Calculus Cheat Sheet

Calculus Cheat Sheet Trigonometric Formulas 1. 2. 3. 4. 5. 6. 7. 8. 9. sin   cos   1 1  tan 2   sec2  1  cot2   csc2  sin( )   sin  cos( )  cos tan( )   tan sin( A  B)  sin A cos B  sin B cos A sin( A  B)  sin A cos B  sin B cos A cos(A  B)  cos A cos B  sin A sin B 2 2 sin  1  cos cot cos 1 14. cot   sin  tan 1 15. sec  cos 1 16. csc  sin  13. tan   17. cos( 18. sin(  10. cos(A  B)  cos A cos B  sin A sin B 11.  2   )  sin  sin 2  2 sin cos 2 2 2 2 12. cos 2  cos   sin   2 cos   1  1  2 sin  2   )  cos Differentiation Formulas 1. 2. 3. 4. 5. 6. 7. 8. 9. d n ( x )  nx n 1 dx d ( fg)  fg   gf  dx d f gf   fg  ( ) dx g g2 d f ( g ( x))  f ( g ( x))g ( x) dx d (sin x)  cos x dx d (cos x)   sin x dx d (tan x)  sec2 x dx d (cot x)   csc2 x dx d (sec x)  sec x tan x dx 10. 11. 12. 13. 14. d (cscx)   csc x cot x dx d x (e )  e x dx d x (a )  a x ln a dx d 1 (ln x)  dx x d 1 ( Arc sin x)  dx 1 x2 d 1 ( Arc tan x)  dx 1 x2 d 1 ( Arc sec x)  16. dx | x | x2 1 dy dy du   17. Chain Rule dx dx dx 15. Integration Formulas 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.  a dx  ax  C n  x dx  x n 1  C , n  1 n 1  x dx  ln x  C  e dx  e  C x x 1  sin x dx   cos x  C  cos x dx  sin x  C  tan x dx  ln sec x  C or  ln cos x  C  cot x dx  ln sin x  C  sec x dx  ln sec x  tan x  C  cscx dx  ln cscx  cot x  C  sec x d x  tan x  C  sec x tan x dx  sec x  C  csc x dx   cot x  C  cscx cot x dx   cscx  C  tan x dx  tan x  x  C 2 2 2 ax  a dx  ln a  C  ln x dx  x ln x  x  C x a 2  x dx 1 x  Arc tan   C 2 a x a dx  x  Arc sin   C a a2  x2 20. dx x2  a2  x 1 1 a Arc sec  C  Arc cos  C a a a x Formulas and Theorems 1a. Definition of Limit: Let f be a function defined on an open interval containing c (except possibly at c ) and let L be a real number. Then lim f ( x )  L means that for each   0 there xa exists a   0 such that f ( x)  L   whenever 0  x  c   . 1b. A function y  f (x) is continuous at x  a if i). ii). iii). 2. f(a) exists lim f ( x) exists xa lim  f (a ) xa 3. Even and Odd Functions 1. A function y  f (x) is even if f ( x)  f ( x) for every x in the function’s domain. Every even function is symmetric about the y-axis. 2. A function y  f (x) is odd if f ( x)   f ( x) for every x in the function’s domain. Every odd function is symmetric about the origin. Periodicity A function f (x) is periodic with period p ( p  0) if f ( x  p)  f ( x) for every value of x. Note: The period of the function y  A sin( Bx  C ) or y  A cos(Bx  C ) is The amplitude is 2 . B A . The period of y  tan x is  . 4. Intermediate-Value Theorem A function y  f (x) that is continuous on a closed interval a, b takes on every value   between f (a) and f (b) . Note: If f is continuous on a, b and f (a) and f (b) differ in sign, then the equation 5.   f ( x)  0 has at least one solution in the open interval (a, b) . Limits of Rational Functions as x   f ( x) lim  0 if the degree of f ( x)  the degree of g ( x) i). x   g ( x) Example: x 2  2x lim 0 x   x3  3 ii). iii). f ( x) is infinite if the degrees of f ( x)  the degree of g ( x) x   g ( x ) x3  2x  Example: lim x   x2  8 f ( x) lim is finite if the degree of f ( x)  the degree of g ( x) x   g ( x ) lim 2 x 2  3x  2 2 lim  5 x   10 x  5 x 2 Example: 6. Horizontal and Vertical Asymptotes 1. A line y  b is a horizontal asymptote of the graph y  f (x) if either 2. lim f ( x)  b or lim f ( x)  b . x x   A line x  a is a vertical asymptote of the graph y  f (x) if either lim f ( x)   or lim   . x  a x  a- 7. Average and Instantaneous Rate of Change i). Average Rate of Change: If x , y y  f (x) , then the average rate of change of y with respect to x over the interval x0 , x1  is f ( x1 )  f ( x0 )  y1  y 0  y . x1  x0 x1  x0 x ii). Instantaneous Rate of Change: If the instantaneous rate of change of 8. 9.  0 0  and x1, y1 are points on the graph of x0 , y 0  is a point on the graph of y  f (x) , then y with respect to x at x 0 is f ( x0 ) . f ( x  h)  f ( x ) f ( x)  lim h h0 The Number e as a limit n  1 i). lim 1    e n   n  1  nn ii). lim 1    e n  0 1  Rolle’s Theorem If f is continuous on a, b and differentiable on 10. f (a)  f (b) , then there is at least one number c in the open interval a, b  such that f (c)  0 . 11. Mean Value Theorem If f is continuous on a, b and differentiable on     a, b such that a, b , then there is at least one number c in a, b  such that 12. f (b)  f (a)  f (c) . ba Extreme-Value Theorem If f is continuous on a closed interval a, b , then f (x) has both a maximum and minimum on a, b .     13. To find the maximum and minimum values of a function y  f (x) , locate 1. 2. the points where f (x) is zero or where f (x) fails to exist. the end points, if any, on the domain of f (x) . 14. Note: These are the only candidates for the value of x where f (x) may have a maximum or a minimum. Let f be differentiable for a  x  b and continuous for a a  x  b , 1. 2. If f ( x)  0 for every x in If f ( x)  0 for every x in a, b , a, b , then f is increasing on a, b . then f is decreasing on   a, b .