Diploma Programme Mathematics SL formula booklet For use during the course and in the examinations First examinations 2014 Edited in 2015 (version 2) © International Baccalaureate Organization 2012 5045 Contents Prior learning 2 Topics 3 Topic 1—Algebra 3 Topic 2—Functions and equations 4 Topic 3—Circular functions and trigonometry 4 Topic 4—Vectors 5 Topic 5—Statistics and probability 5 Topic 6—Calculus 6 Mathematics SL formula booklet 1 Formulae Prior learning A= b × h Area of a parallelogram Area of a triangle = A 1 (b × h) 2 Area of a trapezium = A 1 ( a + b) h 2 Area of a circle A = πr 2 Circumference of a circle C = 2πr Volume of a pyramid = V 1 (area of base × vertical height) 3 Volume of a cuboid (rectangular prism) V =l × w × h Volume of a cylinder V = πr 2 h Area of the curved surface of a cylinder A= 2πrh Volume of a sphere V= 4 3 πr 3 Volume of a cone V= 1 2 πr h 3 Distance between two points ( x1 , y1 , z1 ) and d= ( x1 − x2 ) 2 + ( y1 − y2 ) 2 + ( z1 − z2 ) 2 ( x2 , y2 , z2 ) Coordinates of the midpoint of a line segment with endpoints ( x1 , y1 , z1 ) and ( x2 , y2 , z2 ) Mathematics SL formula booklet x1 + x2 y1 + y2 z1 + z2 , , 2 2 2 2 Topics Topic 1—Algebra 1.1 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 )= (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 1.2 1.3 1− r The sum of an infinite geometric sequence S∞ = Exponents and logarithms ax = b ⇔ Laws of logarithms log c a + log c b = log c ab a log c a − log c b = log c b r log c a = r log c a Change of base log b a = Binomial coefficient n n! = r r !(n − r )! Binomial theorem n n (a + b) n = a n + a n −1b + + a n − r b r + + b n 1 r Mathematics SL formula booklet u1 , r <1 1− r x = log a b log c a log c b 3 Topic 2—Functions and equations 2.4 Axis of symmetry of graph of a quadratic function b f ( x) = ax 2 + bx + c ⇒ axis of symmetry x = − 2a 2.6 Relationships between logarithmic and exponential functions a x = e x ln a log a a x= x= a loga x 2.7 Solutions of a quadratic equation ax 2 + bx + c= 0 ⇒ Discriminant ∆= b 2 − 4ac x= −b ± b 2 − 4ac , a≠0 2a Topic 3—Circular functions and trigonometry 3.1 3.2 3.3 Length of an arc l =θr Area of a sector 1 A = θ r2 2 Trigonometric identity tan θ = Pythagorean identity cos 2 θ + sin 2 θ = 1 Double angle formulae sin 2θ = 2sin θ cos θ sin θ cos θ cos 2θ = cos 2 θ − sin 2 θ = 2cos 2 θ − 1 = 1 − 2 sin 2 θ 3.6 Cosine rule c 2 = a 2 + b 2 − 2ab cos C ; cos C = Sine rule a b c = = sin A sin B sin C Area of a triangle 1 A = ab sin C 2 Mathematics SL formula booklet a 2 + b2 − c2 2ab 4 Topic 4—Vectors 4.1 4.2 Magnitude of a vector Scalar product v = v12 + v2 2 + v32 v⋅w = v w cos θ v ⋅ w= v1w1 + v2 w2 + v3 w3 4.3 v⋅w v w Angle between two vectors cos θ = Vector equation of a line r = a + tb Topic 5—Statistics and probability 5.2 Mean of a set of data n ∑fx i i x= i =1 n ∑f i i =1 5.5 5.6 5.7 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) = P (A) P (B | A) Independent events P ( A ∩ B) = P ( A) P ( B) Expected value of a discrete random variable X E(X = ) µ= ∑ x P ( X= x) x Binomial distribution n r n−r X ~ B(n , p ) ⇒ P ( X = r) = 0,1, , n p (1 − p ) , r = r Mean E ( X ) = np Variance Var (= X ) np (1 − p ) Standardized normal variable z= 5.8 5.9 n ( A) n (U ) Probability of an event A Mathematics SL formula booklet x−µ σ 5 Topic 6—Calculus 6.1 6.2 y = f ( x) ⇒ Derivative of x n f ( x) = xn ⇒ 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) = ex ⇒ f ′( x) = ex Derivative of ln x f ( x) = ln x ⇒ 1 f ′( x) = x Chain rule y = g (u ) , u =f ( x) ⇒ Product rule y =uv ⇒ Quotient rule 6.4 dy f ( x + h) − f ( x ) = f ′( x) = lim h → 0 dx h Derivative of f ( x) u y= v n dx ∫x= 1 cos 2 x dy dy du = × dx du dx dy dv du =u + v dx dx dx du dv v −u dy = dx 2 dx dx v ⇒ Standard integrals f ′( x) = nx n −1 x n +1 + C , n ≠ −1 n +1 1 ∫ x dx =ln x + C , x>0 − cos x + C ∫ sin x dx = dx ∫ cos x= ∫e 6.5 Area under a curve between x = a and x = b x Total distance travelled from t1 to t 2 Mathematics SL formula booklet x ex + C d= b A = ∫ y dx a Volume of revolution V= about the x-axis from x = a to x = b 6.6 sin x + C ∫ b a πy 2 dx distance = ∫ t2 t1 v(t ) dt 6