Formulas to remember

Transcription

1 Complex numbers Let z = x + iy be a complex number The conjugate z = x iy Formulas to remember The real part Re(z) = x = z+z The imaginary part Im(z) = y = z z i The norm z = zz = x + y The reciprocal z = x iy x +y 3 The direction from (for a non zero z) z z 4 The Euler representation z = r exp(iθ) where r = z and θ = Arg(z) = arctan(y/z) The angle θ can be replaced by any angle in the set arg(z) = {Arg(z) + nπ n Z} Finite series formulas Arithmetic series a + (a + d) + + (a + (n )d) = Geometric series formula (a + (i )d) = n(a + (n )d) a + ar + ar n = ar i = a rn r 3 Telescoping series formula If g(x) = f(x) f(x ), then g(i) = f(n) f() 4 Sums of powers (a) = n

2 (b) (c) (d) (e) i = i = n(n + ) n(n + )(n + ) 6 i 3 = n (n + ) 4 i 4 = n(n + )(n + )(3n + 3n ) 3 3 Power series Geometric series Exponential series ax i = a + ax + ax + = a x where x < exp(x) = x i = + x + x! + + xn + for all x R n! 3 Basic Trigonometric series sin(x) = ( ) i x i+ (i + )! = x x3 3! + x5 5! for all x R cos(x) = ( ) i xi (i)! = x! + x4 4! for all x R 4 Generalized Binomial series ( + x) n = nc i x i = (n)(n ) (n i + ) x i i! where x < or n is a non negative integer and x is any complex number

3 5 Log series log( + x) = x x + x3 xn+ + + ( )n + where x is real with x < 3 n + 6 Arctan series where x is real with x < 7 Arcsine series where x is real with x < arctan(x) = x x3 3 + x5 xn+ + + ( )n 5 n + + arcsin(x) = x + x ()(3) ()(4) x ()(3)(5) ()(4)(6) x Geometry The determinant of a matrix: a b = ad bc c d Cramer s Rule: To solve ax + by = e and cx + dy = f define: = a b c d = ad bc, x = e b f d = ed fc and y = a e c f = af ec If then the answer is x = x, y = y If = and at least one of x, y is non zero then there is no solution If all, x, y are zero, then there are infinitely many solutions provided each of the two equations has at least one solution 3 The distance between two points A and B on the real line is d(a, B) = A B 3

4 4 For two points P (a, b ) and P (a, b ) in the xy-plane: The distance between the points is d(p, Q) = (a a ) + (b b ) 5 The parametric two point form of the line containing the points P (a, b ) and P (a, b ) is x = a + t(a a ), y = b + t(b b ) 6 The compact parametric two point form of the line is (x, y) = P + t(p P ) or (x, y) = ( t)p + tp 7 The midpoint of two points A, B is A+B 8 The two point form of the equation through the points P (a, b ), P (a, b ) is (a a )y (b b )x = a b a b 9 The slope of the line through two points P (a, b ), P (a, b ) is b b a a where the slope is infinite and the line is vertical if a = a The slope intercept form of the line joining P (a, b ), P (a, b ) is y = mx + c where m = b b a a is the slope and c = a b a b a a is the y-intercept If p is the x intercept and q is the y-intercept, then the line is: x p + y q = A line parallel to ax + by = c is ax + by = k for some k 3 A line perpendicular to ax + by = c is bx ay = k for some k 4 The quadratic function Q(x) = ax + bx + c has extremum value at x = b The a value is 4ac b It is a maximum if a < and minimum if a > 4a 5 The equation ax + bx + c = with a has solutions: Quadratic Formula x = b ± b 4ac a 4

5 6 The equation of a circle with center (h, k) and radius r is (x h) + (y k) = r 7 For the circle x + y + ux + vy = w the center is ( u/, v/) and radius is w + u + v A parametric form of the circle x + y = r is: where m is the parameter (x, y) = ( r ) m m + m,r + m The trigonometry parameterization for the same circle is: z = r exp(iθ) or with parameter θ (x, y) = (r cos(θ), r sin(θ)) 9 A circle with diameter P (a, b ), P (a, b ) is: (x a )(x a ) + (y b )(y b ) = The distance from a point (p, q) to the line ax + by + c = is: ap + bq + c a + b The expression ap + bq + c = if and only if the point lies on the line Moreover, the sign of the expression ap + bq + c changes if the point moves from one side of the line to the other 5 Trigonometry 3 tan(x) = sin(x) cos(x), cot(x) = cos(x) sin(x), sec(x) = cos(x), csc(x) = sin(x) sin (x) + cos (x) =, sec (x) = + tan (x), csc (x) = + cot (x) sin(x + y) = sin(x) cos(y) + cos(x) sin(y) 5

6 cos(x + y) = cos(x) cos(y) sin(x) sin(y) tan(x + y) = tan(x) + tan(y) tan(x) tan(y) cos( x) = cos(x), sin( x) = sin(x) cos(x) = cos (x) sin (x) = cos (x) = sin (x) sin(x) = sin(x) cos(x) cos( x ) = ± + cos(x) sin( x cos(x) ) = ± The signs need to be fixed by the position of the locator point P( t ) Sine Law sin(a) a = sin(b) b = sin(c) c Cosine Law a = b + c bc cos(a) 6 Derivative Formulas If c is a constant, then D x (c) = Moreover, D x (cy) = cd x (y) Sum Rule D x (y + z) = D x (y) + D x (z) 3 Power Rule For any real n, D x (y n ) = ny n D x (y) As a special case: D x (x n ) = nx n 6

7 4 Product Rule 5 Quotient Rule 6 Chain Rule D x (yz) = D x (y)z + yd x (z) D x ( y z ) = D x(y)z yd x (z) z D x (f(g(x)) = f (g(x))g (x) 7 Linear approximation formula: Linear approximation to f(x) at x = a is: 8 Newton s Method for finding a root L(x) = f(a) + f (x)(x a) Start with some convenient guess x This is the current guess If x n is the current guess then the new current guess shall be x n+ = x n f(x n) f (x n ) Stop when successive current guesses are close to each other (within the set accuracy level) or when the value of the function is sufficiently close to zero (within the set accuracy) 7