pre calc final exam cheat sheet, Cheat Sheet of Mathematics

Download pre calc final exam cheat sheet and more Cheat Sheet Mathematics in PDF only on Docsity! Dr. Adler SPSU Math 1113 Cheat Sheet: Page 1 SPSU Math 1113: Precalculus Cheat Sheet §5.1 Polynomial Functions and Models (review) Steps to Analyze Graph of Polynomial 1. y-intercepts: f (0) 2. x-intercept: f(x) = 0 3. f crosses / touches axis @ x-intercepts 4. End behavior: like leading term 5. Find max num turning pts of f: (n – 1) 6. Behavior near zeros for each x-intercept 7. May need few extra pts to draw fcn. §5.2 Rational Functions Finding Horizontal/Oblique Asymptotes of R where degree of numer. = n and degree of denom. = m 1. If n < m, horizontal asymptote: y = 0 (the x-axis). 2. If n = m, line
=  is a horizontal asymptote. 3. If n = (m + 1), quotient from long div is ax + b and line y = ax + b is oblique asymptote. 4. If n > (m + 1), R has no asymptote. §7.6 Graphing Sinusoidals Graphing y = A sin (ωx) & y = A cos (ωx) |A| = amplitude (stretch/shrink vertically) |A| < 1 shrink |A| > 1 stretch A < 0 reflect Distance from min to max = 2A ω = frequency (stretch/shrink horizontally) |ω| < 1 stretch |ω| > 1 shrink ω < 0 reflect period = T =  §7.8 Phase Shift = y = A sin (ωx – φ) + B y = A cos (ωx – φ) + B §8.1 Inverse Sin, Cos, Tan Fcns y = sin-1 (x) Restrict range to [-π/2, π/2] y = cos-1 (x) Restrict range to 0,  y = tan-1 (x) Restrict range to −  ,  §8.2 Inverse Trig Fcns (con’t) y = sec-1 x where |x| ≥ 1 and 0 ≤ y ≤ π, y ≠  y = csc-1 x where |x| ≥ 1 and −  ≤ y ≤ , y ≠ 0 y = cot-1 x where -∞ < x < ∞ and 0 < y < π §8.3 Trig Identities tan  =    cot  =    csc  = #  sec  = #  cot  = #%&  Pythagorean: sin2 θ + cos2 θ = 1 tan2 θ + 1 = sec2 θ cot2 θ + 1 = csc2θ §8.4 Sum & Difference Formulae cos'α ± β+ = cos α cos β ∓ sin α sin β sin'α ± β+ = sin α cos β ± cos α sin β tan'α ± β+ = %&'.+±%&'/+#∓%&'.+ %&'/+ §8.5 Double-Angle & Half-Angle Formulae sin '2α+ = 2sin α cos α cos '2α+ = cos α − sin α cos '2α+ = 1 − 2 sin α = 2 cos α −1 tan'2α+ = %&'.+#2%&3 '.+ sin 4 = ±5#2'4+ cos 4 = ±5#6'4+ tan 4 = ±5#2'4+#6'4+ = #2 4 4 =  4#6 4 §9.2 Law of Sines  7 =  8 =  9: §9.3 Law of Cosines a2 = b2 + c2 – 2bc cos A c2 = a2 + b2 – 2ab cos C b2 = a2 + c2 – 2ac cos B §9.4 Area of Triangle K = # ;< sin = = # >< sin ? = # >; sin @ Heron’s Formula A = # '> + ; + <+ K = CA 'A − >+'A − ;+'A − <+ §9.5 Simple & Damped Harmonic Motion Simple Harmonic Motion d = a cos(ωt) or d = a sin(ωt) Damped Harmonic Motion D'E+ = >F−';E+ '2G+⁄ cos I5J − 3KL3 EM where a, b, m constants: b = damping factor (damping coefficient) m = mass of oscillating object |a| = displacement at t = 0  = period if no damping §10.1 Polar Coordinates Convert Polar to Rectangular Coordinates x = r cos θ y = r sin θ Convert Rectangular to Polar Coordinates If x = y = 0 then r = 0, θ can have any value else N = CO +
  = PQR QS tan−1 TUtan−1 TU +  2⁄−  2⁄ V QX YN QXZQXX YN QXXXO = 0,
> 0O = 0,
< 0 §10.3 Complex Plane & De Moivre’s Theorem Conjugate of z = x + yi is ]^ = x + yi Modulus of z: |]| = √] ]^ = CO +
 Products & Quotients of Complex >bs (Polar) z1 = r1 (cos θ1 + i sin θ1) z2 = r2 (cos θ2 + i sin θ2) ]#] = N#N cos'# + + + a sin'# + + bcb3 = dcd3 cos'# − + + a sin'# − + z2 ≠ 0- De Moire’s Theorem z = r (cos θ + i sin θ) ]e = Ne cos 'f+ + a sin'f+ n ≥ 1 ghijklm nhhop n ≥ 2, k = 0, 1, 2, …, (n – 1)) ]q = √N rcos e + q e  + a sin e + q e s where k = 0, 1, 2, …, (n – 1)