SPSU Math 1113: Precalculus Cheat Sheet §5.1 Polynomial Functions and Models (review) Steps to Analyze Graph of Polynomial tan 2α = 1. y intercepts: f (0) 2. x intercept: f(x) = 0 3. f crosses / touches axis @ x intercepts sin = ± cos = ± 4. End behavior: like leading term 5. Find max num turning pts of f: (n – 1) tan = ± = = 6. Behavior near zeros for each x intercept §9.2 Law of Sines 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 is a horizontal asymptote. Heron’s Formula sin sin sin 3. If n = (m + 1), quotient from long div is = ax + b and line y = ax + b is oblique asymptote. K = = + + 4. If n > (m + 1), R has no asymptote. − − − §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 − ⁄ 2 = − ω = frequency (stretch/shrink horizontally) where a, b, m constants: cos |ω| < 1 stretch |ω| > 1 shrink ω < 0 reflect b = damping factor (damping coefficient) period = T = m = mass of oscillating object |a| = displacement at t = 0 = period if no damping §7.8 Phase Shift = y = A sin (ωx – φ) + B 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 0, y = tan (x) Restrict range to else , − = + −1 §8.2 Inverse Trig Fcns (con’t) tan Q Q 1 y = sec x where |x| ≥ 1 and 0 ≤ y ≤ π, y ≠ −1 = tan + Q Q y = csc 1 x where |x| ≥ 1 and ≤ y ≤ , y ≠ 0 = 0, > 0 ⁄ 2 1 y = cot x where ∞ < x < ∞ and 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: tan = cot = Products & Quotients of Complex 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 = sin θ + cossec =θ = 1 cot = tan 2 θ + 1 = sec 2 θ cot 2 θ + 1 = csc 2θ = cos + + sin + §8.4 Sum & Difference Formulae z2 ≠ 0 De Moire’s Theorem = cos − +z = r (cos θ + i sin θ) sin − cos α ± β = cosα cosβ∓sinα sin β n ≥ 1 sin α ± β = sinα cosβ±cosα sin β ± = cos n ≥ 2, k + sin = 0, 1, 2, …, (n – 1)) tan α ± β = ∓ §8.5 Double Angle & Half Angle Formulae where = k√ = 0, 1, 2, …, cos + (n – 1) + sin + sin 2α = 2sinα cosα cos 2α = cos α − sin α cos 2α = 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 + = 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 v + w = ( a1 + a2)i + (b1 + b2)j = (a1 + a2, b1 + b2) Cramer’s Rule: etc. v – w = ( 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|| = + + = + + = + §10.5 The Dot Product D = v = a i + b j w = a i + b j ≠ 0 1 1 2 2 the unique soln of system given by v · w = a1 a2 + b 1 b2 Angle between 2 Vectors ∙ = = = Decompose a Vector into Orthogonal Vectorscos = Properties of Determinates Vector projection of v onto w Value of D changes sign if 2 rows interchanged. ∙ Value of D changes sign if 2 columns interchanged. Draw v & w with same initial pt = If all entries in any row are zero, then D = 0 From terminal pt of v drop ┴ to w = − 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. = … … = + + ⋯ + 2. Multiply (or divide) each side of eqn by same non zero Product of rectangular 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. 5. Now in row echelon form. Analyze resulting system of eqns for Can also use substitution & or elimination just like systems of solns to original system of eqns. linear eqns. Beware of extraneous solns.
Page 2: Cheat Sheet SPSU Math 1113 Dr. Adler