Mathematics Applications and Interpretation Formula

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 =π r2 , where r is the radius Circumference of a circle C = 2π r , 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 =π r 2 h , 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 A = 2π rh, 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 )  Area of the curved surface of a cylinder x1 + x2 2 , y1 + y2 2 Prior learning – HL only Solutions of a quadratic equation The solutions of Mathematics: applications and interpretation formula booklet ax 2 + bx+ c= 0 are x = −b ± b 2− 4ac , a≠ 0 2a 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 = vA vE ×100% , where vE is the exact value and SL 1.3 SL 1.6 n ( 2 u1 + ( n −1) d) ; Sn = n ( u1 + un ) 2 2 u1 ( r n −1) u1 (1 − r n ) = , r ≠1 1− r r −1 kn vE the approximate value of v Mathematics: applications and interpretation formula booklet vA is Topic 1: Number and algebra – HL only AHL 1.9 log a xy =log a x+ log a y x log a = log a x− log a y y Laws of logarithms log a x m = mlog 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 θ z = r (cosθ+ isinθ )= r ei = r cisθ AHL 1.15 Power formula for a matrix n n −1 −1 = 1 det d −c −b , ad ≠ bc a  a b  = ⇒  c d  = ad − bc  Inverse of a 2 × 2 matrix a b  = ⇒ det =  c d   2×2  Determinant of a matrix  AHL 1.14  AHL 1.13 u1 , r <1 1−r = , where is the matrix of eigenvectors and the diagonal matrix of eigenvalues Mathematics: applications and interpretation formula booklet is 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) = ax2 + bx+ c ⇒ axis of symmetry is y2 − y1 x2 − x1 Topic 2: Functions – HL only AHL 2.9 Logistic function f ( x) = Mathematics: applications and interpretation formula booklet L L , k, C > 0 kx , 1 + Ce − x =− b 2a Topic 3: Geometry and trigonometry – SL and HL SL 3.1 Distance between two points ( x1, y1, z1 ) and d = ( x1 −x2 ) 2 + ( y1 − y2 ) 2 + ( z1 − z2 ) 2 x1 +x2 ,  2  Coordinates of the midpoint of a line segment with endpoints ( x1, y1, z1 )  ( x2 , y2 , z2 ) y1 + y2 2 , z1 + z2 2 and ( x2 , y2 , z2 ) Volume of a right-pyramid 1 V = Ah , where A is the area of the base, h is the height 3 Volume of a right cone 1 V = π r 2h , where r is the radius, h is the height 3 Area of the curved surface of a cone Volume of a sphere Surface area of a sphere SL 3.2 SL 3.4 A =π rl , where r is the radius, l is the slant height 4 V = π r 3 , where r is the radius 3 2 A = 4 r , where r is the radius Sine rule a b c = = sin A sin B sin C Cosine rule c2 = a 2 + b 2 − 2ab cosC ; cos C = a Area of a triangle A= Length of an arc l= 2 +b2 −c2 2ab 1 ab sin C 2 θ × 2π r , where θ is the angle measured in degrees, 360 r is the radius Area of a sector A= θ 360 × π r 2, where θ is the angle measured in degrees, the radius Mathematics: applications and interpretation formula booklet r is 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 A= 1 2 r 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   −sin θ cos θ  (θ > 0 ) Mathematics: applications and interpretation formula booklet θ about the origin AHL 3.10 AHL 3.11 Vector equation of a line Parametric form of the equation of a line AHL 3.13 = v12 + v22 + v32 , where Magnitude of a vector Scalar product x = x 0 + λ l, y = y0 + λ m, z= z0 +λ n ⋅ =v1w1 + v2 w2 + v3 w3 , where ⋅ =  w1   , = w2    w3      and vw+v w+ vw cos θ = 1 1 2 2 3 3 v2 w3 − v3 w2 = v3 w1− v1 w3 , where v1 w2 − v2 w1  ×   × Area of a parallelogram v1  = v 2  v 3 cos θ , where θ is the angle between  Vector product      = +  Angle between two vectors v1 = v 2 v 3 where parallelogram Mathematics: applications and interpretation formula booklet   ,   w1  =w2  w3      sin θ , where θ is the angle between = A= × v1  = v 2  v 3 and form two adjacent sides of a and Topic 4: Statistics and probability – SL and HL SL 4.2 SL 4.3 SL 4.5 IQR = Q3 − Q1 Interquartile range k ∑fx i i x= Mean, x , of a set of data Probability of an event A i =1 n P ( A) = , where n = k ∑f i i =1 n ( A) n( U ) 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) Mathematics: applications and interpretation formula booklet Topic 4: Statistics and probability – HL only AHL 4.14 Linear transformation of a single random variable E (aX + b) = a E ( X )+ b Linear combinations of n independent random variables, X1, X2 , ..., Xn E (a1X1 ± a2 X2 ± ...± an Xn )= a1 E ( X1) ± a2 E( X2) ± ...± anE( Xn) Var ( aX + b) = a 2 Var ( X) Var ( a1 X1 ± a2 X2 ± ...± an Xn) = a1 2 Var ( X1) + a2 2 Var ( X2) + ...+ an2Var ( Xn) Sample statistics Unbiased estimate of 2 population variance sn −1 AHL 4.17 AHL 4.19 n 2 s 2n −1 = sn n −1 Poisson distribution X ~ Po( m) Mean E ( X) = m Variance Var ( X ) = m Transition matrices n 0 = Mathematics: applications and interpretation formula booklet n , where 0 is the initial state Topic 5: Calculus – SL and HL SL 5.3 SL 5.5 n 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 f ( x) = xn ⇒ f ′( x) = nxn The trapezoidal rule dx = n+1 x + C, n +1 1− n≠ − 1 b A = ∫ yd x a 1 h (( y0 + yn ) + 2( y1 + y2+ ...+ yn−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 y= dy dv du =u + v dx dx dx du − dv v u u ⇒ d y = dx dx 2 v x v d Mathematics: applications and interpretation formula booklet dy dy d u = × dx du dx AHL 5.11 Standard integrals 1 ∫ x dx = ln x+ C ∫ sin x d x = −cos x + C ∫ cos x d x =sin x+ C 1 ∫ cos ∫e AHL 5.12 AHL 5.13 AHL 5.16 Area of region enclosed by a curve and x or y-axes x 2 x = tan x+ C d x = ex + C b b a a b b A = ∫ y d x or A = ∫ x d y Volume of revolution about x or y-axes V = ∫ y2 x or V = ∫ x2 y a a Acceleration a= Distance travelled from t1 to t2 distance = Displacement from t1 to t2 displacement Euler’s method dv d2 s dv = =v dt dt 2 ds ∫ t2 t1 v( t) d t t2 = ∫ v( t) d t t1 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+ yn +1 = yn+ h× f1 ( xn , yn , tn ) h× f2 ( xn , yn , tn ) tn +1 = tn + h where h is a constant (step length) AHL 5.17 Exact solution for coupled linear differential equations = Ae λ1t 1+ Be λ 2t Mathematics: applications and interpretation formula booklet 2