Math AA Formula Booklet

Mathematics: analysis and approaches formula booklet For use during the course and in the examinations First examinations 2021 Version 1.1 Contents Prior learning SL and HL 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 12 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 a 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 2 Topic 1: Number and algebra – SL and HL SL 1.2 SL 1.3 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 = n n ( 2u1 + (n − 1) d ) ; Sn = (u1 + un ) 2 2 u1 ( rn − 1) u1 (1 − rn ) , r ≠1 = r −1 1− r SL 1.4 Compound interest SL 1.5 Exponents and logarithms a x = b ⇔ x = log a b , where a > 0, b > 0, a ≠ 1 SL 1.7 Exponents and logarithms log a xy = log a x + log a y x log a = log a x − log a y y kn 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 log a x m = m log a x log a x = logb x logb a SL 1.8 The sum of an infinite geometric sequence S∞ = SL 1.9 Binomial theorem ( a + b)n = an + n C an − 1b + + n C an −r br + + bn 1 r u1 , r <1 1− r n! nC = r r !( n − r )! 3 Topic 1: Number and algebra – HL only AHL 1.10 Combinations n! nC = r r !( n − r)! Permutations nP = n! r (n − r )! AHL 1.12 Complex numbers z = a + bi AHL 1.13 Modulus-argument (polar) and exponential (Euler) form z = r (cosθ + isin θ ) = reiθ = r cisθ AHL 1.14 De Moivre’s theorem [r (cosθ + isinθ ) ] n = r n (cos nθ + isin nθ ) = r n ei nθ = r n cis nθ 4 Topic 2: Functions – SL and HL SL 2.1 SL 2.6 SL 2.7 SL 2.9 Equations of a straight line y = mx + c ; ax + by + d = 0 ; y − y 1 = m ( x − x 1) Gradient formula m= Axis of symmetry of the graph of a quadratic function f ( x) = ax2 + bx + c ⇒ axis of symmetry is x = − Solutions of a quadratic equation ax 2 + bx + c = 0 ⇒ Discriminant ∆ = b 2 − 4ac Exponential and logarithmic functions a x = e x ln a ; log a a x = x = alog a x where a , x > 0, a ≠ 1 y2 − y1 x2 − x1 x= b 2a −b ± b 2 − 4 ac , a≠ 0 2a Topic 2: Functions – HL only AHL 2.12 Sum and product of the roots of polynomial equations of the form n ∑a x r r Sum is −a n −1 ; product is an ( −1) n a0 an =0 r =0 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 c 2 = a 2 + b 2 − 2 ab cos C ; cos C = Area of a triangle 1 A = ab sin C 2 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 a 2 + b2 − c2 2ab radians 6 sinθ cosθ SL 3.5 Identity for tan θ tan θ = SL 3.6 Pythagorean identity cos2 θ + sin 2 θ = 1 Double angle identities sin 2θ = 2sin θ cosθ cos 2θ = cos 2 θ − sin 2 θ = 2 cos 2 θ −1 = 1 − 2sin 2 θ Topic 3: Geometry and trigonometry – HL only AHL 3.9 Reciprocal trigonometric identities secθ = 1 cosθ cosecθ = Pythagorean identities 1 sinθ 2 2 1 + tan θ = sec θ 1 + cot 2 θ = cosec2 θ AHL 3.10 Compound angle identities sin ( A ± B ) = sin A cos B ± cos A sin B cos ( A ± B) = cos A cos B  sin A sin B tan ( A ± B ) = Double angle identity for tan AHL 3.12 Magnitude of a vector tan 2θ = tan A ± tan B 1  tan A tan B 2 tan θ 1 − tan 2θ  v1    v = v12 + v22 + v32 , where v =  v2  v   3 7 AHL 3.13 Scalar product  v1   w1      v ⋅ w = v1w1 + v 2w 2 + v 3w3 , where v =  v2  , w =  w2  v  w   3  3 v ⋅ w = v w cosθ , where θ is the angle between v and w AHL 3.14 AHL 3.16 v1 w1 + v2 w2 + v3 w3 v w Angle between two vectors cos θ = Vector equation of a line r = a + λb Parametric form of the equation of a line x = x0 + λl , y = y0 + λ m, z = z0 + λ n Cartesian equations of a line x −x0 y − y0 z −z0 = = l m n Vector product  v2 w3 − v3 w2   v1   w1  v × w =  v3 w1 − v1 w3 , where v =  v2  , w = w2  w   v w −v w  v   3  1 2 2 1  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 AHL 3.17 Vector equation of a plane r = a + λb + µc Equation of a plane (using the normal vector) r ⋅n = a⋅n Cartesian equation of a plane ax + by + cz = d 8 Topic 4: Statistics and probability – SL and HL SL 4.2 Interquartile range SL 4.3 k Mean, x , of a set of data SL 4.5 SL 4.6 IQR = Q3 − Q1 x= ∑fx i i i =1 k , where n = n 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 ) P ( A ∩ B) 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 4.12 ∑f X ~ B (n , p ) Mean E ( X ) = np Variance Var ( X ) = np (1 − p ) Standardized normal variable z= x− µ σ 9