 1/3` sq.units]
2. Solve the following simultaneous equation using row reduced echelon form:
x + y + z = 2; 2x + y – z = 5; x + 2y + 3z = 2.
[Ans: x = 1, y = 2, z = -1]
There are many topic involved in advanced math. Here we consider only three topics. They are quadratic functions, matrices and Integration. Here in quadratic functions, we need to write the equation in the form of y = A [x – h]2 + k. Here the vertex of the parabola or quadratic function is [h, k]. Based on the value of A that is, based on its sign, the parabola will open up or down. This will help us to find either minimum value or maximum value of the function.
To form that standard form we need to be aware of completing the square method. In matrices, we can learn how to find the solution of the variable using row reducing echelon method. So a set of simultaneous equations can be solved using that method in matrices Integration will help us to find the area under the curve.
Example Problems on Advanced Math Problems.
Ex 1: Find the vertex of the quadratic function y = x2 + 6x + 7.
Solution: Given: y = x2 + 6x + 7.
`=> ` y = [x + 3]2 – 32 + 7 = [x +3]2 – 9 + 7.
`=>` y = [x + 3]2 – 2.
Therefore The vertex is [-3, -2]
Ex 2: Solve using row reduced echelon method: 2x + y – z = 3; x + y + z = 1; x – 2y – 3z = 4.
Solution: 2x + y – z =3
x + y + z = 1
x -2y -3z = 4
`=>` ` [[2,1,-1],[1,1,1],[1,-2,-3]] [[x],[y],[z]] = [[3],[1],[4]]`
therefore By row reduced echelon form,
let us find the values of x, y, z.
`[[2,1,-1, 3],[1,1,1, 1],[1,-2,-3,4]]`
= `[[2, 1,-1, 3], [0, 1, 3, -1], [0,-5,-5, 5]] R2 -> 2R_2 ** R_1`, R3 `-> ` 2R2 – R1
= `[[2, 1,-1, 3], [0, 1, 3, -1], [0, 1, 1, -1]]` `R_2 -> R_2 / -5`
= `[[2, 0,-4, 4], [0, 1, 3 , -1], [0, 0,-2 , 0] ]` R1 `-> ` R1 – R2, R3 -> R3 – R2
= `[[2,0,-4 , 4],[0,1,3 , -1],[0,0,1, 0]]` `R_3 -> R_3 / -2` .
`=>` 2x – 4x = 4
y + 3x = -1
z = 0
` =>` y = -1, x = 2
Therefore x = 2, y = -1, z = 0.
Ex 3: Find the area under the curve y = x + 3 between 0 to 5.
Solution:
Area = `int_ 0^5` (x +3) dx
= 0 [`x^2 /2` + 3x`] _ 0^5` = [`25/2` + 15 – 0]
=` 55/2` sq.units.Having problem with trig help online keep reading my upcoming posts, i will try to help you.
Practice Problems on Advanced Math Problems.
1. Find the area under the curve y = x2 + 6x + 5. between 0 to 4.
[Ans: Area = `49 1/3` sq.units]
2. Solve the following simultaneous equation using row reduced echelon form:
x + y + z = 2; 2x + y – z = 5; x + 2y + 3z = 2.
[Ans: x = 1, y = 2, z = -1]
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