Diploma Programme Mathematics: analysis and approaches formula booklet For use during the course and in the examinations First examinations 2021 Version 1.0 © International Baccalaureate Organization 2019 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 = 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 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= 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 Mathematics: analysis and approaches formula booklet 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 The sum of n terms of a u1 (r n − 1) u1 (1 − r n ) , r ≠1 = S = n finite geometric sequence 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= log a x + log a y a xy x log = log a x − log a y a 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 = log b x log b a SL 1.8 u1 The sum of an infinite = S∞ , r <1 geometric sequence r −1 SL 1.9 Binomial theorem (a + b) n = a n + n C a n −1b +… + n C a n − r b r +… + b n 1 r n! nC = r r !(n − r )! Mathematics: analysis and approaches formula booklet 3 Topic 1: Number and algebra – HL only AHL 1.10 Combinations n! nC = r r !(n − r )! Permutations n P = n! r (n − r )! AHL 1.12 Complex numbers z= a + bi AHL 1.13 Modulus-argument (polar) and exponential (Euler) form z= r (cos θ + isin θ ) = re iθ = r cis θ AHL 1.14 De Moivre’s theorem [ r (cosθ + isin θ )] Mathematics: analysis and approaches formula booklet n = r n (cos nθ + isin nθ ) = r n einθ = 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 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 = − 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= a loga x where a , x > 0, a ≠ 1 x= b 2a −b ± b 2 − 4ac , 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 =0 r r ( −1) a0 − an −1 ; product is Sum is an an n =0 Mathematics: analysis and approaches 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 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 ( x1 − x2 ) 2 + ( y1 − y2 ) 2 + ( z1 − z2 ) 2 ( x2 , y2 , z2 ) and ( x2 , y2 , z2 ) SL 3.2 SL 3.4 4 3 πr , where r is the radius 3 a 2 + b2 − c2 2ab radians Mathematics: analysis and approaches formula booklet 6 SL 3.5 Identity for tan θ tan θ = SL 3.6 Pythagorean identity cos 2 θ + sin 2 θ = 1 Double angle identities sin 2θ = 2sin θ cos θ sin θ cos θ cos 2θ = cos 2 θ − sin 2 θ = 2cos 2 θ − 1 = 1 − 2sin 2 θ Topic 3: Geometry and trigonometry – HL only AHL 3.9 Reciprocal trigonometric identities secθ = 1 cos θ cosecθ = Pythagorean identities 1 sin θ 1 + tan 2 θ = sec 2 θ 1 + cot 2 θ = cosec 2θ 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 ± tan B tan ( A ± B ) = 1 ∓ tan A tan B Double angle identity for tan AHL 3.12 Magnitude of a vector tan 2θ = v = Mathematics: analysis and approaches formula booklet 2 tan θ 1 − tan 2 θ v1 v + v2 + v3 , where v = v2 v 3 2 1 2 2 7 AHL 3.13 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 AHL 3.14 AHL 3.16 v1w1 + 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 λ l , y =+ y0 λ m, z =+ z0 λ n Cartesian equations of a line x − x0 y − y0 z − z0 = = l m n Vector product w1 v1 v2 w3 − v3 w2 v ×= w v3 w1 − v1w3 , where v = v2 , w = w2 v w −v w w v 1 2 2 1 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 AHL 3.17 Vector equation of a plane r = a + λb + µ c Equation of a plane (using the normal vector) r ⋅n =a⋅n Cartesian equation of a plane ax + by + cz = d Mathematics: analysis and approaches 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 = Q3 − Q1 IQR k Mean, x , of a set of data x= ∑fx i =1 , 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) 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 = E(X ) discrete random variable X SL 4.8 Binomial distribution SL 4.12 i i x P(X ∑= x) X ~ B (n , p) Mean E ( X ) = np Variance Var (= X ) np (1 − p ) Standardized normal variable z= Mathematics: analysis and approaches formula booklet x−µ σ 9 Topic 4: Statistics and probability – HL only AHL 4.13 AHL 4.14 Bayes’ theorem P ( B | A) = P ( B) P ( A | B) P ( B ) P ( A | B) + P ( B′) P ( A | B′) P ( Bi | A) = P( Bi ) P( A | Bi ) P( B1 ) P( A | B1 ) + P( B2 ) P( A | B2 ) + P( B3 ) P( A | B3 ) k Variance σ 2 k f i ( xi − µ ) f i xi 2 ∑ ∑ = = = i1 − µ2 σ 2 i 1= n n 2 k ∑ f (x i i − µ) 2 Standard deviation σ σ= Linear transformation of a single random variable b ) aE ( X ) + b E ( aX += Expected value of a continuous random variable X i =1 n a 2 Var ( X ) Var ( aX + b ) = E(X = ) µ= ∫ ∞ −∞ x f ( x) dx Variance Var ( X ) = E ( X − µ ) 2 = E ( X 2 ) − [ E (X ) ] Variance of a discrete random variable X Var ( X ) = x) = x) − µ 2 ∑ ( x − µ )2 P ( X = ∑ x2 P ( X = Variance of a continuous random variable X 2 ∞ ∞ −∞ −∞ 2 2 2 Var ( X ) = ∫ ( x − µ ) f ( x) dx = ∫ x f ( x) dx − µ Mathematics: analysis and approaches 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 n dx ∫x= Area between a curve y = f ( x) and the x-axis, where f ( x) > 0 SL 5.6 SL 5.9 x n +1 + C , n ≠ −1 n +1 b A = ∫ y dx a Derivative of sin x f ( x) =sin x ⇒ f ′( x) =cos x Derivative of cos x − sin x f ( x) =⇒ cos 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 d x x y= ⇒ = 2 dx v v dy dv du =u + v dx dx dx dv d 2 s = dt dt 2 Acceleration = a Distance travelled from t1 to t 2 distance = Displacement from t1 to t 2 displacement = Mathematics: analysis and approaches formula booklet dy dy du = × dx du dx ∫ t2 t1 v(t ) dt ∫ t2 t1 v (t )dt 11 SL 5.10 Standard integrals 1 dx ∫ x= ln x + C − cos x + C ∫ sin x dx = dx ∫ cos x= ∫e SL 5.11 Area of region enclosed by a curve and x-axis x sin x + C d= x ex + C b A = ∫ y dx a Topic 5: Calculus – HL only AHL 5.12 Derivative of f ( x) from first principles AHL 5.15 Standard derivatives y = f ( x) ⇒ dy f ( x + h) − f ( x ) = f ′( x)= lim h → 0 h dx tan x f ( x) =tan x ⇒ f ′( x) =sec 2 x sec x f ( x) =sec x ⇒ f ′( x) =sec x tan x cosec x f ( x) = cosec x ⇒ f ′( x) = −cosec x cot x cot x −cosec 2 x f ( x) =⇒ cot x f ′( x) = ax f ( x) = a x ⇒ f ′( x) = a x (ln a ) log a x f ( x) = log a x ⇒ f ′( x) = arcsin x f ( x)= arcsin x ⇒ f ′( x)= arccos x 1 f ( x) = arccos x ⇒ f ′( x) = − 1 − x2 arctan x f ( x)= arctan x ⇒ f ′( x)= Mathematics: analysis and approaches formula booklet 1 x ln a 1 1 − x2 1 1 + x2 12 AHL 5.15 Standard integrals a dx ∫= x 1 x a +C ln a 1 1 x = ∫ a 2 + x 2 dx a arctan a + C 1 x d= x arcsin + C , a a −x ∫ AHL 5.16 AHL 5.17 Integration by parts Area of region enclosed by a curve and y-axis Volume of revolution about the x or y-axes AHL 5.18 Euler’s method 2 dv ∫ u dx d=x uv − ∫ v x <a du dx or ∫ u d= v uv − ∫ v du dx b A = ∫ x dy a b b a a V = ∫ πy 2 dx or V = ∫ πx 2 dy xn + h , where h is a constant yn += yn + h × f ( xn , yn ) ; xn += 1 1 (step length) Integrating factor for e∫ Maclaurin series f ( x) =f (0) + x f ′(0) + Maclaurin series for special functions e x =1 + x + y ′ + P ( x) y = Q ( x) AHL 5.19 2 P ( x )dx x2 + ... 2! ln (1 + x) =x − x 2 x3 + − ... 2 3 sin x =x − x3 x5 + − ... 3! 5! cos x =− 1 x2 x4 + − ... 2! 4! arctan x =x − Mathematics: analysis and approaches formula booklet x2 f ′′(0) + … 2! x3 x5 + − ... 3 5 13