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Friday, 28 April 2023

Tuesday, 18 April 2023

Integration with trigonometric substitutions - Several Examples

 








Integration - Trigonometric - Several Examples

 










HSC Mathematics Ext1 Exam 3U 2016

Question 1: ===========================







 
Question 11(a,b,c): =============================



Question 11(d,e): =============================



Question 11(f), 12(a) i ii: =========================



Question 12(a) iii iv: =============================



Question 12b: =============================



Question 12c: =============================





Question 13: =============================


Question 14: =============================







Harder Inequalities question from Cambridge Book Show x^2+xy+y^2>0 for x, y\ne 0.

Harder Inequalities question from Cambridge Book Show x^2+xy+y^2>0 for x, y\ne 0.

3 Methods:

Method 1: Consider two cases. (Keep in mind x,y are both non zero)
Case 1: xy>0. In this case the result is trivial as each summand is positive. That is since x^2>0, y^2>0 and xy>0 then their sum is also positive!//

Case 2: xy<0. Then -xy>0.
So x^2+xy+y^2=(x+y)^2-xy. But (x+y)^2>0 and so the LHS is the sum of two positive numbers and so is itself positive.//

Method 2: Use completing the square

\begin{array}{l} x^2+xy+y^2&=x^2+xy+y^2/4+3y^2/4\\ &=\left( x+y/2 \right)^2 +3y^2/4\end{array}

Each summand is clearly positive and hence the sum is also.//

Method 3: Use the discriminant by considering x^2+xy+y^2 as a quadratic in x.
\Delta = y^2-4y^2=-3y^2<0.
Since the coefficient of x is 1>0 and \Delta<0 then the expression must be positive.

[Remember that if you have a quadratic Ax^2+Bx+C then \Delta=B^2-4AC]


FRACTIONS - Basic Operations

{2\over 7}+{3\over 7}={2+3\over 7}={5\over 7}

 {7\over 10}-{4\over 10}={7-4\over 10}={3\over 10}

 {4\over 5}\times{5\over 6}={4\times 5\over 5\times 6}={20\over 30}={2\over 3}

 {4\over 5}\div{5\over 6}= {4\over 5}\times{6\over 5}={4\times 6\over 5\times 5}={24\over 25}

{{2\over 5}\over 6} ={2\over 5}\div 6={2\over 5} \times {1\over 6}={2\over 30}={1\over 15}

{2\over {5\over 6}} =2\div {5\over 6}=2\times {6\over 5}={12\over 5}=2 {2\over 5}

HSC Mathematics Exam 2U 2016

Question 1: Answer (B)
Noting that sine is positive in Quadrants 1 & 2 and cosine is negative in quadrants 2 & 3.
Hence \theta must lie in quadrant 2. 

Question 2: Answer (C)
p={6\over 30}={1\over 5}

Question 3: Answer (B)
The first clue is the minus sign which indicates the coefficient of x^2 is negative, hence its a concave down parabola, thus B or C. If y=0, 3-(x-2)^2=0\implies x=2\pm \sqrt{3} so it has two zeros, so it must be (B) since (C) has no zeros as it doesn't cross the x-axis.

Question 4: Answer (A)
The only graph that satisfies f(-x)=-f(x) is (A). Or the only graph to go onto itself under rotation f 180^o about the origin is (A).

Question 5: Answer (B)
{d\over dx} \ln(\cos x)={ -\sin x\over \cos x}=-\tan x


Question 6: Answer (A)
The normal period of y=\tan x is \pi. So in this case the period is {\pi\over 3}.

Question 7: Answer (A)
With \theta radians, arc length is l=r\times \theta.
\theta={l\over r}

Area is A={1\over 2}r^2\theta={1\over 2}.5^2.{7\over 5}={32\over 5}

Question 8: Answer (D)
From the graph of y=\cos 2x it is seen there are 5 solutions to the equation |\cos (2x)|=1.


Question 9: Answer (C)
Drawing a picture we see two triangles whose combined area is A=0.5\times 2\times 2+0.5\times 3\times 3={13\over 2}.



Or if you happen to know the formula for integral of absolute value we have
\int^2_{-3} |x+1| dx =  \left.{ (x+1) |x+1|\over 2}\right|^{x=2}_{x=-3}=9/2+2={13\over 2}.


Question 10: Answer (D)
4+\log_2x=4\log_22+\log_2x=\log_22^4+\log_2x=\log_2 16x

Question 11(a) ====================================




Question 11(b)
y={x+2\over 3x-4}

y'={vu'-uv'\over v^2}

y'={ (3x-4)\times 1-(x+2)\times 3\over (3x-4)^2}

y'=-{ 10\over (3x-4)^2}


Question 11(c)  Solve |x-2|\le 3

-3\le x-2 \le 3

-3+2\le x-2+2 \le 3+2

-1\le x \le 5



Question 11(d)
\int_0^1(2x+1)^3 dx=\left. {(2x+1)^4\over 8}\right|^1_0
3^4/8-1/8=10

Question 11(e)

Solve the equations
\begin{array}{rl} y= & -5-4x\\ y= & 3-2x-x^2\\ \therefore -5-4x & =3-2x-x^2\\x^2-2x- 8 & =0\\ (x-4)(x+2) & =0\\ x=4  & \& \quad x=-2 \end{array}

When x=-2, y=3 and when x=4y=-21
Points of intersection are: (-2,3)\quad , \quad (4,-21)


Question 11(f)
y=\tan x
y'=\left. \sec^2x\right|_{x=\pi/8}=1.17

Question 11(g)

Solve \sin {x\over 2}={1\over 2}.

0\le {x\over 2}\le \pi

{x\over 2}={\pi\over 6}, \pi-\pi/6={\pi\over 6}, {5\pi\over 6}

x={\pi\over 3}, {5\pi\over 3}


Question 12(a): ===================================




Question 12(b):  ===================================



Question 12(c):  ===================================

Question 12(d):  ===================================



Question 13(a): ==================================




Question 13(b(c): ================================




Question 13(d): ==================================


Question 14(a): ==================================



Question 14(b): ==================================





Question 14(c): ==================================





Question 14(d,e): ==================================





Question 15(a):  =================================


Question 15(b):  =================================



Question 15(c):  =================================

(from BOS notes)


Question 16(a): ================================




Question 16( b): ================================