SL Formulabooklet
- 1. Diploma Programme Mathematics SL formula booklet For use during the course and in the examinations First examinations 2014 Published March 2012 © International Baccalaureate Organization 2012 5045 Mathematical studies SL: Formula booklet 1
- 3. 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
- 4. Formulae Prior learning Area of a parallelogram A = b× h Area of a triangle A = 1( b× h ) 2 Area of a trapezium 1( ) A = a + b h 2 Area of a circle A = πr2 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 = πr2h Area of the curved surface of a cylinder A = 2πrh Volume of a sphere 4 3 V = πr 3 Volume of a cone 1 2 V = πr h 3 Distance between two points 1 1 1 (x , y , z ) and 2 2 2 (x , y , z ) 2 2 2 1 2 1 2 1 2 d = (x − x ) + ( y − y ) + (z − z ) Coordinates of the midpoint of a line segment x x , y y , z z with endpoints (x , y z 1 2 1 2 1 2 , ) and (x , y , z ) 1 1 1 2 2 2 + + + 2 2 2 Mathematics SL formula booklet 2
- 5. Topics Topic 1—Algebra 1.1 The nth term of an u = u + ( n −1) d arithmetic sequence n 1 The sum of n terms of an arithmetic sequence S = n (2 u + ( n − 1) d ) = n ( u + u ) n 1 1 n 2 2 The nth term of a geometric sequence = n− n u u r 1 1 The sum of n terms of a S = u ( r n − 1) − n finite geometric sequence 1 = u (1 r ) 1 r − 1 1 − r n , r ≠1 The sum of an infinite S u geometric sequence 1 = ∞ 1 − r , r <1 1.2 Exponents and logarithms x log a a = b ⇔ x = b Laws of logarithms log log log c c c a + b = ab a b a log log log c c c b − = log r log c c a = r a Change of base log log a a log c b c b = 1.3 Binomial coefficient ! n ! ( )! r n r n r − = Binomial theorem ( ) 1 n n n n n − n − r r n a b a a b a b b + = + + + + + 1 r Mathematics SL formula booklet 3
- 6. Topic 2—Functions and equations 2.4 Axis of symmetry of graph of a quadratic function f ( x ) ax 2 bx c axis of symmetry x b 2 a = + + ⇒ =− 2.6 Relationships between logarithmic and exponential functions ax = ex ln a log x loga x aa = x = a 2.7 Solutions of a quadratic equation 2 ax bx c x b b ac a 2 − ± − + + = 0 ⇒ = 4 , ≠ 0 2 a Discriminant Δ = b2 − 4ac Topic 3—Circular functions and trigonometry 3.1 Length of an arc l =θ r Area of a sector 1 2 A = θ r 2 3.2 Trigonometric identity sin tan θ cos θ θ = 3.3 Pythagorean identity cos2θ + sin2θ =1 Double angle formulae sin 2θ = 2sinθ cosθ cos 2θ = cos2θ − sin2θ = 2cos2θ −1=1− 2sin2θ 3.6 Cosine rule c2 = a2 + b2 − 2abcosC ; 2 2 2 cos + − C a b c 2 ab = Sine rule a = b = c A B C sin sin sin Area of a triangle 1 sin A = ab C 2 Mathematics SL formula booklet 4
- 7. Topic 4—Vectors 4.1 Magnitude of a vector 2 2 2 v = v1 + v2 + v3 4.2 Scalar product v ⋅w = v w cosθ 1 1 2 2 3 3 v ⋅w = v w + v w + v w Angle between two vectors cosθ ⋅ = v w v w 4.3 Vector equation of a line r = a + tb Topic 5—Statistics and probability 5.2 Mean of a set of data n Σ i 1 Σ n 1 f x i i i = i x f = = 5.5 Probability of an event A P( A ) = n ( A ) n U ( ) Complementary events P(A) + P(A′) =1 5.6 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) 5.7 Expected value of a discrete random variable X E( X ) = μ Σx = x P( X = x ) 5.8 Binomial distribution n X ~ B( np , ) P( X r ) p r (1 p )n − r , r 0,1, , n ⇒ = = − = r Mean E(X ) = np Variance Var(X ) = np(1− p) 5.9 Standardized normal variable − μ σ z = x Mathematics SL formula booklet 5
- 8. Topic 6—Calculus 6.1 Derivative of f (x) y f x y f x f x h f x ( ) d ( ) lim ( ) ( ) ′ + − = ⇒ = = x → 0 h d h 6.2 Derivative of xn f (x) = xn ⇒ f ′(x) = nxn−1 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 ( ) tan ( ) 1 2 cos = ⇒ ′ = f x x f x x Derivative of ex f (x) = ex ⇒ f ′(x) = ex Derivative of ln x 1 f (x) ln x f (x) x = ⇒ ′ = Chain rule u f x y y u y = g(u) , = ( ) ⇒ d = d × d x u x d d d Product rule d d d y = uv ⇒ y = u v + v u x x x d d d Quotient rule u y v d u − u d v y x x d d d d 2 = ⇒ = v x v 6.4 Standard integrals n + 1 xn x x C n + ∫ 1 dx ln x C, x 0 x d = + , ≠ − 1 1 n ∫ = + > ∫sin xdx = −cos x + C ∫cos xdx = sin x + C ∫e d = e + x x x C 6.5 Area under a curve between x = a and x = b d b A = ∫ y x a Volume of revolution about the x-axis from x = a to x = b π 2 d b a V = ∫ y x 6.6 Total distance travelled from 1 t to 2 t ( ) d t t = ∫ v t t distance 2 1 Mathematics SL formula booklet 6
- 9. Mathematics SL formula booklet 7