HSC 3U Maths Formulae

MATHS Yr 12 Formulae Series and Sequences 1. Find the nth term of an AP: Tn = a + (n - 1)d 2. Sum of n terms of an AP: Sn = ½n (a + L) OR Sn = ½n [2a + (n – 1)d] 3. Find the nth term of a GP: Tn = arn-1 4. Sum of n terms of a GP: Sn = a (1 - rn) Sn = a - rL 1–r 1-r (If r 1) 5. Limiting Sum: S= a 1–r (where –1 < r < 1)

n n n

6. Sigma Notation:

Σ means “find the sum of” 7. Compound Interest: A = P (1 + r)n 8. Time Payments (finding the monthly repayment): Amount owing from previous month + Interest Monthly repayment

n

Integration 1. Finding the primitive:

∫x

n

dx = xn+1 + C n+1

n

dx = (ax + b)n+1 + C a(n + 1)

2. The primitive of (ax + b) n:

∫(ax + b) 3. The definite integral: b

b

∫f(x) dx = [ F(x) ] a

a

= F(b) – F(a) 4. The area bounded by the curve y = f(x), the x-axis and the lines x = a and x = b. b

∫f(x) dx

A= a

5. Area between two curves b

∫f(x) – g(x) dx

A= a

6. The area bounded by the curve y = f(x), the y-axis and the lines n n n n n

x = a and x = b. y=b

∫x dy

A=

y=a

7. Trapezoidal rule: (Approximation method) Area ≈ b - a [ f(a) + f(b) ] 2 ≈ h [ first function + last function + 2(sum of remaining function values) ] 2

8. Simpson’s Rule: (Approximation method) Area ≈ b – a [ f(a) +4 f(a + b) + f(b) ] 6 2 9. Volumes of Revolution: When rotated around x-axis: b

V=π

∫ y² dx

a When rotated around y-axis: b

V=π

∫ x² dy

a

10. Volumes of Rotation: Rotating area between 2 curves: When rotated around x-axis: b

V=π

∫ f(x)² - g(x) ² dx

a When rotated around y-axis: b

V=π

∫ fx² - gx² dy

a

(outside – inside curve)

Exponential and Logarithms 1. Derivative of y = ex y = ex dy = ex dx 2. The chain rule (function of a function) dy = dy X du dx du dx 3. Derivative of y = ef(x) y = ef(x) dy = f’(x) ef(x) dx 4. Integration of Exponentials ∫ eax+b dx = eax+b + C a In particular, ∫ ex dx = ex + C 5. Further Integration ∫ f’(x) ef(x) dx = ef(x) + C 6. Log Forms y = ax loga y = x ( y > 0)

x x x f f f a a x x f f x

7. Change of Base Theorem logb a = logx a logx b 8.

The Derivative of y = loge x y = loge x dy = 1 dx x An Important Result: dx = 1 dy dy dx

9.

The Derivative of y = loge f(x) y = loge f(x) dy = f’(x) dx f(x)

10. Sketching log curves -By inspection - By calculus 11. Integration of 1 and f’(x) x f(x)

∫ 1 dx = logex + C x

∫ f’(x) dx = loge f(x) + C f(x) 12. Useful Results loga a = 1 loge 1 = 0 eloge a = a 13. Derivative of y = a x y = ax y’ = logea X a x l x x x

14. Integration of y = a x

∫ ax = 1 ln a

x x x

X ax + C

Trig Function 1. Radians 1 ‫ ת‬rad = 180° Change to rad: x° X ‫ת‬ 180° Change to degrees: ‫ ת‬X rad 180 2. Length of an arc l=rø 3. Area of sector A = ½ r² ø 4. Area of minor segment A = ½ r² (ø – sinø) 5. Cosine rule (handy to find sides and angles of a triangle) a² = b² + c² - 2bc cos A cos A = b² + c² - a² 2 bc 6. Graphs sinx: y = Asinnx Amplitude = A Period = 2 ‫ת‬ n

cosx: y = Acosnx Amp = A Period = 2 ‫ת‬ n

tanx: y = Atannx Amp = A Period = ‫ת‬ n

7. Differentiation of trig function a) y = sin ax y’ = a cos ax b) y = cos ax y’ = - a cos ax c) y = tan ax y’= a sec² ax 8. Integration of trig function a) ∫ cos x dx= sin x + C b) ∫ sin x dx = - cos x + C c) ∫ sec² x dx = tan x + C d) ∫ sin ax dx= -1 cos ax + C a e) ∫ cos ax dx= 1 sin ax + C a f) ∫ sec² ax dx = 1 tan ax + C a Strictly extension work: 9. Compound angle results • • • • •

sin (A+B) = sinAcosB + cosAsinB sin (A-B) = sinAcosB – cosAsinB cos (A+B) = cosAcosB – sinAsinB cos (A-B) = cosAcosB + sinAsinB tan (A+B) = tanA + tanB 1 – tanAtanB • tan (A-B) = tanA – tanB 1 + tanAtanB 10. Double angle results • sin2A = 2sinAcosA • cos2A = cos²A – sin²A = 2cos²A –1 = 1 – 2sin²A • tan2A = 2tanA

1 – tan²A 11.Integration of trig functions

∫sin (ax + b) dx = -cos (ax + b)

+C

a

∫cos (ax + b) dx = sin (ax + b)

+C

a

∫sec² (ax + b) dx = tan (ax + b)

+C

a

12. Integration of sin²x or cos²x

∫ sin²x dx = ½ ∫1 – cos2x dx = ½ ( x – sin2x) + C 2

∫ cos²x dx = ½ ∫cos2x + 1 dx = ½ (sin2x + x) + C 2

∫ sinmxcosx dx =

sinm+1x + C m+ 1

13.Angles between two lines tanα =

m1 –m2 1+ m1m2

N.B: Angle between two curves is defined as the angle between the tangents to the curves of that point.

14.The t results sinø = 2t 1 + t² cosø = 1 - t² 1 + t² tanø = 2t 1 - t² m m

15. Using subsidiary angle to solve trig equation asinø + bcosø = C Asin (ø + α) Or Acos (ø + α)

Inverse Functions 1. Finding the inverse e.g. f(x) = 2x Domain = -2, -1 , 0 , 1 , 2 Range = -4, -2, 0, 2, 4 f-1(x) = ½x (swap x and y values, make y the subject) Domain = -4, -2, 0, 2, 4 (domain and range are Range = -2, -1 , 0 , 1 , 2 interchanged for inverse) 2. Properties of the function and its inverse a) f-1[f(x)] = f [f-1(x)] = x b) Domain and range are interchanged for the function and its inverse c) A function has an inverse if a horizontal line intersects the graph at one point only, ie: “For every value of x there is one value of y, and for every value of y there is only one value of x… There is a one to one correspondence between the x and y values” 3. The graph of the inverse function a) Same scales are used b) Graph of function reflects inverse about the line y=x c) If function and inverse intersect, they do so on line y=x d) midpoint of a point and its inverse lies on line y=x 4. Special result dy X dx = 1 dx dy -

5. Inverse sine function y = sin-1x ‫ת‬

2 -1

1 -‫ת‬ 2

 Domain= -1 ≤ x ≤ 1  Range= -‫ ≤ ת‬y ≤ ‫ת‬ 2 2  Function is always increasing  Odd function  “y is an angle with a sine of x” 6. Inverse cosine function: y = cos-1x ‫ת‬ ‫ת‬

2 -1  Domain= -1≤x≤1  Range= 0≤y≤‫ת‬  Always decreasing  Neither odd nor even

-

1

7. Inverse tangent function: y = tan-1x

‫ת‬

2

-‫ת‬ 2  Domain = all x  Range = -‫ < ת‬y < ‫ת‬ 2 2  Always increasing  Odd function 8. Differentiation of inverse trig functions: o y = sin-1f(x) y’ = 1 √[1-[f(x)]² o y = cos-1f(x)

y’ = -1 √[1-[f(x)]² o y = tan-1f(x)

y’= 1 1 + [f(x)]² 9. Integration of inverse functions:

∫ dx √(a²-x²)

∫ dx a²+x² -

For circle geometry formulae, see separate booklet…

= sin-1 x + C a -1 = -cos x + C a = 1tan-1 x a a

+C

Polynomials 1. A polynomial is an algebraic expression with many terms. 2. The polynomial P(x) P(x)= Po + P1x + P2x² +….. Pn-1xn-1 + Pnxn 3. Special polynomials: Monic: Coefficient of leading term is 1 Constant: Polynomial of degree 0 Linear: Polynomial of degree 1 Quadratic: Polynomial of degree 2 Cubic: Polynomial of degree 3 Zero Polynomial: Polynomial with all coefficients equal to 0 4. The Zeros and Roots The zeros of a polynomial P(x) are the solutions to P(x) = 0 5. Graphing the polynomial Finding the zeros help to sketch the graph, especially if y = P(x) can be written as a product of linear factors 6. The Zeros of a Polynomial (i) The Quadratic polynomial: ax² +bx + c is a quadratic polynomial The solutions are the zeros o o o

 >0, roots real/different; parabola intersects x-axis at 2 points  = 0, roots real/equal; parabola intersects x-axis at 1 point  < 0, no real roots

(ii) The General polynomial: P(x)= Po + P1x + P2x² +….. Pn-1xn-1 + Pnxn P(x) = 0 can have as many as n distinct roots 7. Division of polynomials P(x) = A(x) . Q(x) + R(x) Polynomial Divisor

Quotient Remainder

[Where degree R(x)< degree A(x)] Use long division! n n n n

8. The Factor Theorem: (x-a) is a factor of the Polynomial P(x) if P(a) = 0 HINT: When factorising look at the factors of the constant term; To find factors of a Polynomial, we usually find a factor by trial and error and then use long division to find any others.

9. Relationship between roots and coefficient: • Quadratic polynomial: ax² +bx + c = 0 Roots are α, β α + β = -b a αβ = c a • Cubic: ax³ + bx² +cx + d = 0 Roots are α, β, γ α + β + γ = -b a αβ + βγ + αγ = c a αβγ = -d a •

Quartic: ax4 + bx³ + cx² +dx + e = 0 Roots are α, β, γ, δ α + β + γ + δ = -b a αβ + βγ + αγ + αδ + βδ + γδ = c a αβγ + βδγ + αδγ = -d a αβγδ = e a

10. Useful results:

4

α² + β² =

(α + β)² - 2 αβ

α² + β² + γ² =

(α + β + γ)² - 2(αβ + βγ + αγ)

α² + β² + γ² + δ² =

(α + β + γ + δ)² - 2(αβ + βγ + αγ + αδ + βδ + γδ)

11. Deductions from the Factor Theorem:

 If a1 and a2 are zeros of P(x), then (x - a1)(x - a2) is a factor of P(x)

 If P(x) has n distinct zeros, then the polynomial can be expressed as P(x) = Pn(x - a1)(x - a2) (x – a3)…(x - an)

 If P(x) has degree n, then it cannot have more than n zeros 12. Estimating the roots: (i) Halving the Interval Method

a

c

b

Let the root be c. -1 approximation to x=c is a+b 2 st

If f(a+b) < 0 2 then root lies between x = a+b and x=b 2 -2nd approximation will be the midpoint of x = a+b and x=b 2 etc! (ii) Newton’s Method of Approximation If x=a is the root to f(x), and a1 is a given close approximation a2 (2nd approximation) = a1 – f(a1) f’(a1) etc! 13. Unfavourable approximations for Newton’s Method: → Choosing an approximation on wrong side of root → If a root is a point of inflection → Approximation is at or near a stationary point

Mathematical Induction 1. Definition: Mathematical induction is a process used to prove certain statements or results. 2. Process of mathematical induction Step 1: Show result is true for n = 1 Step 2: Assume result is true for n = k Step 3: Show result is true for n = k + 1 Step 4: “Since the result is true for n = 1, then from step 3 the result will be true for n = 1+1=2, n= 2+1=3 and so on, for all positive integral values of n.” 3. Proving results of series HINT: Sk+1 = Sk + Tk+1 4. Proving divisibility e.g. Prove 32n – 1 is divisible by 8 Use steps, and for n = k assume: 32k – 1 =M 8 [Where M is a whole number.] 5. Proving inequalities e.g. Prove 5n ≥ 1 + 4n Use steps, then for step 3, to show 5k+1 ≥ 1+ 4(k+1): 5k+1 = 5. 5k 5k+1 ≥ 5 (1+4k) 5k+1 ≥ 20k +5 5k+1 ≥ 1+ 4(k+1) + 16k Therefore, 5k+1 ≥ 1+ 4(k+1) 6. Misc. Examples e.g Prove 2 X 1! + 5 X 2! + 10 X 3! +... (n² + 1)n! = n(n+1)! Use steps, then from step 3 show that Sk+1 = Sk + Tk+1 ie. (k+1)(k+2)! = k(k+1)! + (k²+2k+2) (k+1)! NOTE: n! means factorial n: n! = n (n-1)(n-2)(n-3) etc!

2 2

Methods of Integration 1. Integration by Substitution: If u = f(x) then du = f’(x) dx and du = f’(x) dx

∫

e.g. Solve e5x + 2 dx using substitution u = 5x+2

∫

= eu du 5

∫

=1 eu du 5 =1 e5x + 2 + C 5

u = 5x + 2 du = 5 dx dx = du 5

2. Substitution with definite integrals Change the limits of the new variable To evaluate, there is no need to revert to the “old” variable

5 u u 5

Parametric Equations of the Parabola 1. Recall: Equation of parabola with focus at S (0,a) and equation of directrix y = -a x² = 4ay [Cartesian Equation] 2. x² = 4ay can also be represented by two parametric equations: x = 2at y = at² where t is a third variable called a parameter and gradient at this point = t 3. Equation of chord joining P (2ap, ap²) and Q (2aq, aq²) y-y1 = m(x-x1) where m = ap² - aq² 2ap – 2aq =p+q 2 Eqn: y - ap² = p + q (x – 2ap) 2 (p+q)x -2y -2apq = 0 4. Focal chord: (0,a) satisfies equation of chord pq = -1 5. Equation of tangent and normal to x² = 4ay at a point P (2ap, ap²) x² = 4ay y = x² 4a y’ = x 2a y’ = 1 X 2ap 2a =p TANGENT Eqn: y - ap² = p(x - 2ap) y – px +ap² = 0 NORMAL Eqn: y - ap² = -1 (x - 2ap) p py + x -2ap –ap³ = 0

6. The Chord of Contact PQ (pts P, Q of contact of the 2 tangents drawn from T {x0,y0} to the parabola x² = 4ay) x x0 = 2a (y + y0) NOTE: T (x0,y0) is an external pt to x² = 4ay if x0² > 4ay0

7. Other helpful formulae: _______________ Distance: √(x1² - x2²) + (y1² - y2²) Midpoint: x1 + x2, y1 + y2 2 2 Gradient: y1 - y2 x1 - x2 8. Locus Problems: Use all formulae and substitution into Cartesian equations to find locus. Always describe the locus.

Rates of Change 1. 2 Unit (2 variables): ~ Rate of change refers to how one quantity changes in relation to the other. i.e. Rate of change of a quantity, Q, is given by: dQ dt To find the initial quantity, INTEGRATE To find the rate of change, DIFFERENTIATE 2. Extension (more than 2 variables): Use the chain rule, i.e. dQ = dQ X dx dt dx dt [You may need to use it twice sometimes]

3. Area/Volume Equations: Cylinder: V = π r²h SA = 2 π r h Cone: V = ⅓ π r²h Sphere: V = 4 π r³ 3 SA = 4 π r² 4. Exponential Growth and Decay: When dQ = kQ (the rate of change is directly proportional to quantity) dt The solution to this equation is: Q = Aekt A and k are constants (k = rate constant) t = time

k

5. Further Exponential Growth and Decay: In ideal situations, N = Aekt In practical situations, a more realistic model is: N = P + Aekt ~ k and P are which is a solution to constants. dN = k (N – P) ~ Where k0, N→∞ 6. Motion in a straight line: Displacement (s or x): Position of the particle from 0 Velocity (V or x) : Rate of change of displacement (derivative) If particle is travelling → it is positive If particle is travelling ← it is negative If particle is stationary/at rest, velocity is zero.

Acceleration (a or x): brings about a change in velocity (second derivative of s, first derivative of V) If a is positive, particle is accelerating → If a is negative, particle is accelerating ← If a = 0 particle is moving at a constant/uniform velocity If decelerating, particle is moving in opposite direction If accelerating, particle is moving in same direction

7. Using primitives x= ∫ V dt V = ∫ a dt 8. Distance travelled Distance = area under a velocity time curve 9. Acceleration as a function of Displacement or Velocity (strictly extension): a = V dV dx a= d (½ V²) dx

k k

10.Simple Harmonic Motion: The particle moves backwards and forwards along a straight line about a central position (say, the origin) e.g. mass on a spring, pendulum _________________________l_________________________ 0 a = - n²x Where n is a constant

11. Further SHM equations: Amplitude = a Period = 2 π n

V² = n² (a²- x²) x = a cos nt 12.Notes on SHM:

 -a ≤ x ≤ a 

Starts at x=a, moves with increasing speed towards 0, moves to x=-a with decreasing speed, stops then reverses direction

 Max displacement: x=a  Max speed: x=0  Max acceleration: x=a  Least acceleration: x=0 13. Centre of motion not at x = 0: a = -n² (x – C) 14.Projectile Motion: ~OA = range ~Time taken for projectile to reach A is called time of flight ~ h is the greatest height. At h, y = 0 ~ θ = the angle projected

y V h

0

θ

A

x

 Motion in x direction and y direction considered separately  x=0 y = -g (gravity)  at t=0, V y x= Vcosθ y = Vsinθ  Therefore these should be

x

incorporated and derived into eqns

15. Projectile motion from a height: At t = 0, x= 0 y=h (Where h = constant height, e.g. building) 16. Projectile motion where projectile is thrown horizontally x=V y=0

The Binomial Theorem 1. Pascal’s Triangle: Can be used to determine the coefficients of x in the expansion of (1 + x)n.

2. (1 + x)n has (n + 1) terms 3. Expansion of (a + x)n: Same coefficients as (1 + x)n  (n +1) terms  Power of a decreases by one, while power of x increases by one  For every term, the sum of the powers of a and x is n 

e.g. (a + x)4 = [a(1 + x/a)] 4 a4 (1 + x/a) 4 a4 + 4a³x +6a²x² + 4ax³ +x4 4. Determining the term independent of x (ie, x0): a) Expand binomials b) Multiply terms that gives independent term n n n n 4 4 4 4 4 4 0

5. nCr is the coefficient of xn in the expansion of (1 + x)n. Formula: n! r! (n - r)! NOTE: Can also be written in sigma notation.

6. Other important results: n

Co = 1 Cn = 1 n C1 = n n  Cr = nCn-r   

n

7. More notation: (a + x)n: Tk+1 = nCk an – k x k NOTE: To find term independent of x, let sum of powers equal zero, then solve for k!

8. Greatest coefficient:

Tk+1 Tk

>1

9. Relations between coefficients: e.g. By comparing coefficients of (1 + x)6 and (1 + x)³ (1 + x)³, find coefficient of x² in (1 + x)6:

10. Proving results: n n n n n n n n n n n 6 6

Many of these results can be proved by first writing (1 + x)n = nCo + nC1 x + nC2 x² +….. nCn xn and then… a) substituting x = 1 or x = -1 to both sides OR b) differentiating or integrating both sides and then substituting x = 1 or x = -1. 11. Counting Techniques:  If one event can happen in p different ways and after this, another event can happen in q different ways, then the two successive events can happen in pq ways.  If repetition is not allowed, the number of ways is usually n! (n factorial ways) 12. Permutations:  Permutations refer to the number of arrangements that is possible when order is important n  Pr is used, when n is the total number and r is the number “counted at a time”  nPr formula: n! (n - r)! 13. Arranging n items with r repetitions: If n items must be arranged where say, three items are repeated, p, q and r times, then the number of arrangements possible will be: n! p! q! r! 14. Combinations: Arrangements of items where order is not important n Pr r! 15.Counting Techniques and Probability: n n n n n n n n n

Use nCr or nPr, depending on the question, and divide by total ways it can happen. 16. Binomial Probability: Occasions where just two possible outcomes exist from an event, e.g. tossing a coin or dice. For n trials, No. of trials

(p + q)n = nCoqn p0 + nC1 qn-1 p1 +….. nCr qn-r pr +….. nCn q0 pn Failure

Success

P ( no success)

P (1 success)

P (r successes)

P (n successes)

The probability that an event will occur exactly r times in n independent trials is given by n

Cr qn-r pr

17. Arrangements about a circle or ring: The number of arrangements of n items around a circle/ring is (n – 1)! NOTE: For beads or keys, clockwise or anti-clockwise, arrangements are the same so arrangements of n items will be ½ x (n – 1)!

General Solutions to Trig Equations n n n n n 0 n n 1 n n n 0 n n n

1. tan θ tan θ = a θ = nπ + tan-1a 2. cos θ cos θ = a θ = 2nπ + cos-1a 3. sin θ sin θ = a θ = nπ + (-1)n sin-1a NOTE: Solutions can be found by either adding or subtracting 2π

4. Important Results: ♣ tan-1(-x) = - tan-1x ♣ cos-1(-x) = π - cos-1x

♣ sin-1(-x) = - sin-1x

n -