SPSU Math 1113: Precalculus Cheat Sheet §5.1 Polynomial Functions and Models (review) ͦ΢ΏΜ ʚρʛ Steps to Analyze Graph of Polynomial tan ʚ2αʛ = ͥͯ΢ΏΜ v ʚρʛ 1. y-intercepts: f (0) 2. x-intercept: f(x) = 0 P ͥͯΑΝΡ ʚPʛ P ͥͮΑΝΡ ʚPʛ 3. f crosses / touches axis @ x-intercepts sin ʠͦʡ = ±ǯ ͦ cos ʠͦʡ = ±ǯ ͦ 4. End behavior: like leading term P ͥͯΑΝΡ ʚPʛ ͥͯΑΝΡ P ΡΗΜ P 5. Find max num turning pts of f: (n – 1) tan ʠͦʡ = ±ǯͥͮΑΝΡ ʚPʛ = ΡΗΜ P = ͥͮΑΝΡ P 6. Behavior near zeros for each x-intercept §9.2 Law of Sines ΡΗΜ ΡΗΜ ΡΗΜ 7. May need few extra pts to draw fcn. = = §9.3 Law of Cosines a2 = b 2 + c 2 – 2bc cos A §5.2 Rational Functions c2 = a 2 + b 2 – 2ab cos C b2 = a 2 + c 2 – 2ac cos B Finding Horizontal/Oblique Asymptotes of R where degree of numer. = n and degree of denom. = m §9.4 Area of Triangle 1. If n < m, horizontal asymptote: y = 0 (the x-axis). K = ͥ = ͥ = ͥ 2. If n = m, line ġ is a horizontal asymptote. Heron’s Formulaͦ ͖͗ sin ̻ ͦ ͕͗ sin ̼ ͦ ͕͖ sin ̽ 3. If n = (m + 1),ͭ quotient = Ġ from long div is ax + b and line y = ax ͥ + b is oblique asymptote. K = ͧ = ͦ ʚ͕ + ͖ + ͗ʛ 4. If n > (m + 1), R has no asymptote. ǭͧ ʚͧ −͕ʛʚͧ −͖ʛʚͧ −͗ʛ §9.5 Simple & Damped Harmonic Motion §7.6 Graphing Sinusoidals Simple Harmonic Motion Graphing y = A sin (ωx) & y = A cos (ωx) d = a cos(ωt) or d = a sin(ωt) |A| = amplitude (stretch/shrink vertically) |A| < 1 shrink |A| > 1 stretch A < 0 reflect Damped Harmonic Motion Distance from min to max = 2A v −ʚ͖ͨ ʛ⁄ ʚ2͡ʛ ͦ ͘ʚͨʛ = ͕ ʦǯ! − v ͨʧ ω = frequency (stretch/shrink horizontally) where a, b, m͙ constants: cos ͨ( |ω| < 1 stretch |ω| > 1 shrink ω < 0 reflect b = damping factor (damping coefficient) period = T = ͦ_ m = mass of oscillating obJect ʺ h ʺ |a| = displacement at t = 0 = period if no damping §7.8 Phase Shift = Ā ͦ_ y = A sin (ωx – φ) +ă B h y = A cos (ωx – φ) + B §10.1 Polar Coordinates Convert Polar to Rectangular Coordinates §8.1 Inverse Sin, Cos, Tan Fcns x = r cos θ y = r sin θ y = sin -1 (x) Restrict range to [-π/2, π/2] Convert Rectangular to Polar Coordinates y = cos -1 (x) Restrict range to If x = y = 0 then r = 0, θ can have any value -1 ʞ0, ʟ y = tan (x) Restrict range to _ _ else , ͦ ͦ ʠ− ͦ ͦʡ ͦ = ǭͬ + ͭ −1 4 §8.2 Inverse Trig Fcns (con’t) ˫ tan ʠ3ʡ Qͽ ͣͦ QͽΊ -1 4 y = sec x where |x| ≥ 1 and 0 ≤ y ≤ π, y ≠ ˮ −1 ͽͽ ͽͽͽ _ = tan ʠ3ʡ + Q ͣͦ Q y = csc -1 x where |x| ≥ 1 and ≤ y ≤ , y ≠ 0ͦ ˬ ͬ = 0, ͭ > 0 _ _ ˮ ⁄ 2 -1 y = cot x where -∞ < x < ∞ −andͦ 0 < y ͦ< π ˭ − ⁄ 2 ͬ = 0, ͭ < 0 §10.3 Complex Plane & De Moivre’s Theorem §8.3 Trig Identities Conjugate of z = x + yi is = x + yi Modulus of z: ͮ¿ ΡΗΜ W ΑΝΡ W ͦ ͦ tan = ΑΝΡ W cot = ΡΗΜ W Products & Quotients|ͮ| = √ͮ of ͮ¿ =Complexǭͬ + ͭ bs (Polar) ͥ ͥ ͥ 2 2 z1 = r 1 (cos θ 1 + i sin θ 1) z2 = r 2 (cos θ 2 + i sin θ 2) Pythagorean:csc = ΡΗΜ W sin θsec + cos =θΑΝΡ = W1 cot = ΢ΏΜ W tan 2 θ + 1 = sec 2 θ cot 2 θ + 1 = csc 2θ ͮͥͮͦ = ͦͥͦͦʞcos ʚͥ + ͦʛ + ͝ sin ʚͥ + ͦʛʟ §8.4 Sum & Difference Formulae 5u -u z2 ≠ 0- ͥ ͦ ͥ ͦ De Moire’s5v = -v ʞcos Theoremʚ − ʛ +z = ͝ sinr (cosʚ θ− + i ʛsinʟ θ) cos ʚα ± βʛ = cosα cosβ∓sinα sin β n ≥ 1 sin ʚα ± βʛ = sinα cosβ±cosα sin β ) ) ΢ΏΜ ʚρʛ±΢ΏΜ ʚςʛ ͮ = ͦ ʞcos ʚ͢ nʛ ≥+ 2, ͝ sin k ʚ=͢ 0,ʛ ʟ1, 2, …, (n – 1)) tan ʚα ± βʛ = ͥ∓΢ΏΜ ʚρʛ ΢ΏΜ ʚςʛ ̸̯̭̰̬̥̉ ̴̘̯̯̳ §8.5 Double-Angle & Half-Angle Formulae ġ W ͦ&_ W ͦ&_ & ͦ ͦ whereͮ = k√ =ͦ 0,ʢcos 1, 2, ʠ) …,+ (n) –ʡ 1) + ͝ sin ʠ) + ) ʡʣ sin ʚ2αʛ = 2sinα cosα cos ʚ2αʛ = cos α − sin α ͦ ͦ cos ʚ2αʛ = 1 − 2 sin α = 2 cos α −1 Dr. Adler SPSU Math 1113 Cheat Sheet: Page 1 §10.4 Vectors §12.3 Systems of Linear Eqns: Determinants Unit Vectors D = = (ad – bc) ≠ 0 unit vectors: i, j, k in direction x-axis, y-axis, z-axis ͕ͬ + ͖ͭ = ͧ ͕ ͖ Ƥ ʺ ʺ Add & Subtract Vectors Algebraically ͗ͬ + ͭ͘ = ͨ Dx = Dy = ͗ ͘ v = (a , b ) = a i + b j w = (a , b ) = a i + b j ͧ ͖ ͕ ͧ 1 1 1 1 2 2 2 2 ʺ ʺ ʺ ʺ v + w = ( a1 + a2)i + (b1 + b2)j = (a1 + a2, b1 + b2) Cramer’sͨ Rule: ͘ ͗ ͨ etc. v – w ī Ĭ = ( a1 – a2)i + (b1 – b2)j = (a1 – a2, b1 – b2) ͬ = ͭ = v α = (α a 1)i + (α b 1)j = (α a 1, α b 1) ͕ͥͥ ͬ + ͕ͥͦ ͭ + ͕ͥͧ ͮ = ͗ͥ ͦͥ ͦͦ ͦͧ ͦ || v|| = ƥ͕ ͬ + ͕ ͭ + ͕ ͮ = ͗ ͦ ͦ ͧͥ ͧͦ ͧͧ ͧ ǯ ͥ ͥ ͕ ͬ + ͕ ͭ + ͕ ͮ = ͗ ͕ + ͖ ͥͥ ͥͦ ͥͧ §10.5 The Dot Product D = ͕ ͕ ͕ ͕ͦͥ ͕ͦͦ ͕ͦͧ v = a i + b j w = a i + b j ɵ ɵ ≠ 0 1 1 2 2 the unique͕ solnͧͥ ͕ofͧͦ system͕ͧͧ given by v · w = a1 a2 + b 1 b2 ī Ĭ ĭ Angle between 2 Vectors Ë∙Ì ͬ = ͭ = ͮ = Decompose a Vector into Orthogonalcos = VectorsĹËĹĹÌĹ Properties of Determinates Vector proJection of v onto w Value of D changes sign if 2 rows interchanged. Ë∙Í Value of D changes sign if 2 columns interchanged. ̋ - Draw v & w with same initial pt Ì̊ = ĹÍĹ Í If all entries in any row are zero, then D = 0 - From terminal pt of v drop ┴ to w Ì̋ = Ì − Ì̊ If all entries in any column are zero, then D = 0 - This creates rt triangle with v as hypotenuse. If any 2 rows have identical corresponding values then D = 0 - Legs of triangle are decomposition If any 2 columns have identical corresponding values then D = 0 If any row multiplied by (nonzero) number k, D is multiplied by k. §12.1 Sys of Linear Eqns; Substitution/Elimination If any column multiplied by (nonzero) k, D is multiplied by k. Solve Systems of Equations by Substitution If entries of any row multiplied by nonzero k and result added to 1. Solve 1 eqn for 1 variable in terms of others. corresponding entries of another row, value of D is unchanged. 2. Substitute result in remaining eqns. If entries of any column multiplied by nonzero k and result added 3. If have eqn in 1 variable, solve it, otherwise loop back to 1 to corresponding entries of another column, D is unchanged. above. 4. Solve remaining variables, if any, by substituting known values §12.4 Matrix Algebra in remaining eqns. Product of Row x Column: 5. Check soln in original system of eqns. ͥ ͗ Solve Systems of Eqns by Elimination ͦ ͥ ͦ ) ͗ ͥ ͥ ͦ ͦ ) ) 1. Interchange any 2 eqns. ͌̽ = ʞͦ ͦ … ͦ ʟ ʮ…ʯ = ͦ ͗ + ͦ ͗ + ⋯ + ͦ ͗ 2. Multiply (or divide) each side of eqn by same non-zero Product of rectangular͗) matrices: constant. A is m x r matrix, B is r x n matrix. 3. Replace any eqn in system by sum (or difference) of that eqn & Aij = Σk Aik Bkj nonzero multiple of another eqn in system. Finding Inverse of onsingular Matrix §12.2 Systems of Linear Eqns: Matrices To find inverse of n x n nonsingular matrix A: Row Operations on the Matrix: 1. Form the matrix [ A | I n]. 1. Interchange any 2 rows. 2. Transform [ A | I n] into reduced row echelon form. 2. Replace a row by nonzero multiple of that row. 3. Reduced row echelon form of [ A | I n] will contain identity 3. Replace a row by sum of that row and a nonzero multiple of matrix In left of vertical bar; the n x n matrix on right of vertical some other row. bar is inverse of A. Matrix Method for Solving System Linear Eqns Solve System Linear Eqns Using Inverse Matrix 1. Write augmented matrix that represents the system. Can write system of eqns as AX = B. 2. Perform row operations that place “1” in locn 1, 1: Perform If have inverse A -1 then multiply by it. row operations that place “0” below this. -1 3. Perform row operations that place “1” in locn 2, 1, leaving X = A B entries to left unchanged. If this is not possible, move 1 cell to §12.6 Matrix Algebra right and try again. Perform row operations that place “0” below it & to left. Solving by Substitution 4. Repeat step 4, moving one row down and 1 col right. Repeat For system of eqns, pts whose coordinates satisfy all eqns are until bottom row or vertical bar reached. represented by intersections of the graphs of eqns.