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