IB Math AI Formula Booklet

Diploma Programme Mathematics: applications and interpretation formula booklet For use during the course and in the examinations First examinations 2021 Version 1.1 Contents Prior learning SL and HL 2 HL only 2 Topic 1: Number and algebra SL and HL 3 HL only 4 Topic 2: Functions SL and HL 5 HL only 5 Topic 3: Geometry and trigonometry SL and HL 6 HL only 7 Topic 4: Statistics and probability SL and HL HL only 9 10 Topic 5: Calculus SL and HL 11 HL only 11 Prior learning – SL and HL Area of a parallelogram A = bh , where b is the base, h is the height Area of a triangle 1 A = (bh) , where b is the base, h is the height 2 Area of a trapezoid 1 A = ( a + b) h , where a and b are the parallel sides, h is the height 2 Area of a circle A = πr 2 , where r is the radius Circumference of a circle C = 2πr , where r is the radius Volume of a cuboid V = lwh , where l is the length, w is the width, h is the height Volume of a cylinder V = πr 2h , where r is the radius, h is the height Volume of prism V = Ah , where A is the area of cross-section, h is the height Area of the curved surface of a cylinder A = 2πrh , where r is the radius, h is the height Distance between two points ( x1, y1 ) and ( x2 , y2 ) d = ( x1 − x2 )2 + ( y1 − y2 )2 Coordinates of the midpoint of a line segment with endpoints ( x1 , y1 ) and ( x2 , y2 )  x1 + x2 y1 + y2  ,   2   2 Prior learning – HL only Solutions of a quadratic equation The solutions of ax 2 + bx + c = 0 are x = −b ± b 2 − 4ac ,a ≠0 2a 2 Topic 1: Number and algebra – SL and HL SL 1.2 The nth term of an arithmetic sequence u n = u1 + (n − 1) d The sum of n terms of an arithmetic sequence Sn = The nth term of a geometric sequence u n = u1 r n −1 The sum of n terms of a finite geometric sequence Sn = SL 1.4 Compound interest r   FV = PV × 1 +  , where FV is the future value,  100 k  PV is the present value, n is the number of years, k is the number of compounding periods per year, r% is the nominal annual rate of interest SL 1.5 Exponents and logarithms a x = b ⇔ x = loga b , where a > 0, b > 0, a ≠ 1 Percentage error ε= SL 1.3 SL 1.6 n n ( 2u1 + (n − 1) d ) ; S n = (u1 + u n ) 2 2 u1 ( rn −1) u1 (1 − rn ) , r ≠1 = 1− r r −1 kn vA − vE ×100% , where vE is the exact value and vA is vE the approximate value of v 3 Topic 1: Number and algebra – HL only AHL 1.9 Laws of logarithms log a xy = log a x + log a y x log a = log a x − log a y y log a x m = m log a x for a, x, y > 0 AHL 1.11 The sum of an infinite geometric sequence S∞ = AHL 1.12 Complex numbers z = a + bi Discriminant ∆ = b 2 − 4ac Modulus-argument (polar) and exponential (Euler) form z = r (cosθ + isin θ ) = reiθ = r cisθ AHL 1.13 AHL 1.14 AHL 1.15 u1 , r <1 1−r Determinant of a 2 × 2 matrix a b A =  ⇒ det A = A = ad − bc  c d Inverse of a 2 × 2 matrix 1  d a b −1 A =  ⇒ A =  det A  −c  c d Power formula for a matrix −b  , ad ≠ bc a  M n = PD n P −1 , where P is the matrix of eigenvectors and D is the diagonal matrix of eigenvalues 4 Topic 2: Functions – SL and HL SL 2.1 SL 2.5 Equations of a straight line y = mx + c ; ax + by + d = 0 ; y − y1 = m ( x − x1) Gradient formula m= Axis of symmetry of the graph of a quadratic function f ( x) = ax 2 + bx + c ⇒ axis of symmetry is x = − y2 − y1 x2 − x1 b 2a Topic 2: Functions – HL only AHL 2.9 Logistic function f ( x) = L , L , k,C > 0 1 + Ce − kx 5 Topic 3: Geometry and trigonometry – SL and HL SL 3.1 Distance between two points ( x1, y1, z1 ) and d = ( x1 − x2 ) 2 + ( y1 − y 2 ) 2 + ( z1 − z 2 ) 2 ( x2 , y2 , z2 ) Coordinates of the midpoint of a line segment with endpoints ( x1, y1, z1 )  x1 + x2 y1 + y2 z1 + z2  ,  2 , 2 2   and ( x2 , y2 , z2 ) V= Volume of a right cone 1 V = πr 2h , where r is the radius, h is the height 3 Area of the curved surface of a cone SL 3.2 SL 3.4 1 Ah , where A is the area of the base, h is the height 3 Volume of a right-pyramid A = πrl , where r is the radius, l is the slant height Volume of a sphere 4 V = πr 3 , where r is the radius 3 Surface area of a sphere A = 4πr 2 , where r is the radius Sine rule a b c = = sin A sin B sin C Cosine rule c2 = a2 + b2 − 2ab cos C ; cos C = Area of a triangle 1 A = ab sin C 2 Length of an arc l= θ 360 2 2 2 a +b −c 2ab × 2 πr , where θ is the angle measured in degrees, r is the radius Area of a sector A= θ 360 × πr 2, where θ is the angle measured in degrees, r is the radius 6 Topic 3: Geometry and trigonometry – HL only AHL 3.7 Length of an arc l = rθ , where r is the radius, θ is the angle measured in radians Area of a sector 1 A = r2 θ , where r is the radius, θ is the angle measured in 2 radians AHL 3.8 Identities 2 2 cos θ + sin θ = 1 tan θ = AHL 3.9 Transformation matrices sinθ cos θ  cos 2 θ sin 2 θ   sin 2θ − cos 2θ  , reflection in the line y = (tan θ) x   k 0   , horizontal stretch / stretch parallel to x-axis with a scale  0 1 factor of k 1 0    , vertical stretch / stretch parallel to y-axis with a scale 0 k  factor of k  k 0  0 k  , enlargement, with a scale factor of k, centre (0, 0)    cosθ  θ  sin − sinθ   , anticlockwise/counter-clockwise rotation of cosθ  angle θ about the origin (θ > 0 )  cosθ sinθ    , clockwise rotation of angle θ about the origin  −sin θ cos θ  (θ > 0 ) 7 AHL 3.10 AHL 3.11 AHL 3.13 Magnitude of a vector  v1  v = v12 + v22 + v32 , where v =  v 2  v   3 Vector equation of a line r = a + λb Parametric form of the equation of a line x = x 0 + λl , y = y 0 + λm, z = z 0 + λn Scalar product  v1   w1      v ⋅ w = v1w1 + v 2w 2 + v 3w 3 , where v =  v 2  , w =  w2  v  w   3  3 v ⋅ w = v w cos θ , where θ is the angle between v and w v1 w1 + v2 w2 + v3 w3 v w Angle between two vectors cos θ = Vector product  v2 w3 − v3 w2  v × w =  v3w1 − v1w3 v w − v w  1 2 2 1   v1   w1       , where v = v 2 , w = w2      w   v   3   3 v × w = v w sin θ , where θ is the angle between v and w Area of a parallelogram A = v × w where v and w form two adjacent sides of a parallelogram 8 Topic 4: Statistics and probability – SL and HL SL 4.2 Interquartile range IQR = Q3 − Q1 SL 4.3 k i i Mean, x , of a set of data SL 4.5 ∑fx x= i =1 , where n = n k ∑f i i =1 n ( A) n (U ) Probability of an event A P ( A) = Complementary events P ( A) + P ( A′) = 1 Combined events P ( A ∪ B) = P ( A) + P ( B) − P ( A ∩ B ) Mutually exclusive events P ( A ∪ B) = P ( A) + P ( B) Conditional probability P ( A B) = Independent events P ( A ∩ B) = P ( A) P ( B) SL 4.7 Expected value of a discrete random variable X E ( X ) = ∑ x P ( X = x) SL 4.8 Binomial distribution SL 4.6 P ( A ∩ B) P (B) X ~ B ( n , p) Mean E ( X ) = np Variance Var ( X ) = np (1 − p) 9 Topic 4: Statistics and probability – HL only AHL 4.14 Linear transformation of a single random variable Linear combinations of n independent random variables, X 1, X 2 , ..., Xn E (aX + b ) = a E ( X ) + b Var ( aX + b) = a 2 Var ( X ) E (a1 X 1 ± a2 X 2 ± ... ± a n X n ) = a1 E (X 1 ) ± a2 E (X 2 ) ± ... ± a nE (X n ) Var ( a1 X1 ± a2 X 2 ± ... ± an Xn ) = a1 2 Var ( X1 ) + a2 2 Var ( X2 ) + ... + an2 Var ( X n ) Sample statistics Unbiased estimate of 2 population variance sn −1 AHL 4.17 AHL 4.19 s 2n −1 = n 2 sn n −1 Poisson distribution X ~ Po( m) Mean E(X ) = m Variance Var ( X ) = m Transition matrices T n s 0 = sn , where s0 is the initial state 10 Topic 5: Calculus – SL and HL SL 5.3 SL 5.5 Derivative of x Integral of x n ∫x n Area of region enclosed by a curve y = f ( x) and the x-axis, where f ( x) > 0 SL 5.8 n n 1 f ( x) = x ⇒ f ′( x) = nx − The trapezoidal rule x + + C , n ≠ −1 n +1 n 1 n dx = b A = ∫ yd x a 1 h (( y0 + y n ) + 2( y1 + y2 + ...+ y n−1 ) ) , 2 b −a where h = n ∫ b a y dx ≈ Topic 5: Calculus – HL only AHL 5.9 Derivative of sin x f ( x) = sin x ⇒ f ′( x) = cos x Derivative of cos x f ( x) = cos x ⇒ f ′( x) = −sin x Derivative of tan x f ( x) = tan x ⇒ f ′( x) = Derivative of e x 1 cos 2 x f ( x) = e x ⇒ f ′( x) = e x 1 x Derivative of ln x f ( x) = ln x ⇒ f ′( x) = Chain rule y = g (u ) , where u = f ( x) ⇒ Product rule y = uv ⇒ Quotient rule du dv −u v u dy x x d d y= ⇒ = v v2 dx dy dy du = × dx du dx dy dv du =u +v dx dx dx 11 AHL 5.11 Standard integrals 1 ∫ x dx = ln x +C ∫ sin x dx = − cos x + C ∫ cos x dx = sin x + C 1 ∫ cos ∫e AHL 5.12 Area of region enclosed by a curve and x or y-axes Volume of revolution about x or y-axes AHL 5.13 AHL 5.16 x 2 x = tan x + C d x = ex + C b b b b A = ∫a y d x or A = ∫a x dy 2 2 V = ∫ a πy dx or V = ∫a πx dy dv d2 s dv = =v dt dt 2 ds Acceleration a= Distance travelled from t1 to t2 distance = Displacement from t1 to t2 displacement = Euler’s method ∫ t2 t1 v(t ) d t ∫ t2 t1 v( t) d t yn +1 = yn + h × f (xn , yn ) ; xn +1 = xn + h , where h is a constant (step length) Euler’s method for coupled systems xn +1 = xn + h × f1 (xn , yn , tn ) yn +1 = yn + h × f2 (xn , y n , t n ) tn +1 = tn + h where h is a constant (step length) AHL 5.17 Exact solution for coupled linear differential equations x = Ae λ1t p1 + Be λ2t p 2 12