Textbook Notes (363,420)
MATH 264 (2)
Iforgot (1)
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School
McGill University
Department
Mathematics & Statistics (Sci)
Course
MATH 264
Professor
Iforgot
Semester
Fall

Description
Table of Common Derivatives By David Abraham Product and Quotient Rules: d [ f x g x = d [f x g x + f x( ) d [g(x)] dx dx dx d d d f x [ ( )] ( ) f ( ) [g(x)] = dx 2 dx dx x( ) g(x) Trigonometric Functions: d d d sin(x)= cos(x) cos(x)= sin(x) tan(x)= sec (x) dx dx dx d d d 2 dxsec(x) =sec(x)tan(x) dx csc(x) = csc(x)cot(x) dx cot(x)= csc (x) Inverse Trigonometric Functions: d sin (x) = 1 d cos (x) = 1 d tan ( ) = 1 dx 2 dx 2 dx 1+ x 2 1 x 1 x d sec ( ) = 1 d csc ( ) = 1 d cot ( ) = 1 dx x x 1 dx x x 1 dx 1+ x 2 Exponential and Logarithmic Functions: d x x d x x d 1 d 1 a = a ln(a) e = e ln( ) = loga( x) = dx dx dx x dx x ln( ) Table of Common Integrals Standard Integration Techniques: Integration by parts: f (x)g(x)dx = f (x) g(x)dx d [f x)] g x dx x dx 2 2 2 a 2 2 2 a Trig Substitution: a b x x = b sin() b x a x = b sec() 2 2 2 a a + b x x = tan() b Trigonometric Identities: Pythagorean Identities: sin ( + cos ( = 1 1+ tan () = sec () 1+ cot () = csc () Sum-Difference Formulae: sin(u v) = sin(u)cos(v) cos(u)sin(v) cos(u v) = cos(u)cos(v) sin(u)sin(v) Half Angle and Power Reducing Formulae: 2 1 cos(2) 2 1+ cos(2) sin ( = 2 cos ( = 2 2 sin(2) = 2sin()cos() cos(2)=12sin () Trigonometric Functions: sin(u)du = cos(u) + c cos(u)du = sin(u) + c tan(u)du = lnsec(u) + c 2 2 sec (u)du = tan(u) + c csc (u)du = cot(u) + c sec(u)tan(u)du = sec(u) + c csc(u)cot(u)du = csc(u) + c sec(u)du = lnsec(u) + tan(u) + c csc(u)du = lncsc(u) cot(u) + c Exponential and Logarithmic Functions: u e du = e + c a du = a +c ln(u)du = uln(u) u + c ln( ) Inverse Trigonometric Functions: 1 1 1 1 1 2 2 du =sin a + c 2 2du = a tan a + c a u a +u 1 1 1 du = sec + c u u 2 a2 a Advanced Calculus Double Integrals: The volume of a region D, bounded above by the function f(x,y) can be calculated via the following double integral in Cartesian coordinates: = , = , The equivalent in polar coordinates is then: = , = , Where: = + = = If the area of the region D is desired instead of the volume of the region, the integrand f(x,y) or f(r,) = 1. Triple Integrals: Triple integrals over a three dimensional region can be computed in the following way: ,, = ,, The volume of the region can be computed if the integrand is made to be 1. The equivalent in Cylindrical Coordinates is then: ,, = ,, Where: = + = = = The equivalent in Spherical Coordinates is then: ,, = ,, sin Where: = sin cos = sin sin = cos = + + Change of Variable: The Jacobian When performing a double or triple integral, sometimes the region D or V can be simplified by use of a change of variable. In such a case, a transformation factor must be added to the integration. This factor is equal to the determinant of the Jacobian matrix: , = = , Therefore: = Vector Fields: A vector field is a function that assigns a vector, having both magnitude and direction to any point in 2 or 3 dimensional space. It has the general form in three dimensions: = ,, + ,, + ,,
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