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a definitive sheet by chad valencia, ucla mathematics major version 2.0.2000, rev 1

A B BCcalculusy Sample Function f(x)

trig in a nutshellB c a x C b A

personal notes

ABkx->c

sin x =a/c= opposite/hypotenuse cos x = b/c = adjacent/hypotenuse sec x = c/b = hypotenuse/adjacent csc x = c/a = hypotenuse/opposite tan x = a/b = sin x/cos x = opposite/adjacent cot x = b/a = cos x/sin x = adjacent/opposite

absolute maximum

Range: ( - , k ] - < y k

3 step test for continuity: relative maximum 1. f(c) exists f '(x)=0 2. lim exists 3. lim = f(c)x->c

Odd/Even Identities sin (- x) = - sin x cos (- x) = cos x tan (- x) = - tan x cot (- x) = - cot x sec (- x) = sec x csc (- x) = - csc x

Double Angle Identities sin 2x = 2 sin x cos x cos 2x = cos x - sin x cos x = 1+ cos 2x 2 sin x = 1- cos 2x 2

inflection point f ''(x)=0 relative minimum f '(x)=0 a b zero f (x)=0 c d e x

sin x + cos x = 1 1 + tan x = sec x 1 + cot x = csc x

cos (a+b) = cos a cos b - sin a sin b sin (a+b) = sin a cos b + cos a sin b cos (a - b) = cos a cos b + sin a sin b sin (a - b) = sin a cos b - cos a sin b log x = log10x logex = ln x ln (xy) = ln x + ln y ln (x/y) = lnx - lny ln x = n ln x ln e = e ln x = x ln 1 = 0 ln e = 1

Domain: ( - , e ] - < x e + b d + e f '(x) b c + e f ''(x)

logs in a nutshellf (x) = a1 1 x

f (x) = log a x

Area = af (x) dx

if: a b = x

log a x = b

n

lim (1+ 1 n

n

)= e

derivativesdefinition of the derivative:

trig derivativesInverse Trig Standard Trig 1 du (d/dx)(csc u) = - csc u cot u d -1 sin u = 1 - u dx (d/dx)(sec u) = sec u tan u dx (d/dx)(cot u) = - csc u 1 -1 d tan u = 1 + u du (d/dx)(tan u) = sec u dx dx (d/dx)(cos u) = - sin u 1 du -1 d (d/dx)(sin u) = cos u dx sec u = |u| u - 1 dx

f ' (x) = lim f (x + h) - f (x)h -> 0

h

Addition Rule f ' (u + v) = f ' u + f ' v Power Rule f ' ( x c ) = c x c -1

Product Rule f ' (u v) = udv + vdu Chain Rule (f o g)' = f ' g '

Quotient Rule v du - u dv u f'( )= v v (Lo D Hi minus Hi D Lo over Lo Lo)

volumes & areas

transcendental derivativesd dx

V(sphere) = 4 p r 3

V(cone) = 1 p r h3

lHpitals RuleWhenxa

lim

0 = 0 OR lim f (x) lim f (x) = x a g '' (x) xaf (x) g(x)

Mean Value Theorem f (b) - f (a) = f ' (c) b-a

ln u

1 du = u dx d u a dx

d u e = dx

e

u du

SA(sphere) = 4 p r

A

=s4 3

dx

g(x)

=a

u

du dx

ln a

integralsFirst Fundamental Theorem of Calculus

trigonometric integralsSecond Fundamental Theorem of Calculus (Leibniz's Rule) Standard Trig sin x dx = - cos x + C tan x dx = - ln |cos x| + C cos x dx = sinx + C cot x dx = ln |sin x| + C sec x dx = tan x + C sin x dx = x - sin 2x + C 2 4 csc x dx = - cot x + C cos x dx = x + sin 2x + C sec x tan x dx = sec x + C 2 4 csc x cot x dx = - csc x + C Inverse Trig du -1 u du -1 u = 1 tan a + C a a - u = sin a + C a + u

velocity & motion a(t) s(t) or x(t) position acceleration v(t) d velocity dthe velocity equations(t) = g t + vO t + sO g = - 32 ft / s , - 9.8 m / s

af (x) dx = F(a) - F(b)where F is the antiderivative of f

b

d dx

v (x)

f (t) dtdxdx

u (x)

= f (v) dv - f (u) duTrapezoidal Rule

disc & shell methodsVolume X-AxisDisc (no hole)b

T = b - a (y0 + 2y1 + 2y2+ ... + 2yn - 1+ yn ) 2n a & b = bounds n = number of intervals

(Use h(b1+b2)/2 for trapezoids of different height)

Disc w/ Holeb 2 2

Shelld c

Rectangular Approximation Methods (RAM)

udu u =

du u - a

=

1 a

sec |

-1

u |+ a

C

(use b x h for approximation)LRAM RRAM MRAM

transcendental integralsln |u| + Cu u

Power Rule: a x dx = xa+1 +C a+1 x-1

Average Value b Avg. (f (x)) = 1 . f (x) dx b-a

ln x dx = x ln x - x + Ca a du = ln a+ Cu u

r dx p(R - r )dx 2pr h dy Y-Axis p r dy p(R - r )dy 2pr h dxp2 a d c a d b 2 2 2 c a r = radius R = Outside radius r = inside radius r = radius h = height

a

e du = e + C

integration by parts udv = uv - vduPriority:

partial fractionspx + q = A + (x+a)(x+b) (x+a) B (x+b)

Logarithmic Inverse Trig Algebraic Trigonometric Euler's Constant (e)

Tabular Integration [(algebraic)(trigonometric/e)]dx ex: 2xcosx dx + cosx 2x - sinx 6x 12x + -cosx 12 - -sinx cosx 0 2xsinx + 6xcosx - 12x sin x - 1 x cos x +C 2Alternating Signs

d

px + q = A + B (x+a)2 (x+a) (x+a)2 px2 - qx + r = A + Bx + C (x+a)(x2+bx+c) (x+a) (x2+bx+c)

trig substitutions Use: If you see: 2 2 a +x x = a tan q 2 2 a -x x = a sin q 2 2 x -a x = a sec qsolved through trig substitution: sec u du = ln |sec u + tan u| + C

personal notes

improper integralsP Series Test: 1

Comparison Test if f(x) < convergent function, f(x) is convergent if f(x) > divergent function, f(x) is divergent

1 xp

a definitive sheet by chad valencia, ucla mathematics major version 2.0.2000, rev 1

A B BCcalculuslast number

Converges if p > 1 Diverges if 0 < p < 1

Limit Comparison Test: Let f(x) be a known convergent or divergent function: lim f(x)= L 0 < L < x g(x) f(x) & g(x) both converge or both diverge

Greek letter sigma (sum of)

sequencesLimits of Common Sequences: Convergence/Divergence: Let L be a finite numbern

lim ln n =nn

0 01 n

San=0Infinite Series:n

BCsequencen

nNth Term Test for Divergence Given: If:

first number

n Convergent

lim an = Ln Divergent

limn

x = n!

n

lim an L

n

|x| divergent series, San is divergent

Limit Comparison Test Let Sbn be a known convergent or divergent series:

Alternating Series Test (AST)

2 y ( ) ( )

2 x ( ) ( )

(-1) ann=1

n-1

vectorsNotation v = ai + bj Unit Tangent and Unit Normal VectorsT(t) = r ' (t) If T = u1i + u2j ||r ' (t)|| N = -u2i + u1j Vector Valued Functions r(t) = x(t)i + y(t)j r'(t) = x'(t)i + y'(t)j r(t)dt = (x(t)dt + c)i + (y(t)dt + c)j Velocity Equation for Vectors: (- g t + S0)j + V0 t where v0 = #( cos t i + sin t j) # = initial velocity/muzzle speed

n

lim San r =

0 1 into original equation 4. Take of the interval to find radius of convergence

Error (Alternating Series/Taylor) Error = Actual - Approximate Error < First Unused Term

Polar

Basic Shapes (pink = cosine, blue = sine) Circles Lemniscates Spiral of Archimedes r = a cos q r = a cos q r = aq r = a sin q r = a sin q

Rectangular y (x,y)

x

Polar Conversion (x,y) (r,q) x + y = r y x = r cos q tan q =x y = r sin q

taylor/maclaurin seriesTaylor Polynomial P(x) = f (a) + f ' (a)(x - a) + f ''(a)(x - a)2 + f ''' (a)(x - a)3 + + f n (a)(x - a)n 2! 3! n!

S f ( a)(x - a) n!n =0

n

n

Polar

Common MacLaurin Series (r,q)r q2 4 6 ( - 1)n x2n cos x = 1- x + x - x + = (2n)! n =0 2! 4! 6!

Limacons Limacons Cardioids w/ Dimple w/ Inner Loop r = a + b cos q r = a + b cos q r = a + b cos q r = a + b sin q r = a + b sin q r = a + b sin q |a|=|b| |a|>|b| |a| 1 1 du d -1 dx sinh u =u + 1 dx

Hyperbolic Trig Integrals sinh u du = cosh u + C cosh u du = sinh u + C sech u du = tanh u + C csch u du = - coth u + C sech x tanh u du =- sech u + C csch x coth u du = - csch u + C 2000 Chad A. Valencia. All Rights Reserved.

1 -1 d tanh u = 1 - u du , |u|1 dx dx 1 du -1 d sech u =u u - 1 dx , 0