Trigs

Trig deals with 3 lengths placed together to form an enclosed shape called a triangle. This triangle will now have 3 angles. Remember an angle can be formed with 2 lengths like a Vand this angle will create a length which is like the open side of the V.
A triangle can be of 2 types:

• right angle
• non right angle

With the right angle triangle, one angle is 90 degrees. The sides of the triangles are labelled or named with respect to an angle (not the 90) in the triangle.

• Identify the angle
• Longest side hyp
• opp is opposite to the angle
• adj is adjacent to the angle

Anything to be found can be evaluated using sin, cos, tan or pythagoras theorem.

With a non right angle triangle

• label the vertices
• label the angles
• Label the length associated with the angle

Use sine rule or cosine rule

In trig questions it is important to identify all the triangles in the figure and draw the separately

1. DEFG is a trapezium with DX a perpedicular line to GF. DE = 10cm, DG = 13cm, Angle EFX and DXF are right angles. Find the length DX. The area of the trapezium.
2. ABC and PCD are right angle triangles. Angle ABC = 40 degrees, AB = 10 cm, PD = 8 cm and BD = 15cm. Angles BCA and DCP are right angles. Find the length BC the angle PDC.
3. A plane takes off at an angle of elevation 17 degrees to the ground. After 25 seconds the plane has travelled a horizontal distance of 1400. Calculate the height of the plane above the ground after 25 seconds.
4. STW is a triangle with the angle STW is 52 degrees. ST = 5cm, TW is 9cm. Calculate the lenght = SW and the area of triangle STW.

Calculus Revision questions

Differentiation deals with change change of something with respect to another thing
y = 4 has no change since 4 is a constant
y = 4x is a straight line which means steady or fixed change y = mx + c so fixed change is gradient which is m which is 4
y = 4x^2 means this is a curve so there is change at every point(different grad at every point) so calculas helps in approximating by giving the following rate of change

• y=ax^n dy/dx = anx^(n-1)
• y = sin x dy/dx = cos x
• y = cos x dy/dx = - sin x

If gradient or tangent or normal is to be found, it must be at a specific point since each point has its own gradient or tangent or normal.

Polynomial differentiation

1. Find dy/dx of the equation y = 2x + 8/(x^2). Find the coordinates of the turning point, is it max or min and verify your answer. Find the normal to the curve AT THE POINT (-2, -2)
2. A curve has equation y = k/x where k is a constant. Given that the gradient of the curve is -3 when x = -2, find the value of the constant k.
3. The equation of a curve is y = x + 2cos x. Find the x-coordinates of the stationary points of the curve and determine the nature of the turning points.
4. The equation of a curve is y = x^3 - 8. Find the equation of the normal to the curve AT THE POINT where the curve crosses the x-axis.
5. 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.
6. A curve is such that dy/dx = 2x^2 - 5 and (3, 8) is a point on the curve. Find the equation of the curve.

Integration deals with area. Area must be bounded otherwise area cannot be found. Hence to be bounded, normally 4 lines must be known to give an enclosed region. Integration of a curve gives the area under the curved trapped towards the x-axis. Two other lines are given by the lines x = 7 to x = 1 which is called upper bound and lower bound. WHen 2 curves are given the bounds are found by determining the points of intersection.

1. The curve y = 3x^.5 and the line y = x, find the two x bounds (ie the points of intersection) Find the area under the curve but above the line.
2. The region P is bounded by the curve y = 5x - x^2, the x=axis and the line x = h. The region Q is bounded by the curve y = 5x - x^2, the xaxis and the lines x = h and x = 2h. Given that the area of Q is twice the area of P, find the value of h.

Revision questions

1. f(x) = 2 - 3x -2x^2 bring to the form a(x + b)^2 + c and hence determine the coordinates of the turning point. Is the t.p. a max or min justify your answer.
2. Solve the equations 3x +y = 14 and 2x^2 -xy = 3
3. Express 1 - 4x - 2x^2 in the form a - b(x + c) hence when ix the function a maximum
4. CDB is a right angle triangle, BC is 5 metres and BCD is 40 degrees and BDC is 90 degrees. Calculate the length of BD and DC. Show that the area of the triangle is 12.5sin 40cos40.
5. Solve the equation 5y^2 = 8y - 2
6. MLO is a triangle with MNL a right angle triangle and NLO another right angle triangle. ML is 26 cm, NL is 10 cm, MOL is 35 degrees. Calculate the length of MN and MO.
7. Solve the simultaneous equations 2x^2 + y^2 = 33 and x + y = 3
8. Solve the simultaneous equations x + 1 = 2y and x^2 - 3y = 4
9. EFGH is a parallelogram EFI is 40 degrees, EF = 8m, EI is a perpendicular line to FG such that IG = 5m. Calculate the length FI, EI and the area of EFGH.
10. ABCD is a trapezium. AB = 12m, AD = 1.5m, BC = 3m and AD is parallel to DC. I is a point on BC such that CID is 90 degrees. Calculate the angle CDI and the length of DC.

Maths quiz mark is out

Marks were not very good
Please check with admin assistant or class rep
If anyone would like to reattempt a quiz
to be instead of their current mark
This wednesday 2nd Dec I will give a chance
must be from 9:00 am - 12:00 noon
Revise and redo
It will be in my staff room

Also see me on blog names
out of the 79, I have the real name for 65

What about the rest of the class (120 -79), your deadline for submission
was fri 27 or mon 30 with your at least 60 comments

Do not forget graph projects due nov 30th

Integration

1. Integrate (1 + 2 sin x ) wrt x with 45 and 0 degrees as bounds8.
2. Integrate 2x + 3 cos x ) wrt x with 90 and 60 degrees as bounds

These questions has bounds of degrees since trig are involved.

1. Integrating term by term gives,
2. integrate 1 wrt x and integrate 2sin x wrt x
3. this gives x - 2cos x + c
4. Now the bounds are used to get the required bounded region
5. Because the curve is a trig, degrees or radians can be used
6. In this case the curve has a trig part and a polynomial part so radians must be used

1. ABCD of a swimming pool, 12 m long. AB is the horizontal top edge. AD is the depth of the shallow end and BC the depth of the deep end.
i) Calculate the angle that DC makes with the horizontal
ii) Calculate the length of the sloping edge, DC
iii) If the pool is 5m wide, calculate the total surface area.

2.One face of the roof of a house in the shape of a parallelogram ABCD. The angle ABF= 40 degrees and the length AB = 8m. AI represents the rafter placed perpendicular to BC such that FC = 5m.
Calculate i) the length BI
ii) the length AI
iii) the area of ABCD

3. Differentiate the following i) x + 1/x
ii) 2cos x – sin x + 3x^7+ x^4
iii) √(x^3) +11x + 7
b) Integrate 3x^5 – 2sin x dx

Last class questions

1. Solve the following system of equations.
   x^2 + 4y^2 = 20
   xy = 4
2. A vector v has a magnitude of 200m and an angle of elevation of 50o. To the nearest tenth of a meter, find the magnitudes of the horizontal and vertical components of v.
3. Find the area bounded by the region y = x^2 + 1 , y = -x + 3, x = 0 and x = 3
4. Find the derivative of y = x^2 + 1/x
5. Find the derivative of y = 1 + 4x^3
6. Find the derivative of y = x^2 + 54/x
7. Displacement is given by s = 2 + 3t - t^2, find the maximum displacement
8. Find the coordinates of the stationary point of y = x^3 -12x - 12
9. i) Factorise by trial and error the expression f(x) = 2x^2 + x + 15 ii) Write in the form f(x) = a(x + b)^2 + c iii) Hence find the turning point of f(x) iv) Differentiate 2x^2 + x + 15 hence determine the turning point. v)Comment on iii) and iv) v1) Sketch the curve 2x^2 + x + 15 and determine the factors of the curve and the turning point vii) Using te graph solve 2x^2 + x + 15 = 4
10. The curve y^2 = 12x intersects the line 3y = 4x + 6 at 2 points. Find the distance between the 2 points.

1. Two ropes hold a boat at a dock. The tension in the ropes can be represented by 40 + 10j N and 50 – 25j N. Find the resultant force.
2. The total power P (in watts) transmitted by an AM radio station is given by
   
   P = 500 + 250 m^2 ,
   where m is the modulation index. Find the instantaneous rate of change of P with respect to m for m = 0.92
3. Find the derivative of the following polynomials:
   i. y = 4 x^2 + 7 x + 3
   ii. y = 4 x^ -6 - 5 x^3 + x
4. Find the derivative of the following trigs:
   a. y = 6 sin Ө
   b. y = 7 sin Ө + 3 cos Ө
5. Determine the gradient of each of the given functions at the given point.
   a) s = 2t^3 - 5t^2 + 4 (-1,-3)
   b) y = 5 sin Ө where Ө = 38◦
6. Find the area under the curve y = x^3 that is between the lines x = 1 and x = 2.
   
7. Integrate the following
   i) 4x^6 + 3x^ -4 + 1
   ii) 2 cos Ө
8. The electric field E at a distance r from a point charge is E = k/(r2), where k is a constant. Find an expression for the instantaneous rate of change of the electric field with respect to r.
9. The blade of a saber saw moves vertically up and down, and its displacement y (in cm) is given by y = 1.85 sin t, where t is the time (in s). Find the velocity of the blade for t = 0.025 s.
10. An open-top container is to be made from a rectangular piece of cardboard 24 cm by 38 cm. Equal squares of side x cm are to be cut from each corner, then the sides are to be bent up and taped together. Find the instantaneous rate of change of the volume V of the container with respect to x for x = 4 cm.
11. An analysis of a company’s records shows that in a day the rate of change of profit (in dollars) in producing x generators is dp/dx = 60(30 – 4x^5). Find the profit in producing x generators if a loss of $500 is incurred if none are produced.
12. The vertical displacement y (in cm) of the end of an industrial robot arm for each cycle is y = 2t^1.5 – cos t, where t is the time (in s). Find its vertical velocity for t = 15s.

Maths Problems set #3

17. Solve for x and y
x^2 = 4 – y x = y + 2

18. Expand the following
i) 4x(3x +2)
ii) (4x + 7)(4x + 7)
iii) (4x + 7)(3x +2)
b) Find the roots of the equation 2x^2 – 11x -21 = 0

19. Solve, for 0◦ < Ө < 360◦, giving your answer to 1 decimal place where appropriate
i) 2 sin Ө = 3 cos Ө
ii) 2 – cos Ө = 2 sin^2 Ө

20.Using lasers, a surveyor makes the measurements shown below, where B and C are in a marsh. The angle PAC is 90 degrees. Find the distance between B and C.
PA = 265.74
angle APB 21.66 degrees
angle APC 8.85 degrees

Questions 1

10. If Integrate (4x + k ) wrt x with 2 and 1 as bounds = 1 find k

11. Find the area of the region enclosed by y = 4 – x ^2 and the x –axis
from x = -1 to x = 1
12. The amount of liquid V cm ^3 in a leaking tank at time t secs. is given

by V = ( 20 – t ) ^3 for 0 ≤ t ≤ 20
Find the rate at which water leaves the tank when t = 5 secs.
13. Given 3y –x + 6 = 0
(1) Make y the subject of the formula 3x - y + 6 - 0
State the gradient, and the y – intercept
14. Using any suitable method , solve the quadratic equation

2x^2 – 4x + 1 = 0
( give answers to 3 s.f )
c) The floor of a room is in the shape of rectangle . The floor is ‘ C ‘ metres long. The width is 2 metres less tha its length .
(1) State in terms of C
(a) the width of the floor
(b) the area of the floor

(ii ) If the area of the floor is 15 m ^2 write an equation in C to show the information.
(iii) Use the equation to determine the width of the floor

15 Solve for p and r given

3p + 2r = 7
p2 – 2r = 11

1 Find dy/dx if y = 4x ^3 - 3 x^ 2 + 2x
2 Differentiate with respect to x

y = 5 x ^3 - 2 x^ -2 + 1

(3) Find dy/dx at x = ñ/8 if y = 6 x^ 2 + 2 sin x

(4) Differentiate , with respect to ø if

v = 2ø ^ 3 + 2 cos ø + sin ø

(5) Integratewith respect to x

2 x^ 4 + 4/x^ 2+ x

6. Integrate w.r.t. x
2 x ^2( 2x + 3 ) Hint Expand then integrate

7 Integrate (1 + 2 sin x ) wrt x with 45 and 0 degrees as bounds

8. Integrate 2x + 3 cos x ) wrt x with 90 and 60 degrees as bounds

Area using Integration

Explain how you would find the area under a curve by using integration.

What is f(x)? What is a and b?

Triangles

All triangles consists of 3 lengths and 3 angles.

What are the 2 types of triangles?
For each type state the properties or rules that can be applies.
Illustrate with examples

Completing the square Vs differentiation

Use the differentiation approach and the completing the square approach
and comment on your observations

Find the coordinate of the turning point and determine whether it is a maximum or a minimum
and what is its maximum or minimum value.

1. y = x^2 + x -6
2. y = x^2 + 14x -5
3. y = x^2 -23x + 18
4. y = 2x^2 - 8x + 5
5. y = 5x^2 -30x + 12
6. y = 8x^2 -32x + 25

Examining differentiation

Differentiation provides a way to get an approximation for gradient at a given point.

Consider the curve
y= x^2 +6x + 8

Differentiating to check gradient at points is:
dy/dx = 2x + 6
This is the gradient machine, can now find gradient for any point

What is the gradient at a turning point?
It is 0. Explain why?

At what point is this curve turning
when gradient = 0
2x + 6 = 0
so x = -3
curve turns when x = -3
What is the coordinate for this turning point?

when x = -3, what is y
substitute x =-3 into the curve y= x^2 +6x + 8
Show that the co-ordinate is (-3, 1)

Now take the same curve y= x^2 +6x + 8
and complete the square into the form a(x +b)^2 + c
show that the turning point is -3 and that the curve is a minimum with a value of 1.

Relating graphs with other topics in maths

What are common uses of graphs?
Can simultaneous equations be solved by graphs and why?
What does the x-intercept mean that is what is the meaning of y = 0?
Can factoring of quadratic give the x-intercepts?
What does completing the square useful for?
Explain how differentiation can be used to find turning point?
What really does differentiation gives?

Graphs

What really is a graph?
What is a graph used for?
What are the key things to look for in a graph?
What are some other ways of saying x-intercept?
Can you determine or have an idea of how many x-intercepts may be present?

What do these really mean

Explain what is being said in these equations:

1. m = 6 + 13
2. m = 5b + 2
3. h = irs
4. w = 5(7d - 2)
5. p = hv + k
6. y = mx + c
7. g = h/3 + t
8. p = (7f - 9)/5
9. h = (5m + 3) /p
10. m = (hp + 3)5

Searching for fractions 100 different ways

Identify any use of fractions from your day so far?
Why do you associate fraction with that scenario?
What would happen if a whole number was used in your scenario instead of a fraction?

Exploring Fractions

What is a fraction?
What really is the purpose of using fractions?
How can a fraction be identified?
How will you exppalin about fractions to your 9 years old brother?
Explain how necessary are fractions to us?