Diploma Programme Mathematics: applications and interpretation formula booklet For use during the course and in the examinations First examinations 2021 Version 1.1 © International Baccalaureate Organization 2019 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 = A 1 (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 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 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= Coordinates of the midpoint of a line segment with endpoints ( x1 , y1 ) and ( x2 , y2 ) x1 + x2 y1 + y2 , 2 2 ( x1 − x2 ) 2 + ( y1 − y2 ) 2 Prior learning – HL only Solutions of a quadratic equation The solutions of ax 2 += bx + c = 0 are x Mathematics: applications and interpretation formula booklet −b ± b 2 − 4ac ,a≠0 2a 2 Topic 1: Number and algebra – SL and HL SL 1.2 SL 1.3 The nth term of an arithmetic sequence un = u1 + (n − 1) d The sum of n terms of an arithmetic sequence S n= The nth term of a geometric sequence un = u1r n −1 n n ( 2u1 + (n − 1) d ) ; Sn= (u1 + un ) 2 2 u1 (r n − 1) u1 (1 − r n ) The sum of n terms of a , r ≠1 = Sn = finite geometric sequence r −1 1− r SL 1.4 Compound interest r FV = PV × 1 + , where FV is the future value, 100k 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 = log a b , where a > 0, b > 0, a ≠ 1 Percentage error ε= SL 1.6 kn vA − vE × 100% , where vE is the exact value and vA is vE the approximate value of v Mathematics: applications and interpretation formula booklet 3 Topic 1: Number and algebra – HL only AHL 1.9 Laws of logarithms log= log a x + log a y a xy x log = log a x − log a y a 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 (cos θ + isin θ ) = re iθ = r cis θ AHL 1.13 AHL 1.14 AHL 1.15 Determinant of a 2 × 2 matrix u1 , r <1 1− r a b A= A = ad − bc ⇒ det A = c d Inverse of a 2 × 2 matrix a b 1 d −1 = A ⇒ A= det A −c c d −b , ad ≠ bc a Power formula for a matrix M n = PD n P −1 , where P is the matrix of eigenvectors and D is the diagonal matrix of eigenvalues Mathematics: applications and interpretation formula booklet 4 Topic 2: Functions – SL and HL SL 2.1 SL 2.5 Equations of a straight line = 0 ; y − y1= m ( x − x1 ) y mx + c ; ax + by + d = y2 − y1 x2 − 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 = − b 2a Topic 2: Functions – HL only AHL 2.9 Logistic function f ( x) = L , L , k,C > 0 1 + Ce − kx Mathematics: applications and interpretation formula booklet 5 Topic 3: Geometry and trigonometry – SL and HL SL 3.1 Distance between two points ( x1 , y1 , z1 ) and d= Coordinates of the midpoint of a line segment with endpoints ( x1 , y1 , z1 ) x1 + x2 y1 + y2 z1 + z2 , , 2 2 2 Volume of a right-pyramid V= 1 Ah , where A is the area of the base, h is the height 3 Volume of a right cone V= 1 2 πr h , where r is the radius, h is the height 3 Area of the curved surface of a cone A = πrl , where r is the radius, l is the slant height Volume of a sphere V= 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 c 2 = a 2 + b 2 − 2ab cos C ; cos C = Area of a triangle 1 A = ab sin C 2 ( x1 − x2 ) 2 + ( y1 − y2 ) 2 + ( z1 − z2 ) 2 ( x2 , y2 , z2 ) and ( x2 , y2 , z2 ) SL 3.2 SL 3.4 Length of an arc = l 4 3 πr , where r is the radius 3 θ 360 a 2 + b2 − c2 2ab × 2πr , 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 Mathematics: applications and interpretation formula booklet 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 = r 2θ , where r is the radius, θ is the angle measured in 2 radians AHL 3.8 Identities cos 2 θ + sin 2 θ = 1 tan θ = AHL 3.9 Transformation matrices cos 2θ sin 2θ sin θ cos θ sin 2θ , reflection in the line y = (tan θ ) x − cos 2θ 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 , enlargement, with a scale factor of k, centre (0, 0) 0 k cos θ sin θ − sin θ , anticlockwise/counter-clockwise rotation of cos θ angle θ about the origin ( θ > 0 ) cos θ − sin θ (θ > 0 ) sin θ , clockwise rotation of angle θ about the origin cos θ Mathematics: applications and interpretation formula booklet 7 AHL 3.10 AHL 3.11 AHL 3.13 Magnitude of a vector v1 v + v2 + v3 , where v = v2 v 3 2 1 v = 2 2 Vector equation of a line r = a + λb Parametric form of the equation of a line x =+ x0 λ l , y =+ y0 λ m, z =+ z0 λ n Scalar product v1 w1 v ⋅ w= v1w1 + v2 w2 + v3 w3 , where v = v2 , w = w2 v w 3 3 v⋅w = v w cos θ , where θ is the angle between v and w v1w1 + v2 w2 + v3 w3 v w Angle between two vectors cos θ = Vector product w1 v2 w3 − v3 w2 v1 v ×= w v3 w1 − v1w3 , where v = v2 , w = w2 w v 3 v1w2 − v2 w1 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 Mathematics: applications and interpretation formula booklet 8 Topic 4: Statistics and probability – SL and HL SL 4.2 SL 4.3 SL 4.5 SL 4.6 Interquartile range IQR = Q3 − Q1 k Mean, x , of a set of data x= ∑fx i =1 i i , where n = n k ∑f i =1 i 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 E(X ) = discrete random variable X SL 4.8 Binomial distribution P ( A ∩ B) P ( B) x P(X ∑= x) X ~ B (n , p) Mean E ( X ) = np Variance Var (= X ) np (1 − p ) Mathematics: applications and interpretation formula booklet 9 Topic 4: Statistics and probability – HL only AHL 4.14 Linear transformation of a single random variable E ( aX += b ) aE ( X ) + b Linear combinations of n independent random variables, X 1 , X 2 , ..., X n a1E ( X 1 ) ± a2 E ( X 2 ) ± ... ± an E ( X n ) E ( a1 X 1 ± a2 X 2 ± ... ± a= n Xn ) Var ( aX + b ) = a 2 Var ( X ) Var ( a1 X 1 ± a2 X 2 ± ... ± an X n ) = a12 Var ( X 1 ) + a2 2 Var ( X 2 ) + ... + an 2 Var ( X n ) Sample statistics Unbiased estimate of population variance sn2−1 AHL 4.17 AHL 4.19 sn2−1 = n 2 sn n −1 Poisson distribution X ~ Po (m) Mean E(X ) = m Variance Var ( X ) = m Transition matrices T n s0 = sn , where s0 is the initial state Mathematics: applications and interpretation formula booklet 10 Topic 5: Calculus – SL and HL SL 5.3 SL 5.5 Derivative of x n f ( x) = x n ⇒ f ′( x) = nx n −1 Integral of x n dx ∫x= Area of region enclosed by a curve y = f ( x) and the x-axis, where f ( x) > 0 SL 5.8 The trapezoidal rule n x n +1 + C , n ≠ −1 n +1 b A = ∫ y dx a 1 y dx ≈ h ( ( y0 + yn ) + 2( y1 + y2 + ... + yn −1 ) ) , 2 b−a where h = n ∫ b a 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 f ( x) = e x ⇒ f ′( x) = ex Derivative of ln x 1 f ( x) = ln x ⇒ f ′( x) = x Chain rule y = g (u ) , where u = f ( x) ⇒ Product rule y =uv ⇒ Quotient rule du dv v −u u dy d x d x y= ⇒ = 2 v dx v Mathematics: applications and interpretation formula booklet 1 cos 2 x dy dy du = × dx du dx dy dv du =u + v dx dx dx 11 AHL 5.11 Standard integrals 1 dx ∫ x= ln x + C − cos x + C ∫ sin x dx = dx ∫ cos x= 1 ∫ cos= x 2 ∫e AHL 5.12 AHL 5.13 AHL 5.16 x sin x + C tan x + C x ex + C d= b b a a Area of region enclosed by a curve and x or y-axes A = ∫ y dx or A = ∫ x dy Volume of revolution about x or y-axes V = ∫ πy 2 dx or V = ∫ πx 2 dy Acceleration = a Distance travelled from t1 to t2 distance = Displacement from t1 to t2 displacement = Euler’s method b b a a dv d 2 s dv = = v 2 dt dt ds ∫ t2 t1 v(t ) dt ∫ t2 t1 v(t ) dt xn + h , where h is a constant yn += yn + h × f ( xn , yn ) ; xn += 1 1 (step length) Euler’s method for coupled systems xn +1 = xn + h × f1 ( xn , yn , tn ) yn +1 = yn + h × f 2 ( xn , yn , tn ) tn +1= tn + h where h is a constant (step length) AHL 5.17 Exact solution for coupled = x Aeλ1t p1 + Beλ2t p2 linear differential equations Mathematics: applications and interpretation formula booklet 12