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The document provides steps to solve a differential equation to determine if a particle is rising or falling at a given point. It takes the derivative of the function, plugs in values for x, y, and dx/dt, combines like terms, and solves for dy/dt. The solution, dy/dt = -60/7, is negative, indicating the particle is falling at the given point.
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This document provides examples of factoring polynomials of different types:
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4) Recognizing and factoring perfect square trinomials and differences of squares.
The document demonstrates various factoring techniques for polynomials in one and multiple variables.
MATH- FACTORIZE AND EXPAND EASY TO UNDERSTAND
Some example with formulas and solution
QUESTION # 1
If (a + b) 2 = 73 and ab = 65, find the value of a2 + b2.
Solution ; (a + b)2 = a2 + 2ab + b2
73 = a2 + 2(65) + b2
73 = a2 + b2 + 130
73 – 130 = a2 + b2
QUESTION # 2) FACTORISE
(I) (3g - h) 2 – g2
Solution ;
a2 - b2 = (a + b) (a - b)
[ ( 3g – h ) + g ] [ ( 3g – h ) – g ]
(II) f 4 + 8f + 16
Solution ;
(a + b) = (a)2 + 2(a)(b) + (b)2 (f 2)2 + 2(f2)(4) + (4)2
(III) 3x2 – 752
Solution ;
a2 – b2 = (a + b)(a - b)
3(x2 – 25y2)
3[ (x)2 – (5y)2 ]
(IV) 4x2 – y2 – 4 + 4y
Solution ;
(2x)2 - (y)2 + (-1 + y)
QUESTION # 3) EXPAND
(I) (p4 - 2)2
Solution ;
(a - b)2 = a2 – 2ab =b2
(p2)2 - 2(p2)(2) + (2)2
(II) ( x y )
( y x )
Solution ;
(a + b)2 = a2 – 2ab = b2
(x/y)2 + 2(x/y) (y/x) + (y/x)2
(III)( a + b)2
( 3)
Solution ;
(a + b)2 = a2 + 2ab + b2
a2 + 2(a)(b/3) + (b/3)2
(IV) (ef - 2)2
Solution ;
(a - b)2 = a2 - 2ab + b2
(ef)2 – 2(ef)(2) + (2)2
QUESTION # 4) EVALUATE
(I) 412 + 738 + 81
Solution ;
[(40 + 1)2] + 738 + 81
(a + b)2 = a2 + 2(a)(b) + (b)2
[ (40)2 + 2(40)(1) + (1)2] + 738 +81
1600 + 80 + 1 + 738 + 81
1681 + 819
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Solution ;
(a + b)2 = a2 + 2(a)(b) + (b)2
(90 + 2)2 – 368 + 4
[ (90)2 + 2(90)(2) + (2)2 ]– 368 + 4
8100 + 360 + 4 – 368 + 4
8460 – 360
(III) 2012 – 992
Solution ;
a2 – b2 = (a - b)2 (a + b)2
(200 + 1)2 - (100 - 1)2
2002 + 2(200)(1) + (1)2 – (100)2 – 2(100)(1) + (1)2
40000 + 400 + 1 – 10000 – 200 + 1
40401 - 9801
QUESTION # 5) FACTORISE
(I) 3a2 – 36a + 108
Solution ;
A2 – 2ab + b2 = (a-b)2
3(a2 – 362)
(II) a2 – 196
Solution ;
a2 – b2 = (a + b)(a - b)
a2 – 142
(III) p2 – 4q2 + 3(p – 2q)
Solution ;
a2 + b2 = (a + b)(a - b)
(p)2 – (2q)2 + 3(p – 2q)
(p – 2q)(p + 2q) + 3(p – 2q)
(IV) 2a4 – 32b2 c2
Solution ;
2(a4 – 16b2 c2)
2(a2)2 (4bc)2
2(a2 – 4bc) (a2 – 4bc)
2bc(a2 - 4) (a2 + 4bc)
2bc(a)2 – (2)2 (a2 +4bc)
2bc (a - 2) (a + 2)(a2 + 4bc)
(V) p4 – 81q4
Solution ;
(p2)2 – (9q2)2
a2 + b2 = (a + b)(a - b)
(p2 – 9q2) (p2 + 9q2)
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If 4r + 20 (7h) =36, and r=5h, then 20h + 20 (7h) =36 illustrates substituting 5h for r in the original equation.
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This document provides examples of factorizing polynomial expressions in various forms:
1) Factoring the difference of two binomials.
2) Factoring the square of a binomial expression.
3) Factoring a trinomial into two binomial factors.
4) Factoring a perfect square trinomial.
5) Factoring the difference of squares.
6) Factoring a trinomial that is a perfect square by addition and subtraction.
7) Factoring quadratic trinomials in the form x^2 + bx + c.
8) Factoring a perfect cube of binomials.
9) Factoring the difference and sum of perfect cubes.
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The document demonstrates various factoring techniques for polynomials in one and multiple variables.
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This document provides examples of factoring polynomials of different types:
1) Factoring the difference of two squares.
2) Finding the square of a binomial expression.
3) Factoring trinomials and polynomials into two binomial factors.
4) Recognizing and factoring perfect square trinomials and differences of squares.
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MATH- FACTORIZE AND EXPAND EASY TO UNDERSTAND
Some example with formulas and solution
QUESTION # 1
If (a + b) 2 = 73 and ab = 65, find the value of a2 + b2.
Solution ; (a + b)2 = a2 + 2ab + b2
73 = a2 + 2(65) + b2
73 = a2 + b2 + 130
73 – 130 = a2 + b2
QUESTION # 2) FACTORISE
(I) (3g - h) 2 – g2
Solution ;
a2 - b2 = (a + b) (a - b)
[ ( 3g – h ) + g ] [ ( 3g – h ) – g ]
(II) f 4 + 8f + 16
Solution ;
(a + b) = (a)2 + 2(a)(b) + (b)2 (f 2)2 + 2(f2)(4) + (4)2
(III) 3x2 – 752
Solution ;
a2 – b2 = (a + b)(a - b)
3(x2 – 25y2)
3[ (x)2 – (5y)2 ]
(IV) 4x2 – y2 – 4 + 4y
Solution ;
(2x)2 - (y)2 + (-1 + y)
QUESTION # 3) EXPAND
(I) (p4 - 2)2
Solution ;
(a - b)2 = a2 – 2ab =b2
(p2)2 - 2(p2)(2) + (2)2
(II) ( x y )
( y x )
Solution ;
(a + b)2 = a2 – 2ab = b2
(x/y)2 + 2(x/y) (y/x) + (y/x)2
(III)( a + b)2
( 3)
Solution ;
(a + b)2 = a2 + 2ab + b2
a2 + 2(a)(b/3) + (b/3)2
(IV) (ef - 2)2
Solution ;
(a - b)2 = a2 - 2ab + b2
(ef)2 – 2(ef)(2) + (2)2
QUESTION # 4) EVALUATE
(I) 412 + 738 + 81
Solution ;
[(40 + 1)2] + 738 + 81
(a + b)2 = a2 + 2(a)(b) + (b)2
[ (40)2 + 2(40)(1) + (1)2] + 738 +81
1600 + 80 + 1 + 738 + 81
1681 + 819
(II) 922 – 368 + 4
Solution ;
(a + b)2 = a2 + 2(a)(b) + (b)2
(90 + 2)2 – 368 + 4
[ (90)2 + 2(90)(2) + (2)2 ]– 368 + 4
8100 + 360 + 4 – 368 + 4
8460 – 360
(III) 2012 – 992
Solution ;
a2 – b2 = (a - b)2 (a + b)2
(200 + 1)2 - (100 - 1)2
2002 + 2(200)(1) + (1)2 – (100)2 – 2(100)(1) + (1)2
40000 + 400 + 1 – 10000 – 200 + 1
40401 - 9801
QUESTION # 5) FACTORISE
(I) 3a2 – 36a + 108
Solution ;
A2 – 2ab + b2 = (a-b)2
3(a2 – 362)
(II) a2 – 196
Solution ;
a2 – b2 = (a + b)(a - b)
a2 – 142
(III) p2 – 4q2 + 3(p – 2q)
Solution ;
a2 + b2 = (a + b)(a - b)
(p)2 – (2q)2 + 3(p – 2q)
(p – 2q)(p + 2q) + 3(p – 2q)
(IV) 2a4 – 32b2 c2
Solution ;
2(a4 – 16b2 c2)
2(a2)2 (4bc)2
2(a2 – 4bc) (a2 – 4bc)
2bc(a2 - 4) (a2 + 4bc)
2bc(a)2 – (2)2 (a2 +4bc)
2bc (a - 2) (a + 2)(a2 + 4bc)
(V) p4 – 81q4
Solution ;
(p2)2 – (9q2)2
a2 + b2 = (a + b)(a - b)
(p2 – 9q2) (p2 + 9q2)
(p)2 – (3q)2 (p2 – 9q2)
1.funtions (1)
The document contains 13 math problems involving functions. The problems cover topics such as:
- Finding the image and object of elements under a given function
- Finding inverse functions
- Determining the type of relation between sets
- Evaluating composite functions
- Solving for unknown constants in function definitions
- Finding the value of x when a composite function equals a given value
The document provides the problems and blank spaces for the answers. The answers section gives the solutions to each of the 13 problems.
Asymptotes | WORKING PRINCIPLE OF ASYMPTOTES
The document defines asymptotes as straight lines that touch a curve at infinity. It then shows the steps to find the asymptotes of the curve x3+2x2y-xy2-2y3+xy-y2-1=0. The steps involve putting y=mx+c into the equation, setting coefficients of x3 and x2 to zero, and solving the resulting equations for m and c. The asymptotes obtained are y=x, y=-x-1, and x+2y=1.
Squaring binomials
This document discusses squaring binomial expressions. It provides three examples of squaring binomials of the form (x + a), showing that (x + a)2 = x2 + 2a*x + a2. Specifically, it squares (x + 3), (b - 5), and (m - 15) to get x2 + 6x + 9, b2 - 10b + 25, and m2 - 60m + 225 respectively.
Tarea de Matematicas
The document provides examples of mathematical exercises involving factoring polynomials and binomial expressions. Some examples include factoring the difference of squares, perfect square trinomials, and perfect cube binomials. Standard factoring techniques are demonstrated such as grouping like terms, finding common factors, and using the difference and sum of cubes identities.
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The document provides instructions to find the stationary points and determine their nature for various polynomial functions including f(x) = 3x^2 - 12x + 5, f(x) = -x^2 - 4x + 8, and f(x) = 2x^2 - 4, as well as several other polynomial functions involving higher degree terms and multiple roots. Students are asked to analyze where the derivatives of each function are equal to 0 or undefined to find the coordinates of stationary points and then determine whether they are maxima, minima or points of inflection.
Reviewer in math i
The property illustrated by the expressions 8 + [6 + (-7)] = (8+6) + (-7) and 12 + 6 = 18 is the associative property of addition.
If 4r + 20 (7h) =36, and r=5h, then 20h + 20 (7h) =36 illustrates substituting 5h for r in the original equation.
If x+4 = 15, then x+4 – 4 = 15 – 4 illustrates subtracting 4 from both sides of the original equation to isolate the variable x.
Maria
This document provides examples of factorizing polynomial expressions in various forms:
1) Factoring the difference of two squares
2) Factoring a perfect square trinomial
3) Factoring the difference and sum of two cubes
It demonstrates techniques for factoring polynomial expressions involving binomials, trinomials, and higher degree terms.
EJERCICIOS PARA EL EXAMEN
This document provides examples of factorizing polynomial expressions in various forms:
1) Factoring the difference of two binomials.
2) Factoring the square of a binomial expression.
3) Factoring a trinomial into two binomial factors.
4) Factoring a perfect square trinomial.
5) Factoring the difference of squares.
6) Factoring a trinomial that is a perfect square by addition and subtraction.
7) Factoring quadratic trinomials in the form x^2 + bx + c.
8) Factoring a perfect cube of binomials.
9) Factoring the difference and sum of perfect cubes.
C++
This document describes a proposed math operator program that can perform various mathematical calculations. The program would allow users to easily calculate things like the area of a square, square roots, cube roots, and the areas and volumes of spheres and cones. It would save users time by automating common math operations. The program is intended to benefit students at various education levels as well as businessmen by providing a simple and cost-effective way to perform calculations.
001 int techintro
The document discusses techniques for evaluating integrals, including expanding the numerator, separating terms in the numerator, completing the square, dividing improper integrals, using trigonometric identities, and adding or subtracting terms in the numerator. It provides examples of integrals and their solutions using various techniques like expanding the numerator, splitting up fractions, using trig identities, and completing the square. Students are assigned practice problems evaluating integrals using the techniques discussed.
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The document summarizes solving a physics problem about the rate at which the foot of a falling ladder is moving away from a wall. It provides the information given, which is that the top of a 13-foot ladder is slipping down a wall at 2 feet per second when it is 5 feet above the ground. It outlines the steps to set up and solve the kinematic equations to find that the rate the foot is moving away from the wall is 5/6 feet per second.
Emily
The document discusses how the rate of change of volume (dV/dt) of a cylinder is related to the rate of change of its height (dh/dt) and radius (dr/dt). It uses the product rule to derive an equation relating dV/dt, dh/dt, and dr/dt. It then applies this equation to a specific example where the height is increasing at 1 in/s, the radius is decreasing at 1 in/s, and calculates that the volume is decreasing at -20π cubic inches per second.
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The document is a student's work showing the steps to find the derivative dy/dt given the equation y=3x+5 and that dx/dt=2. It first takes the derivative of y to get dy/dt=3(dx/dt). It then plugs in the value of 2 for dx/dt to get dy/dt=6. It then repeats the process for a second part where dx/dt is given as -1/3.
William
Given that a 13-foot ladder is leaning against a wall and the top is slipping down at 2 feet per second, the question asks how fast the foot of the ladder is moving away from the wall when the top is 5 feet from the ground. Using the Pythagorean theorem and taking the derivative of the relationship between x and y, it is found that when the top is 5 feet up, the foot is moving away from the wall at a rate of 5/6 feet per second.
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The document discusses social networking behavior and how social media has changed online interactions. It provides definitions of social media and how platforms allow for communities to form, opinions to spread widely, and transparency. The rest of the document outlines different social media tool families like social networking, blogs, bookmarks, microblogs, video/image sharing, and widgets. It stresses that companies must understand and embrace these changes or risk losing market share to competitors.
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A spherical balloon is deflating, with its radius decreasing at a constant rate of 15cm/min. The document calculates that when the balloon's radius is 9cm, the rate at which air must be removed is 4860π cm^3/min in order for the radius to decrease at the given rate. It uses the formula for the volume of a sphere and takes the derivative of volume with respect to time to determine the required rate of air removal.
Rachel3
This document contains solutions to two calculus problems involving derivatives of parametric equations.
For the first problem, the document finds the derivative dy/dt when (x,y) = (1/2,√3/2) for the equation x2 + y2 = 1. It determines that dy/dt = -√3/3.
For the second problem, it finds the derivative dx/dt when (x,y) = (√2/2,√2/2) for the same equation, given that dy/dt = -2. It determines that dx/dt = 2.
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This math problem involves implicit differentiation of the equation A = x^2. The document draws and labels a diagram in Part A, writes the equation in Part B, performs implicit differentiation in Part C to get dA/dt = 2x * dx/dt, and then in Part D substitutes x = 3 ft and dx/dt = 2 ft/min to solve for dA/dt = 12 ft^2/min.
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A player is running from first to second base at 25 ft/s when they are 10 ft from second base. The distance from home plate to each base is 60 ft. To find the rate of change of the player's distance from home plate, the Pythagorean theorem is used to relate the distances. Plugging in the known values of 50 ft from first base and 10 ft from second base yields that the rate of change of the player's distance from home plate is 125 ft/s.
Pelecup
The document provides instructions for solving related rates problems involving a spherical balloon deflating. It identifies finding the rate air is removed when the radius is 9 cm as the problem. It sets up the equation relating volume, radius, and their rates of change and takes the derivative. Substituting the known radius and radius rate allows solving for the required air removal rate of -4860π cm3/min.
Bad Ass Math Powerpoint
A spherical balloon is deflating, with its radius decreasing at a constant rate of 15cm/min. The document calculates that when the balloon's radius is 9cm, the rate at which air must be removed is 4860π cm^3/min in order for the radius to decrease at the given rate. It uses the formula for the volume of a sphere and takes the derivative of volume with respect to time to determine the required rate of air removal.
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The document discusses the relationships between dV/dt, dh/dt, and dr/dt for a cylinder where V is the volume, h is the height, and r is the radius. It states that dV/dt equals pi times r squared times dh/dt plus h times 2r times dr/dt. It then calculates that given h is increasing at 1 in/sec and r is decreasing at 1 in/sec, dV/dt equals -20pi inches^3/sec, meaning the volume is decreasing at that rate.
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The document provides the solution to a calculus word problem involving a particle moving along a curve.
The problem asks for (a) the rate of change of the y-coordinate when the particle is at (1,2) and the x-coordinate is increasing at 6 units/s, and (b) whether the particle is rising or falling at that instant.
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A spherical balloon is deflating, with its radius decreasing at a constant rate of 15cm/min. The document calculates that when the balloon's radius is 9cm, the rate at which air must be removed is 4860π cm^3/min in order for the radius to decrease at the given rate. It uses the formula for the volume of a sphere and takes the derivative of volume with respect to time to determine the required rate of air removal.
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Euro 2024 Predictions - Group Stage Outcomes
Here are our Euro 2024 predictions for the group stages
Will England make it through the group stages?, Will Germany use the home advantage to full effect?
Follow our progress, see how many we get right
If you want to join in let us know before the first game kick off and we can invite you to our private league
or join in with our friends at DeeperThanBlue
https://www.linkedin.com/posts/activity-7204868572995538944-qejG
https://www.selectdistinct.co.uk/2024/06/13/euro-2024-match-predictions/
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- 1. Number 37 By: Nicole Schreiber xy 3 1+y ² 8 5 = Given: dx/dt= 6 at point (1,2)