HSC 3U Maths Formulae
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3U Maths Formulae HSC for the whole syllabus...
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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 = arn1 4. Sum of n terms of a GP: Sn = a (1  rn) Sn = a  rL 1–r 1r (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 xaxis 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 yaxis 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 xaxis: b
V=π
∫ y² dx
a When rotated around yaxis: b
V=π
∫ x² dy
a
10. Volumes of Rotation: Rotating area between 2 curves: When rotated around xaxis: b
V=π
∫ f(x)²  g(x) ² dx
a When rotated around yaxis: 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 (AB) = sinAcosB – cosAsinB cos (A+B) = cosAcosB – sinAsinB cos (AB) = cosAcosB + sinAsinB tan (A+B) = tanA + tanB 1 – tanAtanB • tan (AB) = 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 f1(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) f1[f(x)] = f [f1(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 = sin1x 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 = cos1x ת ת
2 1 Domain= 1≤x≤1 Range= 0≤y≤ת Always decreasing Neither odd nor even

1
7. Inverse tangent function: y = tan1x
ת
2
ת 2 Domain = all x Range =  < תy < ת 2 2 Always increasing Odd function 8. Differentiation of inverse trig functions: o y = sin1f(x) y’ = 1 √[1[f(x)]² o y = cos1f(x)
y’ = 1 √[1[f(x)]² o y = tan1f(x)
y’= 1 1 + [f(x)]² 9. Integration of inverse functions:
∫ dx √(a²x²)
∫ dx a²+x² 
For circle geometry formulae, see separate booklet…
= sin1 x + C a 1 = cos x + C a = 1tan1 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 +….. Pn1xn1 + 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 xaxis at 2 points = 0, roots real/equal; parabola intersects xaxis at 1 point < 0, no real roots
(ii) The General polynomial: P(x)= Po + P1x + P2x² +….. Pn1xn1 + 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: (xa) 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 (n1)(n2)(n3) 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) yy1 = m(xx1) 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 = nCnr
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 qn1 p1 +….. nCr qnr 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 qnr 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 anticlockwise, 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 + tan1a 2. cos θ cos θ = a θ = 2nπ + cos1a 3. sin θ sin θ = a θ = nπ + (1)n sin1a NOTE: Solutions can be found by either adding or subtracting 2π
4. Important Results: ♣ tan1(x) =  tan1x ♣ cos1(x) = π  cos1x
♣ sin1(x) =  sin1x
n 
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