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Integration errors

when you integrate 4x^5 + 3
you get 4x^6/6 + 3x + c

when you integrate 2
you get 2x + c

Exams is on calculus and matrix only

calculus
differentiation is gradient machine
give a point and you will get the gradient from gradient machine
differentiate polynomial, sin x, cos x, e^x , ln x
real life application RATE OF CHANGE is differentiation
VELOCITY = change in distance/ time
ACCELERATION = change in velocity /time

Integration
polynomial
sin x
cos x
area under curve
definite integral means point given so you can find the value of c

GOOD LUCK

Integration Questios

Integrate the following

1. dy/dx = 5x^3 + 2x^2 + 5

2. dy/dx = 6x + 1

3. dy/dx = 2

4. dy/dx = 8x^5 - 5x^3 + 4x

5. dy/dx = 7x^2 + 3x + 4

Integration +c being forgotten

Too many forgetting the + c in integration

question2
dy/dx=2x(x-3)
dy/dx=2x^2-6x
y=2/3x^3-3x^2

What about the + c
y=2/3x^3-3x^2 + c

To find the value of c use the point provided (3,6) x = 3 and y = 6
so c can be found

What about a question with v= and distance is required integrate yes but do not forget the +c

b)TO find the total distance travelled u need to integrate the (V) velocity.
v = 5t -3t^2 + 2
s= 5/2t^2 - t^3 + 2t

it must be
s= 5/2t^2 - t^3 + 2t + c

Errors

1. y = 6x^3 + 4x^2 -5x
when you differentiate how can the y remain y
dy/dx = 18x^2 + 8x -5

2. y^2 = x^3 + 16 is in the syllabus differentiating with 2 unknowns
some of you missed this class

differentiate term by term as normal and if the current term is a y term differentiate w.r.t y and then multiply by dy/dx

2y dy/dx = 3x^2

3. 1/y = 4x + 4
no downstairs
y^-1 = 4x + 4
differentiate term by term

-y^-2 dy/dx = 4

Common errors being made

1. y = 2x + 8/x^2 find dy/dx no downstairs y = 2x + 8x^-2 BUT when differentiating dy/dx = 2 -16x^-3 NOTE -2 -1 = -3
2. y = ln (2x + 3) use sub for m BUT dy/dm CANNOT be the same as dy/dm instead dy/dx must be= dy/dm * dm/dx

A particle moves in a straight line and at point P, it's velocity is given as v = 7t^2 - 5t +3. The particle comes to rest at point Q.

1. What is the acceleration at Q if it arrives at Q when t=7?
2. How far does the particle travel in t=1 to t=4?

Project (20%)

These are all the projects I received except for Brent and Azad (no ID on project but marked)

107000184
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108002539
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109000055
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109000336
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109003667
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109003729
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109004186
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109000323
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110000236
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Revision

1. Find the trap area from (1,4) to (2,1) and the line y = 7 - 3x and the curve y = 4/x^2
2. Find the equation of the curve which passes through the point (3,6) and for which dy/dx = 2x(x -3). Hint open brackets first then integrate.
3. Differentiate e^(3-2x)
4. A particle moves in a straight line so that, t seconds after passing through a fixed point O, its velocity, is giveb by v = 5t -3t^2 + 2. The particle comes to instantaneous rest at the point Q. Find

a) the acceleration of the particle atQ

b) the total distance travelled in the time interval t = 0 to t =3.

Velocity

A particle moves in a straight line so that at time t seconds after passing through a fixed point, its velocity, v m/s, is given by v = 6 cos 2t. Find

1. the two smallest positive values of t for which the particle is at instanteneous rest
2. the distance between the positions of instantaneous rest corresponding to these two values of t
3. the greatest magnitude of the accelerstion

Find dy/dx

1. y = 6x^3 + 4x^2 - 5x

2. y^2 = x^3 + 16

3. 1/y = 4x + 4

4. y = 3x^4 + 3(x^2 + 5)

5. y + 6 = 4x + 9y - 3x^2

Questions 2

1. A curve has equation y = 4/(2)^.5, find dy/dx
2. A curve is such that dy/dx = 16/x^3 and (1,4) is a point on the curve, find the equation of the curve.
3. y = 6theta - sin 2 theta, find dy/dx

Questions 1

1. The equation of a curve is y = 2x + 8/x^2

find dy/dx and d^2ydx^2

1. Differentiate ln (2x + 3)
2. A curve is such that dy/dx = 2x^2 -5. Given that the point (3,8) lies on the curve, find the equation of the curve.

Matrices

1. ....... is used to represent space in the matrix

14 ................6

5x-3 ............8

What value of x will this matrix be singular?

1. What values of p will make this matrix singular

6p + 2 .........8

5 ...................3p

1. Can a singular matrix have an inverse, justify your answer?
2. Tom bought 6 plums and 5 mangoes for $40. Jane bought 3 similar plums and 4 similar mangoes for $23. Using the matrix method determine the price of 1 plum and the price of 1 mango.

Find the product of the following matricies

Find the product of the following matricies

1.
| 1 5 | | 2 2 |
| 3 8 | . | 4 5 |

2.
| 5 9 | | 2 1 |
| 3 2 | . | 2 7 |

3.
| 4 1 | | 1 2 |
| 0 2 | . | 4 1 |

4.
| 9 7 | | 3 3 |
| 9 8 | . | 3 3 |

5.
| 4 4 | | 5 2 |
| 4 4 | . | 3 5 |

6.
| 1 0 | | 0 5 |
| 0 8 | . | 4 4 |

7.
| 2 3 | | 3 2 |
| 3 6 | . | 3 5 |

8.
| 0 5 | | 0 0 |
| 0 8 | . | 4 5 |

9.
| 2 2 | | 3 3 |
| 2 2 | . | 3 3 |

10.
| 3 6 | | 6 3 |
| 6 3 | . | 3 6 |

diff questions

1. The displacement s of a piston during each 8-s is given by s = 8t -t^2. For what value of t is the velocity of the piston 4?
2. The distance s travelled by a subway train after the brakes are applied is given by s = 20t -2t^2 How far does it travel after the brakes are applied in coming to a stop?
3. Water is being drained from a pond such that the volume V of water in the pond after t hours is given by V = 50(60-t)^2. Find the rate at which the pond is being drained after 4 hours.
4. The electric field E at a distance r from a point charge is E=k/r^2 where k is a constant. Find an expression for the instantaneous rate of change of the electric field with respect to r.
5. The voltage V induced in an inductor in an electric circuit is given by V = L(d^2q/dt^2) where L is the inductance. Find the expression for the voltge induced in a 1.6 H inductor if q = (2t + 1)^.5 -1.
6. The altitude h of a certain rocket as a function of the time t after launching is given by h = 550t - 4.9 t^2. What is the maximum altitude the rocket attains?
7. The blade of a saber saw moves vertically up and down and its displacement is given by y = 1.85 sin 36∏t. Find the velocity of the blade for t=0.025.
8. The charge q on a capacitor in a circuit containing a capacitor of capacitance C, a resistance R, and a source of voltage Eis given by q = CE(1 - e^(-t/RC) ). Show that this equation satisfies the equation Rdq/dt + q/C = E.
9. An earth orbiting satellite is launched such that its altitude is given by y = 240(1 - e^(-.05t)). Find the velocity of the satellite for t= 10.
10. Differentiate y = 7sin x + ln (4x^2 +1)

Function Diff

1. Differentiate m = 8/e^t - e^t
2. Differentiate v = 7 cos 3a + 8a^5. What is dv/da when a = 1/2 ∏
3. ∫5sinb db
4. 
   ∫5cosm dm
5. 
   ∫5sinb + 9b^4 db
6. 
   ∫5/x dx
7. 
   ∫5e^x dx

Question Set 8 (Logs)

Solve the Following:

2 = 1.4^x

15 = 6.1^x-1

6 = 10^x+1

14^x = 6^x

4 = 3.3^x

18 = 3.1^x



Find b in the following:

log[4]16 = b

log[3]81 = b

log[2]16 = b

log[5]125 = b

log[3]9 = b

log[6]216 = b

log[8]64 = b

log[4]64 = b

log[b]625 = 4

log[b]49 = 2

log[b]27 = 3

log[b]81 = 2

log[b]4 = 2

log[b]16 = 4

log[3]b = 6

log[4]b = 3

log[5]b = 3125

log[10]b = 1

log[9]b = 4

log[2]b = 6

log[3]b = 4

log[8]b = 3

Question set 7

1. Let A be the area of a circle with radius r at time t. If the radius changes at a rate of 2 in/sec, at what rate is the circle's area changing when r = 1?
2. Let A be the area of a square with side s at time t. If the side changes at a rate of -4 mm/week, at what rate is the square's area changing when s = 3?
3. Let V be the volume of a sphere with radius r at time t. If the radius changes at a rate of 3 ft/min, at what rate is the sphere's volume changing when r = 2?
4. Let S be the surface area of a sphere with radius r at time t. If the radius changes at a rate of -5 m/hr, at what rate is the sphere's surface area changing when r = 1?
5. Let V be the volume of a cube with side s at time t. If the side changes at a rate of 10 in/hr, at what rate is the cube's volume changing when s = 5?
6. Let S be the surface area of a cube with side s at time t. If the side changes at a rate of 7 cm/sec, at what rate is the cube's surface area changing when s = 3?
7. Let A be the area of a circle with radius r at time t. If the radius changes at a rate of -3 ft/sec, at what rate is the circle's area changing when r = 5?
8. Let A be the area of a square with side s at time t. If the side changes at a rate of 2 m/day, at what rate is the square's area changing when s = 10?
9. Let V be the volume of a sphere with radius r at time t. If the radius changes at a rate of -8 in/min, at what rate is the sphere's volume changing when r = 7?

Question set 6

1. An object is thrown in the air from an initial height of 12 feet, with an initial upward velocity of 16 feet/second?
   How long will the object be in the air?
   What will the velocity of the object be after 1 second?
2. An object is thrown in the air from an initial height of 48 feet, with an initial upward velocity of 32 feet/second?
   How long will the object be in the air?
   What will the velocity of the object be after 2 seconds?
3. An object is thrown in the air from an initial height of 100 feet, with an initial upward velocity of 10 feet/second?
   How long will the object be in the air?
   What will the velocity of the object be after 2 seconds?
4. An object is thrown in the air from an initial height of 400 feet, with an initial upward velocity of 50 feet/second?
   How long will the object be in the air?
   What will the velocity of the object be after 4 seconds?
5. An object is thrown in the air from an initial height of 75 feet, with an initial upward velocity of 200 feet/second?
   How long will the object be in the air?
   What will the velocity of the object be after 5 seconds?
6. Suppose it costs -x^2 + 400x dollars to produce x computers/day. Compute the marginal cost to estimate the cost of producing one more computer each day, if current production is 100 computers/day.
7. If the position of an object at time t is given by the function
   s(t) = 3t + 2 meters what are the velocity and acceleration when t = 3 seconds?
8. If the position of an object at time t is given by the function
   s(t) = t3 - t meters what are the velocity and acceleration when t = 5 seconds?
9. If the position of an object at time t is given by the function
   s(t) = 3t3 - 10t2 meters what are the velocity and acceleration when t = 2 seconds?
10. If the position of an object at time t is given by the function
    s(t) = sin t meters what are the velocity and acceleration when t = pi/4 seconds?
11. If the position of an object at time t is given by the function
    s(t) = cos 2t meters what are the velocity and acceleration when t = pi/4 seconds?
12. If the position of an object at time t is given by the function
    s(t) = sin t + t meters what are the velocity and acceleration when t = pi/2 seconds?
13. If the position of an object at time t is given by the function
    s(t) = cos t + sin t meters what are the velocity and acceleration when t = pi seconds?
14. If the position of an object at time t is given by the function
    s(t) = (1/2)t3 + 2t meters what are the velocity and acceleration when t = 1 second?
15. If the position of an object at time t is given by the function
    s(t) = 6t2 - 8t + 19 meters what are the velocity and acceleration when t = 4 seconds?

Question set 5

1. Let f(x) = sin x. Find the first 3 derivatives of f.
2. Let f(t) = 5 cos t. Find the first 4 derivatives of f.
3. Let f(x) = 3cos x + 5x. Find the first 3 derivatives of f.
4. Let f(t) = 2t + sin t. Find the first 3 derivatives of f.
5. Let f(t) = 1/t2 - sin t. Find the first 2 derivatives of f.
6. Let f(x) = sin x - cos x. Find the first 4 derivatives of f.
7. Let f(x) = 2sin x + 3cos x. Find the first 4 derivatives of f.
8. Let f(x) = 4sin x + 1/x. Find the first 3 derivatives of f.
9. Let f(t) = -3sin t + 1/2 cos t. Find the first 3 derivatives of f.

Question set 4

1. y = e ^(2x). Find dy/dx and d^2y/dx^2
2. y = sin x . Find d^133 y /d x^133

Question set 3

1. A stone is dropped into a pond, the ripples forming concentric circles which expand. At what rate is the area of one of these circles increasing when the radius is 4 m and increasing at the rate of 0.5 ms-1?
2. The tuning frequency f of an electronic tuner is inversely proportional to the square root of the capacitance C in the circuit.
   If f = 920 kHz for C = 3.5 pF, find how fast f is changing at this frequency if dC/dt = 0.3 pF/s.
3. An object falling from rest has displacement s in cm given by s = 490t2, where t is in seconds.
   What is the velocity when t = 10 s?
4. Find the equation of the tangent to the curve y = 3x − x^3 at x = 2.
5. Find the derivative of the function
   y = x^1/4 - 2/x
6. Find dy/dx for y = (5x + 7)^12.
7. Find dy/dx for y = (x^2+ 3)^5.
8. Find if dy/dx y = √(4x^2 -x).
9. Find dy/dx if y = (2x^3 - 1)^4
10. Find dy/dx if y =(4x^5 - 1/(7 x^ 3))^4
    

Question set 2

1. 
   You fire a cannonball upward so that its distance (in feet) above the ground
   t seconds after firing is given by h(t) = −16t^2 + 144t. Find the maximum height (dh/dt = 0) it reaches and the number of seconds it takes to reach that height.
2. The daily profit, P, of an oil refinery is given by
   P = 8x − 0.02x^2,
   where x is the number of barrels of oil refined. How many barrels will give maximum profit (dP/dx = 0) and what is the maximum profit?
3. A rectangular storage area is to be constructed along the side of a tall building. A security fence is required along the remaining 3 sides of the area. What is the maximum area that can be enclosed with 800 m of fencing?
4. A box with a square base has no top. If 64 cm2 of material is used, what is the maximum possible volume for the box?

Questions cosθ sin θ sinθ - cosθ

• Solve x = log[3] 81 + log[3]1/9

For what value of x, the following matrix is singular ?
(5-x) (x + 1)

(2) ( 4)

3.. The matrix A =

3 2 satisfies the relation A2 - 4A + I = 0. Find A-1.
1 1

4. cosθ sin θ sinθ - cosθ
Simpliy cosθ + sinθ
- sinθ cosθ cosθ sinθ

5. Evaluate

-1
∫ 1/x dx
-4

6. A wire of length 28m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum?

7. A window is in the form of a rectangle surmounted by a semi-circular opening.
The total perimeter of the window is 10m. Find the dimensions of the window to
admit maximum light through the whole opening

8. A square piece of tin of side 48 cm is to be made into a box without top, by cutting a square from each corner and folding up the flaps to form the box. What should be the side of the square to be cut off, so that the volume of the box is the maximum possible? Also find the maximum volume.

log set 3

1. log [2] x^2 + log [2] (x - 1) = 1 + log[2] (5x + 4)
2. log [2] (x (x - 1)) = 1
3. log z = log (y + 2) - 2 log y, find z in terms of y
4. 2 log x = 1 + log ((4x -15 )/2)
5. log (2x + 5) = 1 - log x
6. log (x -8) + log (9/5) = 1 + log (x/4)
7. log (2 + 2y) = log 5 + log 3

Log set 5

1. Express the following as a single logarithm:1. log [3] 7 + log [3] 5
2. log [2] 16 + 3log [2] 4 - log [2] 8
3. log [5] 9 + log [5] 24.
4. log [5] 5 + log [5] 10 - log [5] 3 = 0 3.4.

Log set 4

1. Solve log[7](x^2) = log[7](2x – 1).
2. Solve 2log[3](x) = log[3](4) + log[3](x – 1)
3. Solve log2(x) + log2(x – 2) = 3

Real LIfe Differentiation

1. A closed box has a fixed surface area A and a square base with side x .
   (a) Find a formula for the volume V of the box, as a function of x .
   (b) Find the rate of change of V with respect to x.
2. The revenue from selling q items is given by the formula R(q) = 500q - q^2
   and the total cost is given by C(q) = 150 + 10q . Write down a function that gives the total
   profit earned. Find the rate of change of total profit with respect to q.
3. For positive constants A and B, the force between two atoms in a molecule is given by
   f(r) = A/r^2 + B/r^3 where r > 0 is the distance between the atoms.
   What is the expression for the instantenous rate of change of force between the atoms with respect to distance?

Log set 2

1. 
   Find the value of the unknown variable in each of these:(i) 2^x = 128 , (ii) 3^y = 1/9 , (iii) 5^x = 625 , (iv) 4^s = 164 , (v) 16^t = 4 ,(vi) 8^ = 1/4 .
2. Solve each of the following equations for the value of x (i) log 3x = 6
   ii) log[3]x + 3log[3](3x) = 3 iii) log (5x - 1) = 2 + log (x - 2) iv) 2logx = log(7x - 12)
3. 3. Write each of the following as a single logarithmic expression:

(i) log[10](x + 5) + 2 log[10] x (ii) log[4](x^3 - y^3) - log[4] (x - y)
iii) 1/2 (3 log[5] (4x) + log [5] (x + 3) - log[5] 9)

Logs set 1

Solve by explaining

1. 2(4 ^x-1) = 17^x
2. log[2]P = log[2]23.9 + 0.45
3. 2log x - 1 = log(1-2x)
4. log[6]y = log[6]4-log[6]2
5. log[3]y = 1/2 log[3] 7 + 1/2log[3]48
6. (log[2]3)(log[2]y) - log[2]16 = 3
7. log[5]x + log[5]30 = log[5]3 + 1
8. 2(log[9] x + 2log[9]18) = 1
9. 3log p = 2 + 3 log 4
10. log[4]x + log[4]6 = log[4]12
11. 2log[3]2 - log[3](x + 1) = log[3]5
12. log[8] (x + 2) = 2 - log[8]2
13. log (x+2) + log x = 0.4771

Complex Numbers Set 13

1. Solve 3 – 4i = x + yi
2. Simplify (2 + 3i) + (1 – 6i)
3. Simplify (5 – 2i) – (–4 – i)
4. Simplify (2 – i)(3 + 4i)
5. Simplify 3/(2i)
6. Simplify 3/(2 + i)

Complex Numbers Set 12

Calculate
(a) (2 − 14j) + (15 + 4j) (b) (12 − 8j) − (5 + 14j) (c) (2 − 6j) × (3 + 2j)
(d) (3 − 4j) × (7 + 2j) (e) (5 − √(-16)) × (2 + √(-9)) (f) (1 +2 j)^2


Plot 1 + 6j, 2 − 9j, −9 +7 j and −11 − 21j on the Argand diagram and for each write as polar coordinates


Calculate the modulus and phase angle of 1 + 12j, 2 − 12j, −14 + j and −5 − 9j.


Solve the quadratic equations
(a) x^2 − x + 5 = 0;
(b) 2x^2 + x + 8 = 0;

Complex Numbers Set 11

Let z1 = 2 − 5j, z2 = 3 + 6j.
Evaluate a) z1 + z2 b) z1 - z2 c) z1z2 d) z1/z2 as x + jy, x, y ∈ R.
Find the modulus and arg(z) for the following complex numbers

1. 25
2. 6j
3. −9,
4. −12j
5. 11 +12 j
6. 41 −6 j
7. −16 + 8j
8. −32 −76 j

b) z = 5 + j√3
c) z = −7 + 6j and mark them on a drawing of the complex plane.

Evaluate a) j^18, b) j^72, c) j^31, d) j^505.

Complex Numbers Set 10

Evaluate

1. j^5 + j^ 3 - j^7
2. j^5 * j^3
3. (j^5 - j^ 7)/j^3
4. (√(-9) + j^5) / (√(-4)
5. (√(-25) + j^3) + (j^7 - √(-9))
6. 4 + 7j added to 11 - 21j
7. (5 - 9j) (6 -3j)
8. (6 - 7j) / (4 - 3j)
9. Express in polar form 10 + 18j
10. Express in polar form -14 - 7j

Complex Numbers Set 9

Evaluate

1. j^6 + j^ 4 - j^2
2. j^8 * j^4
3. (j^6 - j^ 8)/j^3
4. (√(-9) + j^6) / (√(-4)
5. (√(-25) + j^4) + (j^8 - √(-9))
6. 4 + 8j added to 11 - 18j
7. (5 - 3j) (6 -4j)
8. (6 - 2j) / (5 + 3j)
9. Express in polar form -6 + 8j
10. Express in polar form -4 - 7j

K.E. = 1/2 m v squared

If you was the scientist to discover the K.E. formula starting from the fact that K.E. is proportional to the velocity to a power by the mass.
Illustrate your steps to lead to the above formula.

What will this give as a graph?

Voltage (V) Current (mA)
0 0
1 20
2 30
3 65
4 98
5 174
6 280

Your strategy

All students will act as the teacher. Each of you will comment on the strategy
that you will use to teach someone about differentiation of polynomials, logs and ln.
All the common errors you have observed can now be avoided, how?
What is your strategy in your teaching to avoid these errors?

Are these solutions correct?

Solution 1

y = log base 3 (8x^7 + x)
u = 8 x^7 + x
dy/dx = 56 x^6 + 1
= (56 x^6 + 1) / (8 x^7 + x) log base 3 e

Solution 2
P = ln 6s^3 + s
u = 6s^3
dy/dx = 18 s^2
= (18s^2)/(6s^3) + 1

Solution 2
P = ln 6s^3 + s
u = 6s^3 + s
dy/dx = 18 s^2 + 1
= (18s^2 + 1)/(6s^3 + s)

Solution 3
y = 15 x^3 + 7/(2x^3) - x
y = 15x3 + 7 *2 x^-3 - x
dy/dx = 45 x^2 + 42 x^-4 - 1

Solution 3
y = 15 x^3 + 7/(2x^3) - x
dy/dx = 45 x^2 + 7/6 x^-2 - 1

Log checking

Problem 1
log (2x^2 + 6x) - 6 = log (2x)

What can I do now?


Problem 2
Am I going correct?

log (x + 6) = 2 - log (4x)
log (x + 6) = log 100 - log (4x)

Complex Numbers Set 8

Calculate
(a) (2 − 8j) + (5 + 4j) (b) (2 − 8j) − (5 + 4j) (c) (2 − 8j) × (5 + 4j)
(d) (1 − 7j) × (3 + 10j) (e) (5 − j√2) × (2 + 7j√2) (f) (1 + j)^4


Plot 1 + 3j, 2 − 2j, −4 + j and −2 − 2j on the Argand diagram and for each write as polar coordinates


Calculate the modulus and phase angle of 1 + 3j, 2 − 2j, −4 + j and −2 − 2j.


Solve the quadratic equations
(a) x^2 − x + 1 = 0;
(b) 2x^2 + x + 1 = 0;

Complex Numbers Set 7

1. Evaluate (2 + j)(-1 +j/2)
2. Evaluate (1 + 4j)^2
3. Evaluate j(3 + 2j)(6 -4j)
4. Give the modulus and argument of the following complex numbers
   a)1 + j b)1 + 2j c) − 2 + 3j d)√2 − √(-36) e) - 3 - 5j
5. Evaluate a) (−1 + j)/(1 + j) b) j / (1 + 3j) c) (3 − 4j)/(12 + 8j) d) (2 − 3j)/(52 + 13j)

Complex Numbers Set 6

Let z1 = 2 − 3j, z2 = 4 + 6j.
Evaluate a) z1 + z2 b) z1 - z2 c) z1z2 d) z1/z2 as x + jy, x, y ∈ R.

Find r = z and arg(z) for the following complex numbers
z = 1, j, −1, −j, 1 + j, 1 − j, −1 + j, −1 − j, b) z = 1 + j√3 c) z = −2 + j2√3
and mark them on a drawing of the complex plane.

Find the following powers of j: a) j^8, b) j^42, c) j^11, d) j^105.

Complex Numbers Set 5

1. Evaluate (2 + 3j) - (6 –4j)
2. Evaluate (2 + 3j)(1 – 5j)
3. Write the conjugate of a) 3 + 5j b) 2 – 6j
4. Evaluate (3+ 4j)/(3-2j)
5. Evaluate [3 + √(-25) ] + [ 8 - √(-16)]
6. Evaluate [3 + √(-25) ] - [ 8 - √(-16)]
7. Evaluate [3 + √(-25) ] x [ 8 - √(-16)]
8. Evaluate [3 + √(-25) ] / [ 8 - √(-16)]

Complex Numbers Set 4

1. x = 3 - 2j and y = 3 + 2j
   
   Compute:
   x + y
   x - y
   x^2
   y^2
   xy
   (x + y)(x - y)
2. Write the complex number in standard form (i.e. in the form a + b i.)
   1. 3+ √(-16)
   2. -5j + 3 j^2
3. Evaluate (2 + 3j) - (6 -4j)

b) (2 + 3j)(1 - 5j)

c) 7j(7 - 3i)

d) (3 + 2j)^2 + (4 - 3j)

4. Write the conjugate of the complex number.
a) 3 + 5j
b) 2 - 6j
c) 15i
5. Perform the operation and write the result in standard form.
a) 4/(2+3j)
b) (3 + 4j)/(3-2j)
c) 3/(3 + 2j) - 4/(3 -2j)
V. Solve the quadratic equations using the Quadratic Formula.
a) 3x^2 + 9x +7 = 0
b) y^2 -2y + 2 = 0

complex numbers questions 3

What is the magnitude of √(-8)?
What is the magnitude of √(-14)?
What is the magnitude of 5 + 6j?
Is this real or imaginery √(-16)?
What is the phase angle of 7 + 9j?
What is the magnitude of 5 + 6j?

complex numbers questions 2

1. Evaluate the following: √(-25) + √(64) added to √(81) + √(-9)
2. Evaluate the following: √(-25) + √(64) divide by √(81) + √(-9)
3. Evaluate the following: √(-25) + √(64) minus √(81) + √(-9)
4. Evaluate the following: √(-25) + √(64) multiply by √(81) + √(-9)
5. Evaluate the following: j^6 + j^7 added to j^3 + j^4

complex numbers questions

1. A circuit in parallel has Z1 = 2 - 3j and Z2 = 4 + j. Find the total impedance, phase angle and magnitude
2. A circuit in parallel has Z1 = 5 - j and Z2 = 4 + 3j. Find the total impedance in polar form
3. A circuit in series has Z1 = 2 - 3j and Z2 = 4 + j. Find the total impedance, phase angle and magnitude
4. A circuit in series has Z1 = 5 - j and Z2 = 4 + 3j. Find the total impedance in polar form
5. A car is being pulled with 2 forces. One force is 5 -3j and the other is 6 + j. What is the resultant force and show this is correct graphically.

Revision

What are the strategies that can be used for the following:

1. complex numbers
2. logs
3. differentiation
4. real life calculus

Application of simple differentiation

Question 1
Water is being drained from a pond such that the volume V (in m^3) of water in the pond after t hours is given by V = 5000(60 - t)^2. Find the rate at which the pond is being drained after 4 h.

Question 2The velocity of an object moving with constant acceleration can be found from the equation v = (v[0] ^2 + 2as)^2, where v[0] is the initial velocity, a is the acceleration, and s is the distance traveled. Find dv/ds.

Example 3The electric field E at a distance r from a point charge is E = k/r^2, where k is a constant. Find an expression for the instantaneous rate of change of the electric field with respect to r.

Example 4The distance s (in m) traveled by a subway train after the brakes are applied is given by s = 20t - 2t^2. How far does it travel, after the brakes are applied, in coming to a stop?

Look at the following and made some comments to assist.
Remember the log base is in [] brackets

Example 1
log [8] (x + 2) = 2 - log [8] 2
log [8] 1 + log [8] 2 = 2 / (x + 2)

Example 2
log [8] (x + 2) = 2 - log [8] 2
log [8] (x + 2) + log [8] 2 = 2
log (2x + 4) = 2
log (2x + 8) = 2
log 8
2x + 4 = 1.8
2x = 1.8 - 4

Example 3
log [8] (x + 2) = 2 - log [8] 2
log [8] x + log [8] 2 = 2 - log [8] 2
log [8] x + 2 log [8] 2 = 2

Example 4
log [8] (x + 2) = 2 - log [8] 2
log [8] (x + 2) = 1.67
log [8] 1.67 = x + 2

Example 5
log [8] (x + 2) = 2 - log [8] 2
log [8] (x + 2) + log [8] 2 = 2
log [8] 2(x + 2) = 2
log [8] (2x + 4) = 2
log [8] 2x = 2 - 4

Example 6
log [8] (x + 2) = 2 - log [8] 2
log (x + 2) / log 8 = 2 - log 2 / log 8
log (x + 2) / 0.903 = 2 - 0.33
log (x + 2) 0.903 * 1.667
log x + 2 = 1.5
log x = 1.5 - 2
log x = 0.5
x = 0.32

Simple logs

Examine the following statements.
Remember that the base will be represented in (). Help these students.

Example 1
log (4) x = 12
log x = log 12
log 4
log x = 1.08 * 0.6
log x = 0.65
x = 4.5

Example 2
log (4) x = 12
log 4 = 12
x = 12 / log 4
x = 20

Example 3
log (4) x = 12
log (4) x = log 12

Example 4
log (4) x = 12
log (4)12 = x

Example 5
log (8) 32 = x
x = 8/32
x = 1/4

The [] is used since subscript and superscript are not allowed.
2 [3] = 8 can also be expressed as log [2] 8 = 3
Is this correct?
0.035 [x] = 2.74
Find x?

Re-express as a log which is log [0.035] 2.74 = x

1. Draw line
2. log 2.74__ = x
   log 0.035

Or is this correct?
0.035 [x] = 2.74 Find x?

1. Log both sides
2. log 0.035 [x] = log 2.74
3. Re-express to remove power
   x log 0.035 = log 2.74
4. Make x the subject of the equation
   x = log 2.74 / log 0.035

Logs

What really is logs?
Do you think this is an important aspect of maths?
How is logs related to exp?
What is the basic strategy in logs?

Logs

What really is logs?
Do you think this is an important aspect of maths?
How is logs related to exp?
What is the basic strategy in logs?

Exponent

What is the purpose of exponent?
Is a quadratic an exponent?
It is said that everything in life involves some aspect of Maths, give some real life scenarios thanvolves some aspect of Maths, give some real life scenarios that involves the exponent aspect of maths.

A curve graph

What is the difference between a straight line and a curve graph?

Can one graph consist of a straight line graph and a curve graph?

If yes, can they ever intersect?

How can one determine by calculation if these two graphs intersect?

Intersection of two straight lines

Can a graph consist of more than one straight lines?

If yes, why would someone put more than one straight line graphs on the same graph?

In the event that the straight lines intersect, what does this signify?

How can one calculate that point to verify that the graph is accurate?

Gradient and intercepts

How do you find the gradient of a straight line?

What is the use of finding gradient?

How can you explain gradient so that the ratio of change in y over change in x is not mixed up?

What is the purpose of finding y-intercept?

How do find the y-intercept?

What is the meaning of intercept?

A graph commonly consists of two axes called the x-axis (horizontal) and y-axis (vertical). Each axis corresponds to one variable. This variable represent something that a human can relate to for example the axes can be labelled with different names, such as velocity, time, height, temperature, price or quantity.
The place where the two axes intersect is called the origin. The origin is also identified as the point (0,0). Parts of a graph
- x-axis

- y-axis

- origin

A point on a graph represents a relationship. Each point is defined by a pair of numbers containing two co-ordinates (x and y). A co-ordinate is one of a set of numbers used to identify the location of a point on a graph. The co-ordinate is measured from the origin. First the x co-ordinate is given which states movement to the left and to the right. Second the y co-ordinate is given which states movement up and down.

Rate of change

What is meant by change in x-axis

What is meant by change in y-axis

What is meant by change in y-axis to change in x-axis?

What does the change in distance w.r.t. time signify?

How is this depicted or represented on a graph paper?

What is the gradient?

The use of drawing of a graph

Why do we need to draw a graph?

What is meant when a curve or stright line cuts the x-axis?

When a curve or stright line cuts the x-axis, does that have a significance in real life?

What is meant when a curve or stright line cuts the y-axis?

When a curve or stright line cuts the y-axis, does that have a significance in real life?

What is meant by y-intercept, x-intercept?

Features of a graph

The main features of a graph are as follows:
- the understanding of vertical and horizontal
- x-axis
- y-axis
- a point
- resulting shape by connecting points
- what does x = 0 mean
- what does y = 0 mean

What really is a graph

A graph is usually define as an x-axis against a y-axis.
What do you understand by a graph and the use of a graph?

What is a variable

How will you explain the following
1.) 14 - 8
2.) What is my change, if I had $14 and I bought a sandwich for $8?
3.) What is my change, if I had $14 and I bought two CDs, each $4 each?
4.) If I had $14 and I bought 2 blank CDs and received $6 change, what is the price of a CD?