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Calculus Reference Sheet ๐ ๐ขโฒ [log ๐ ๐ข] = (ln ๐)๐ข ๐๐ฅ ๐ ๐ข ๐ขโฒ ๐ฃ โ ๐ข๐ฃ โฒ [ ]= ๐๐ฅ ๐ฃ ๐ฃ2 ๐ [๐ข๐ฃ] = ๐ขโฒ ๐ฃ + ๐ข๐ฃโฒ ๐๐ฅ ๐ ๐ข [|๐ข|] = (๐ขโฒ ) |๐ข| ๐๐ฅ ๐ ๐ข [๐ ] = (ln ๐)๐๐ข ๐ขโฒ ๐๐ฅ ๐ ๐ขโฒ [arcsin ๐ข] = ๐๐ฅ โ1 โ ๐ข2 ๐ โ๐ขโฒ [arccos ๐ข] = ๐๐ฅ โ1 โ ๐ข2 โฒ ๐ ๐ข [ln ๐ข] = ๐๐ฅ ๐ข ๐ ๐ข [๐ ] = ๐ ๐ข ๐ขโฒ ๐๐ฅ Area of a Triangle 1 ๐ด๐๐๐ = ๐โ 2 1 ๐ด๐๐๐ = ๐๐ โ sin ๐ด 2 ๐ด๐๐๐ = โ๐ (๐ โ ๐)(๐ โ ๐)(๐ โ ๐) 1 ๐คโ๐๐๐ ๐ = (๐ + ๐ + ๐) 2 Reciprocal Identities: sin ๐ฅ = 1 csc ๐ฅ csc ๐ฅ = 1 sin ๐ฅ cos ๐ฅ = 1 sec ๐ฅ sec ๐ฅ = 1 cos ๐ฅ 1 tan ๐ฅ = cot ๐ฅ 1 cot ๐ฅ = tan ๐ฅ Odd-Even Identities: csc(โ๐ฅ) = โ csc ๐ฅ cos(โ๐ฅ) = cos ๐ฅ sec(โ๐ฅ) = sec ๐ฅ tan(โ๐ฅ) = โ tan ๐ฅ cot(โ๐ฅ) = โ cot ๐ฅ Sum and Difference Identities: sin(๐ข ± ๐ฃ) = sin ๐ข cos ๐ฃ ± cos ๐ข sin ๐ฃ cos(๐ข ± ๐ฃ) = cos ๐ข cos ๐ฃ โ sin ๐ข sin ๐ฃ tan(๐ข ± ๐ฃ) = ๐ ๐ขโฒ [arctan ๐ข] = ๐๐ฅ 1 + ๐ข2 Arc Length: ๐ = ๐๐ (ฮธ must be in radians) ๐ [csc ๐ข] = โ(csc ๐ข cot ๐ข)๐ขโฒ ๐๐ฅ ๐ [sec ๐ข] = (sec ๐ข tan ๐ข)๐ขโฒ ๐๐ฅ ๐ [cot ๐ข] = โ(csc 2 ๐ข)๐ขโฒ ๐๐ฅ Circle: ๐ด๐๐๐ = ๐๐ 2 ๐๐๐๐๐ข๐๐๐๐๐๐๐๐ = ๐๐ Sphere: Cone: 4 ๐ฃ๐๐๐ข๐๐ = ๐๐ 3 3 tan ๐ข ± tan ๐ฃ 1 โ tan ๐ข tan ๐ฃ 1 ๐ฃ๐๐๐ข๐๐ = ๐๐ 2 โ 3 ๐๐ข๐๐๐๐๐ ๐ด๐๐๐ = 4๐๐ 2 Quotient Identities: tan ๐ฅ = Pythagorean Identities: sin ๐ฅ cos ๐ฅ cos ๐ฅ cot ๐ฅ = sin ๐ฅ Law of Sines: sin(โ๐ฅ) = โ sin ๐ฅ ๐ [sin ๐ข] = (cos ๐ข)๐ขโฒ ๐๐ฅ ๐ [cos ๐ข] = โ(sin ๐ข)๐ขโฒ ๐๐ฅ ๐ [tan ๐ข] = (sec 2 ๐ข)๐ขโฒ ๐๐ฅ sin ๐ด sin ๐ต sin ๐ถ = = ๐ ๐ ๐ Area of a Trapezoid: ๐1 + ๐2 ๐ด๐๐๐ = (โ) 2 Area of a Prism: ๐๐๐๐ ๐๐ ๐กโ๐ ๐๐๐ ๐ โ โ๐๐๐โ๐ก sin2 ๐ฅ + cos2 ๐ฅ = 1 tan2 ๐ฅ + 1 = sec 2 ๐ฅ 1 + cot 2 ๐ฅ = csc 2 ๐ฅ Law of Cosines: ๐2 = ๐ 2 + ๐ 2 โ 2๐๐ โ cos ๐ด ๐ 2 = ๐2 + ๐ 2 โ 2๐๐ โ cos ๐ต ๐ 2 = ๐2 + ๐ 2 โ 2๐๐ โ cos ๐ถ Double Angle Identities: sin 2๐ข = 2 sin ๐ข cos ๐ข cos 2๐ข = cos2 ๐ข โ sin2 ๐ข cos 2๐ข = 2 cos2 ๐ข โ 1 cos 2๐ข = 1 โ 2sin2 ๐ข tan 2๐ข = 2 tan ๐ข 1 โ tan2 ๐ข Conics: Parabolas 2 Std. Eqn (๐ฅ โ โ) = 4๐(๐ฆ โ ๐) (๐ฆ โ ๐)2 = 4๐(๐ฅ โ โ) Opens Focus Up/down (h, k+p) Left/right (h+p, k) Directrix axis y=k-p x=h x=h-p y=k Conics: Ellipses (๐ฅ โ โ) (๐ฆ โ ๐)2 (๐ฅ โ โ)2 (๐ฆ โ ๐) + = 1 + =1 ๐2 ๐2 ๐2 ๐2 y = k (horizontal) x = h (vertical) (โ ± ๐, ๐) (โ, ๐ ± ๐) (โ ± ๐, ๐) (โ, ๐ ± ๐) ๐2 = ๐ 2 + ๐ 2 ๐2 = ๐ 2 + ๐ 2 Conics: Hyperbolas 2 2 (๐ฅ โ โ) (๐ฆ โ ๐)2 (๐ฅ โ โ)2 (๐ฆ โ ๐) โ = 1 โ =1 ๐2 ๐2 ๐2 ๐2 y = k (horizontal) x = h (vertical) 2 Std. Eqn a>b Focal axis Foci vertices pythagorean Std. Eqn a>b Focal axis 2 Foci (โ ± ๐, ๐) (โ, ๐ ± ๐) vertices pythagorean Asymptotes (โ ± ๐, ๐) ๐ 2 = ๐2 + ๐ 2 ๐ ๐ฆ = ± (๐ฅ โ โ) + ๐ ๐ (โ, ๐ ± ๐) ๐ 2 = ๐2 + ๐ 2 ๐ ๐ฆ = ± (๐ฅ โ โ) + ๐ ๐ Center is at (h, k) 2a = length of major axis 2b = length of minor axis ๐ = ๐โ๐ 2 ๐ = ๐2 โ ๐ 2 Center is at (h, k) a = dist. From center to vertex c = dist. From center to focus ๐ = ๐โ๐ โซ ๐๐(๐ข)๐๐ข = ๐ โซ ๐(๐ข)๐๐ข โซ sin ๐ข ๐๐ข = โ cos ๐ข + ๐ โซ cot ๐ข ๐๐ข = ln | sin ๐ข| + ๐ โซ[๐(๐ข) ± ๐(๐ข)] ๐๐ข โซ cos ๐ข ๐๐ข = sin ๐ข + ๐ โซ csc 2 ๐ข ๐๐ข = โ cot ๐ข + ๐ โซ tan ๐ข ๐๐ข = โ ln|cos ๐ข| + ๐ โซ sec 2 ๐ข ๐๐ข = tan ๐ข + ๐ โซ csc ๐ข ๐๐ข = ln|sin ๐ข| + ๐ ๐ข = arcsin ( ) + ๐ ๐ โ๐2 โ ๐ข2 ๐๐ข 1 |๐ข| โซ = arcsec ( ) + ๐ ๐ ๐ขโ๐ข2 โ ๐2 a = โซ ๐(๐ข)๐๐ข ± โซ ๐(๐ข)๐๐ข โซ ๐๐ข = ๐ข + ๐ 1 โซ ๐๐ข ๐๐ข = ( ) ๐๐ข + ๐ ๐๐๐ ๐ข ๐ข โซ ๐ ๐๐ข = ๐ + ๐ โซ sec ๐ข ๐๐ข = ln|sec ๐ข + tan ๐ข| + ๐ โซ โซ ๐๐ข ๐๐ข 1 ๐ข = ( ) arctan ( ) + ๐ ๐2 + ๐ข 2 ๐ ๐