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Tangents and normals

- Tangent= y-y1=m(x-x1) - Normal= y-y1=-1/m(x-x1)

Area under a curve

- basically do it like definite integration - x axis is b and a (b as the larger value)

Rules of Logarithms

- y=a^x then x=loga(y) loga(1) = 0 loga(A) = 1 loga(A^x)=x x= a^loga(x) Product rule: log (AB) = log A + Log B Division rule: log (A/B) = log A - Log B Power rule: loga(x)^m= mloga(x) Change of base: logb(a)= logc(a) / logc(b) Special formulas: -log a(1/x)= -loga(x) -logb(a)= 1/loga(b)

Equation of a straight line

- y=mx+c - y-y1=m(x-x1) , where x1,y1 is the known point - bisector (bisects at 90 degree)

circular measure

-2π=360 , π=180 1. Change from degrees-->radians multiply π/180 2. Change from radians-->degrees multiply 180/π (as long as pi at opp direction then divisable)

position vector

-Always start from the first alphabet Eg: Find BA =BO+OA (the O is near each other, usually)

constant velocity

-The magnitude of the velocity is the speed Velocity= displacement/time taken Position vector(r)= Initial position(a) + time(t)velocity(v)

natural logarithm

-a logarithm with base e, written as ln - if y=e^x then x=In y - In(e^x)=x - In(e)= 0 - In(1)=0 , e(0)=1

Binomial Theorem

-igcse got provide formula - The term independent of x means that the term in which power of x= 0.

combination

-order doesn't matter - nCr (n is total) - object, student

permutation

-order matters - 0!= 1 - nPr (n is total, bigger no) - number,letter,code

Further definite integration

-smth like indefinite integration Eg: y=3x/√x^2+5 dy/dx= 15/√(x^2+5)^3 Hence evaluate ∫2,0 3/ √(x^2+5)^3 dx ∫15/√(x^2+5)^3= 3x/√x^2+5 = 1/5 [3x/√x^2+5] 2,0 (insert X value inside) =2/5

Basics of vectors

-use pythagorean theorem to count magnitude -vector -a and a is same length, opposite direction - a+b, a-b - AB = 1/2AC (collinear point) Unit vector 1. Find magnitude 2. 1/answer

Small increments and approximations

-δy/δx= dy/dx

Length, gradient and midpoint

1. Length= √(x2-x1)^2 + (y2-y1)^2 2. Gradient= y2-y1/x2-x1 3. Midpoint= (x1+x2/2 , y1+y2/2)

cubic expression

1. Let f(x)=equation 2. Positive & negative factors of number(c) 3. Find which one will make the equation=0 and that is the factor 4. (factor)(ax^2+bx+c) 5. equate coefficient of x^2 (b=?) 5. Find value of x

Factor Theorem

1. P(c)=0 , x-c is a factor of P(x) -->P(2)=0, x-2 2. P(b/a), ax-b is a factor of P(x) -->P(7/5)=0, (5x-7)

All Students Trust Cambridge (trigo)

1. Sin=180minus 2. Tan=180plus 3. Cos=360minus -if got minus for equation, use the rest two

Parallel and perpendicular lines

1. Two lines parallel=gradients are equal 2. Two perpendicular lines= m1 X m2= -1 3. Line has gradient m, line perpendicular to it has gradient -1/m

Graphs of cubic polynomials

1. When x=0 2. When y=0 3. y=k(x-a)(x-b)(x-c) 4. (a,0) (b,0) (c,0) -if k is positive, n shape -if k is negative, invert n shape

Inverse functions (written as f⁻¹(x))

1. Write function as y 2. Interchange the x and y variables 3. Rearrange to make y the subject - domain of f(x) is the range of f⁻¹(x) - range of f(x) is the domain of f⁻¹(x) -can only exist if its one-one mapping

solving modulus inequalities

1. [p]≤q --> -q≤p≤q 2. [p]≥p --> p≤-q , p≥q

Rules of Indices

1. a^m X a^n = a^(m+n) 2. a^m/a^n = a^(m-n) 3. (a^m)^n = a^mn 4. a^n X b^n = (ab)^n 5. a^n/b^n = (a/b)^n 6. a^0 = 1 7. a^-n= 1/a^n 8. a^1/n = n√a 9. a^m/n= (n√a^m)

roots of a quadratic equation

1. b2−4ac>0 (2 distinct roots, curve cuts x-axis) 2. b2−4ac=0 (2 equal roots, both same, curve touches one point only, line is tangent) 3. b2−4ac<0 (no roots, 0, entirely above/below x-axis)

Derivatives of Logarithmic Functions

1. d/dx(Inx)= 1/x 2. d/dx(In(ax+b))= a/ax+b -differentiate inside first

Derivatives of Exponential Functions

1. d/dx(e^x)= e^x Eg: e^5x =5 X e^5x =5e^5x 2. d/dx(e^ax+b)= ae^ax+b

Derivatives of Trigonometric Functions

1. d/dx(sin x)= cosx d/dx(sin(ax+b))= a cos (ax+b) -differentiate inside first, then write back ori equation 2. d/dx(cos x)= -sinx d/dx(cos(ax+b))= a -sin(ax+b) 3. d/dx(tan x)= sec^2x d/dx(tan(ax+b))= a sec^2(ax+b)

Stationary points/ Turning points (dy/dx=0)

1. d^2y/dx^2<0 (maximum point, sad face) 2. d^2y/dx^2>0 (minimum point, smile face) Steps: 1. Do dy/dx 2. When dy/dx=0 (stationary point) 3. Do d2y/dx2, to determine nature, when x=?

rate of change

1. dy/dt (rate of change of y) = dy/dx X dx/dt (rate of change of x) -divisable 2. dx/dy= 1 over dy/dx

Composite fuctions

1. fg(x) --> means do function g first then function f 2. f^2(x) --> ff(x)

Graphs of y=[f(x)]

1. find y intercept (when x=0) 2. find x intercept 3. x coordinates (total x/2) 4. y coordinates 5. min/max point -x will always remain but just need reflect -y will change

Graphs of Trigonometric Functions

1. sinx=0,1,0,-1,0 (curve n shape) 2. cosx=1,0,-1,0,1 (minimum point shape) --> period of 360 3. tanx is the weird looking one (period=180)

trigonometric equation

1. tanx=sinx/cosx 2. cos^2x+sin^2x = 1 3. cosecx= 1/sinx 4. secx= 1/cosx 5. cotx=1/tanx (cosx/sinx) 6. 1+tan^2x= sec^2x 7. cot^2x+1=cosec^2x

Kinematics (distance --> displacement, s speed--> velocity, v )

1. v=ds/dt 2. s=dv/dt 3. v=∫a 4. s=∫v -instantaneous rest (v=0)

Quadratic Inequalities

1. x>/≥ (larger than) --> < , > (side of graph) 2. x</≤ (smaller than) --> only one (</>) (area under graph) -when multiply/divide both sides of an inequality by a negative number than the sign must be reversed

Graphs of simple logarithmic and exponential functions

1. y=e^x when x=0, when y=0, asymptote is y 2. y=Inx when x=0, when y=0, asymptote is x

Differentiation (dy/dx is also gradient)

1. y=x^n, dy/dx= nx^n-1 -bx=b (left number oni) -c=0 (any no turn zero) Eg: 2x^3-5x+4 d/dx=6x^2-5

Integration of functions of the form 1/x and 1/ax+b (In）

1. ∫1/x dx= Inx + c 2. ∫1/ax+b dx= 1/a In(ax+b) + c Eg: ∫4/2-3x dx = 4∫ (1/-3) In2-3x = -4/3 In(2-3x) + c

Definite integration

1. ∫ba f(x) dx = [F(x)]ba =F(b)-F(a) -integrate inside first then only insert number to equation and calculate 2. ∫1/x dx= In|x| + c 3. ∫1/ax+b dx= 1/a In|ax+b| + c -answer always positive

Integration of sine and cosine functions

1. ∫cos x dx= sinx + c ∫cos (ax+b)dx= 1/a sin(ax+b) + c 2. ∫sin x dx= -cosx + c ∫sin (ax+b)dx= -(1/a) cos(ax+b) +c Eg: ∫(3cos2x + 5sin3x) dx = 3(1/2sin2x) + 5(-1/3cos3x) = 3/2(sin2x) - 5/3(cos3x) + c

Integration: Exponential Function

1. ∫e^x dx= e^x + c 2. ∫e^ax+b dx= (1/a)e^ax+b + c Eg: ∫e^7x+4 dx =1/7 e^7x+4 =e^7x+4/7 + C

Rules of Surds

1.√ab = √a X √b 2. √a/b = √a/√b 3. √a X √a = a 4. 1/√a X √a --> plus times minus, minus times plus --> make sure denominator cannot have surd --> (a+b)(a-b)=a^2-b^2

Angles between 0° to 90°

45 degree: sin45°= √2/2 cos45°= √2/2 tan45°= 1 60 degree: sin60°=√3/2 cos60°= 1/2 tan60°= √3 30 degree: sin30°= 1/2 cos30°=√3/2 tan30°= 1/√3

y=Atrig(Bx+C)

A= amplitude (tan no) B= period (period/b) (length of one cycle) c= vertical translation (0 c)

Arthmetic Series

a= first term d=common difference l=last term -nth term= a+(n-1)d -Sum of the terms= Sn=n/2(a+l) (last term given) = Sn=n/2(2a+(n-1)d) (if dk last term) -r =T2-T1

When angle is θ

arc length=rθ area of sector= 1/2r^2θ

Quotient Rule

d/dx (u/v)= v(du/dx) - u(dv/dx) ━━━━━━━━━ v^2

second derivates

d^2y/dx^2 (do two times differentiation)

Chain Rule (dy/dx)

dy/du X du/dx （both du cancel each other) Eg: y=(2x+3)^8 du/dx=2 du/dx=8u^7 dy/dx= 2 X 8u^7 = 2 X 8(2x+3)^7 =16(2x+3)^7 -dy/dx= 8(2x+3)^7(2) =16(2x+3)^7

Product Rule

dy/dx = u(dv/dx) + v(du/dx) eg: y=(x+2)^2(2x+3)^3 Let u= (x+2)^2 Let v= (2x+3)^3

Remainder Theorem

example -when f(x) divided by x-1, remainder is 1, f(1)=1

completing the square

f(x)=a(x-h)^2+k - a>0, minimum point (h,k) - a<0, maximum point (h,k) - max/min point --> stationary points/turning points

factorial notation

n! = nX(n-1)! eg: 4! (4X3X2X1)

Mappings

one-one and many-one are called functions input values --> domain(x) output values-->range(Y)

geometric progression

r= common ratio (divided) -nth term= ar^n-1 -r =T2/T1 -Sn= a(r^n-1)/(r-1) -Sum to infinity= S∞ = a / (1-r )

indefinite integrals (reverse of d/dx)

∫x^n dx = (1/n+1)x^n+1 + c Eg: ∫12x^5 = 12/6 x^6 =2x^6 + c Eg: ∫(3x-8)^5 dx = 1/3(6) (3x-8)^6 =1/18 (3x-8)^6 -differentiate inside first (ax+b) =a

practical minimum and maximum problems

Areas 1. Sphere= 4pi^2 2. Hemisphere= 2pi^2 Volume 1. Sphere= 4/3pi^2 2. Hemisphere= 2/3pi^2

Further indefinite integration

If d/dx [F(x)]= f(x) then ∫f(x) dx= F(x) + c (POPULAR !!!!) Eg: Given that y= x√x^2-4 dy/dx= 2x^2-4/√x^2-4 Hence find ∫x^2-4/ √x^2-4 ∫2x^2-4/ √x^2-4 = x√x^2-4 = 1/2 (x√x^2-4) + c -what times 2 become 1, is 1/2 -basically need to times to make it the ori (y)

Non-linear equations

Y= mX+c -Y and X must be only x or y -m and c must be a constant (a,b..)

Modulus

[-x]=positive 1. [ax+b]--> ax+b=k & ax+b=-k

Solving modulus equations

[a]=[b] --> a^2=b^2