MA1301_AY1415S1

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MA1301 cheat sheet...

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MA1301: AY14/15 S1 elementary f (x) lnx sinx cosx tanx sin−1 x cos−1 x tan−1 x

nth term sum no. of terms test(a,b,c. . . ) Derivative tests f ′ (c) = 0 local max local min saddle point Vectors a∥b a·b

a×b for R3 ||a × b||

derivative

∫integral ∫ sinxdx ∫ cosxdx 2 ∫ sec2 xdx ∫ csc1 xdx √ dx 1−x2 ∫ 1 − √ 2 dx ∫ 1−x 1 1+x2 dx

1 x

cosx −sinx sec2 x √ 1 1−x2 1 − √1−x 2 1 1+x2

Arithmetic an = am + (n − m)d Sn = n(a12+an ) Sn = n(2a+(n−1)d) 2 1 n = an −a +1 d c−b=b−a=d

antiderivative −cosx + C sinx + C tanx + C −cotx + C sin−1 x + C cos−1 x + C tan−1 x + C

Geometric an = arn−1 n ) Sn = a(1−r 1−r −ran Sn = a11−r c b

=

b a

=r

. first (x < c ⇒ f ′ (x) > 0) ∧ (x > c ⇒ f ′ (x) < 0) (x < c ⇒ f ′ (x) < 0) ∧ (x > c ⇒ f ′ (x) > 0) x ̸= c ⇒ (f ′ (x) < 0 ∨ f ′ (x). > 0)

rules of diﬀerentiation (αf + βg)′ = αf ′ + βg ′ (f g)′ = f ′ g + f g ′ ′ g′ ( fg )′ = f g−f , g ̸= 0 g2 (f ◦ g)′ (x) = f ′ (g(x)) · g ′ (x) f (x + h) ≈ f (x) + f ′ (x) · h y = (f (x))g(x) ⇒ lny = g(x)ln(f (x)) dy/dt dy x = x(t), y = y(t) ⇒ dx = dx/dt

Fundamental theorem of calculus: ∫x d d f (t)dt = f (x) F (x) = dx dx a Integration by parts: ∫b ∫b f (x)g ′ (x) = |f (x)g(x)|ba − a g(x)f ′ (x) a Rotation about an axis: ∫b V = π a (f (x) − k)2 dx, about the line y = k second (iﬀ f ′′ ) f ′′ (c) < 0 ... f ′′ (c) > 0 c is an inflection point

convex concave strictly convex strictly concave

f ′′ (x) ≥ 0 f ′′ (x) ≤ 0 f ′′ (x) > 0 f ′′ (x) < 0

Applications of cross product a = αb for some scalar α a1 b1 + a2 b2 + . . . + an bn ||a||||b||cosθ ||a||2 when a = b 0 when a⊥b (a2 b3 − a3 b2 , a3 b1 − a1 b3 , a1 b2 − a2 b1 ) −a × b (skew-symmetric) a × (b + c) = a × b + a × c ||a||||b||sinθ 0 when a ∥ b

Area of parallelogram Area of triangle Shortest distance from A to BC

||a × b|| 1 2 ||a × b||

||AB×AC|| ||AC||

Lines & planes r(t) n of π

r0 + tv, for direction vector v and some constant t r · n = r0 · n u × v, where (v ∥ π) ∧ (u ∥ π) but u ̸∥ v

If a line: r(t)=r0 + tv and plane π: r · n = d, then (v · n = 0) ⇒ r(t) ∥ π, else they intersect. ·n2 θ between two planes: θ = cos−1 | ||nn11||||n | 2 || v·n −1 θ between line and plane: θ = sin | ||v||||n|| | intersection line between two planes: r(t) = r0 + t(n1 × n2 ) Complex numbers (z = a + bi ) |z|, or r a2 + b2 arg(z), or α tan−1 ( ab ) (1st quad) π + tan−1 ( ab ) (2nd quad) tan−1 ( ab ) − π (3rd quad) tan−1 ( ab ) (4th quad) polar form r(cosα + i sinα) Euler formula eiθ = cosθ + i sinθ

De Moivre’s (cosθ + i sinθ)±n = cos(nθ) ± i sin(nθ) θ+2kπ (cosθ + i sinθ)1/n = cos( θ+2kπ n ) + i sin( n ) for k = 0, 1, . . . , n − 1 let z1 = r1 (cosα1 + i sinα1 ), z2 = r2 (cosα2 + i sinα2 ), then z1 z2 = r1 r2 (cos(α1 + α2 ) + i sin(α1 + α2 )) |z1 z2 . . . zn | = |z1 ||z2 | . . . |zn | arg(z1 z2 . . . zn ) = arg(z1 ) + arg(z2 ) + . . . + arg(zn )

Trigo identities sin2 θ + cos2 θ = 1 sec2 θ − 1 = tan2 θ csc2 θ − 1 = cot2 θ sin(A ± B) ≡ sinAcosB ± sinBcosA cos(A ± B) ≡ cosAcosB ∓ sinAsinB tanA±tanB tan(A ± B) ≡ 1∓tanAtanB sin2A = 2sinAcosA cos2A = 2cos2 A − 1 = 1 − 2sin2 A = cos2 A − sin2 A

tan2A =

2tanA 1−tan2 A

A∓B sinA ± sinB = 2sin( A±B 2 )cos( 2 ) A−B cosA + cosB = 2cos( A+B 2 )cos( 2 ) A−B cosA − cosB = −2sin( A+B 2 )sin( 2 ) 1 sinAcosB = 2 [sin(A + B) + sin(A − B)] cosAsinB = 12 [sin(A + B) − sin(A − B)] cosAcosB = 12 [cos(A + B) + cos(A − B)] sinAsinB = − 12 [cos(A + B) − cos(A − B)]