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© International Baccalaureate Organization 2023 Diploma Programme For use during the course and in the examinations First examinations 2021 Version 1.0 Mathematics: analysis and approaches SLformula booklet STANDARD LEVEL
Contents Topic 1: Number and algebra – SL Topic 2: Functions – SL Topic 3: Geometry and trigonometry – SL Topic 4: Statistics and probability – SL Topic 5: Calculus – SL 2 3 4 6 7
Mathematics: analysis and approaches SL formula booklet 2 Topic 1: Number and algebra – SL 1.2 1.3 1.8 1.4 1.7 1.9 1.5 Binomial theorem n Compound interest Exponential and logarithmic functions The sum of an infinite geometric sequence The nth term of a geometric sequence The sum of n terms of a finite geometric sequence The nth term of an arithmetic sequence The sum of n terms of an arithmetic sequence Exponents and logarithms Exponents and logarithms x where FV=PV 1 , where FV is the future value, 100 is the present value, n is the number of years, is the number of compounding periods per year, % is the nominal annual rate of interest ax=b⇔x=log b, where a>0,b>0,a≠1a PV k r loga log log log a a a ( log log b b loga loga −1) loga loga = = + = = + − nu u1 x xy x y m n x a x x d y y ) 1 loga n 2 (2 ( 1 −1) ; ( 2 ) n n x n n S S u1 − , 1 r u1 r n d S n u1 u= = + < = + u x urn1 m a x x a ∞ − = = = =ln 1 ( e ) n − 1) 1 nC1 1(1 1 ) 1 C 0, 1 n x x n a n n n−r r n a S a b u (r1 r a u a rn r b r n r a b a , x a b+ = = = − + = − − + + ≠ > + + ≠; , k n r k × + loga loga ∈ − C !( r)! n r r n! n − =
Mathematics: analysis and approaches SL formula booklet 3 Topic 2: Functions – SL 2.1 2.6 2.7 Gradient formula Axis of symmetry of the graph of a quadratic function Solutions of a quadratic equation Discriminant Equations of a straight line ; ;y =mx c+ 0 0 ax by d b 2a 2 4ac , aax +bx+c=0 ∆=b −4ac 2 2 + ⇒ + = x −b± = − ≠ )1 m y x2 2 y1 x1 y y = m ( x x1 = − − − − ( )f x ax2 bx b 2a = + + c ⇒ x = −axis of symmetry is
Mathematics: analysis and approaches SL formula booklet 4 Topic 3: Geometry and trigonometry – SL 3.1 Prior learning – SL Area of a triangle Area of a trapezoid Area of a parallelogram Area of a circle Circumference of a circle Volume of a cuboid Volume of a cylinder Volume of a prism Area of the curved surface of a cylinder Distance between two points(x ,y )and(x ,y )1 1 2 2 Coordinates of the midpoint of a line segment with endpoints (x ,y )and(x ,y )1 1 2 2 Distance betweentwo points y ,z )and1 1 Coordinates ofthe midpoint ofalinesegment with endpoints(x ,y ,z )1 1 1 and (x ,y ,z )2 2 2 , where , where , where is the base, and is the base, is the height 2 , where r is the radius r, where r is the radius , where l is the length, w is the width,h 2 r h , where r is the radius, h is the height , where A is the area of cross-section,h , where r is the radius, h is the height is the height are the parallel sides, is the height is the height is the height C V A A bh Ah 2 rh = = = π =2π b b a b h h h 1 2 ( 1 2 (A A a bh) b)h = = + 2 2 2 2 1 2 1 2 2 2 1 2 2 V V A d d x1 lwh x x1 x y y y y z z = =πr = = π = − − + + − − + − ( ( ) ( ) ( ) ) ( ) 2 2 2 1 2 1 2 2 1 2 ( , ( , , ) 2 2 2 , 2 , 2 x y x1 z2 x1 x x1 y x y y y z z , + + + + +
Mathematics: analysis and approaches SL formula booklet 5 Sine rule Identity for Cosine rule Area of a sector Length of an arc Area of a triangle Volume of a sphere Volume of a right cone Pythagorean identity Double angle identities Surface area of a sphere Volume of a right-pyramid Area of the curved surface of a cone radians , where , wherer , where 2 , wherer 3 , wherer , where , wherer is the radius, is the radius, is the radius is the radius is the radius, is the radius, is the area of the base, is the height is the slant height is the height is the angle measured in is the angle measured in radians l sin tan 1 3 1 3 = π 1 2 r 4 3 sin sin cos sin V V V A a A Ah r πr h2 b B c C = = = = = = π = A h h = = = + = + − = = + c l A A a rl ab b C r 2ab r C C a b − c2 2 2ab cos 2θ 4πr cos2 sin2 2cos2 1 1 2sin2 = = − = − = − A 3.2 3.4 3.5 3.6 2 2 2 2 cossin 2θ 2 1 2 sin2 sin 1 cos cos; rθ θ θ θ θ 2θ θ θ θ θ θ θ tan 2sin cos θ θ θ θ
Mathematics: analysis and approaches SL formula booklet 6 Topic 4: Statistics and probability – SL 4.3 4.5 4.6 4.7 4.8 4.2 4.12 Mean, Interquartile range Binomial distribution X , ) Mean Variance Standardized normal variable Conditional probability Complementary events Probability of an event Combined events Mutually exclusive events Independent events Expected value of a discrete random variable , of a set of data i , where 1 1 P ( P ( IQR= ) P ( P ( P ( A AB) Q3 A Q1 A ∩ B ) B ) + = − ′)=1 x i k ∑= x f xi 1 n = i k ∑= i k ∑ i A X n(A) n(U ) xi X n xi f = = = = = n p A A B A ∪ B A ∪ B X =np X)=np A p B A B P ( E ( ) ) P ( ) ~ B ( P ( P ( P ( E ( ) Var( ) P ( (1− ) ) P ( ) )=P(A)+P(B)−P( )=P(A)+P(B) ) ∩ = ∩ X z x − μ σ =
Mathematics: analysis and approaches SL formula booklet 7 Topic 5: Calculus – SL 5.9 5.5 5.3 5.6 Chain rule Product rule Acceleration Derivative of Derivative of Derivative of Derivative of Derivative of Quotient rule Distance travelled from t1 to t2 Displacement from t1 to t2 Integral of xn Area between a curve y=f(x) and the x-axis, where f(x)>0 distance displacement ), where x n n x f a y y x x x u v dv dt g(u xn n x y x 2s 2 t f u C , n x f x vdu dx d−u v dx v2 y x y u u x 1 ( ( ) ) e ( ( ) ) ex n t2 t1 t2 t1 x f f x x x f f x x nxn− = = ⇒ ⇒ ′ ′ = = x sinx ( ) sinx ( ) cosf x f x x= ⇒ ′ = cos x e lnx ( d ) cos d d d d ( ( ) sin d d d d d d = = = = = = ⇒ ⇒ = +1 + +1 ′ = ≠−1 ) = − ⇒ = × ( ) lnx d d ( ()dt ) ()dt 1 x f y x uv y x f v t x udv dx v t vdu dx = = ⇒ = ⇒ = = ′ + = ∫ ∫ ∫ b ∫aA ydx=
Mathematics: analysis and approaches SL formula booklet 8 5.11 5.10 Standard integrals Area of region enclosed by a curve and x-axis 1 x∫∫ ∫ ∫ dx ln x C= + cos e dxx sin d e sinx cosx x xdx C x C C = = = − + + + x d b ∫aA y x=