This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Section 6.1 1. F ( x ) is an anti derivative of f ( x ) if F ( x ) = f ( x ). Taking the derivative we find F ( x ) = x 2 + 4 x 1 = f ( x ) so that F ( x ) is an anti derivative of f ( x ) . 3. F ( x ) is an anti derivative of f ( x ) if F ( x ) = f ( x ). Taking the derivative we find F ( x ) = (1 / 2)(2 x 2 1) . 5 (4 x ) = 2 x 2 x 2 1 = f ( x ) so that F ( x ) is an anti derivative of f ( x ) . 5. G ( x ) is an anti derivative of f ( x ) if F ( x ) = f ( x ). Taking the derivative we find G ( x ) = 2 = f ( x ) so that G ( x ) is an anti derivative of f ( x ) . The set of all anti derivatives of f ( x ) is found by adding an arbitrary constant to G ( x ) . Thus any anti derivative of f ( x ) is of the form 2 x + C . 7. G ( x ) is an anti derivative of f ( x ) if F ( x ) = f ( x ). Taking the derivative we find G ( x ) = x 2 = f ( x ) so that G ( x ) is an anti derivative of f ( x ) . The set of all anti derivatives of f ( x ) is found by adding an arbitrary constant to G ( x ) . Thus any anti derivative of f ( x ) is of the form x 2 + C . 9. R 6 dx = 6 x + C. (Rule 1) 11. R x 3 dx = 1 1+3 x 1+3 + C = 1 4 x 4 + C. (Rule 2) 13. R x 4 dx = 1 1+( 4) x 1+( 4) + C = 1 3 x 3 + C. (Rule 2) 15. R x 2 / 3 dx = 1 1+(2 / 3) x 1+(2 / 3) + C = 3 5 x 5 / 3 + C. (Rule 2) 17. R x 5 / 4 dx = 1 1+( 5 / 4) x 1+( 5 / 4) + C = 4 x 1 / 4 + C. (Rule 2) 19. R 2 x 2 dx = R 2 x 2 dx = 2 R x 2 dx = (2) 1 1+( 2) x 1+( 2) + C = (2)( 1) x 1 + C = 2 x 1 + C. (Rules 2 and 3) 21. R t dt = R t 1 / 2 dt = ( ) 1 1+(1 / 2) t 1+(1 / 2) + C = ( ) 2 3 t 3 / 2 + C = 2 3 t 3 / 2 + C. (Rules 2 and 3) 23. R (3 2 x ) dx = R 3 dx 2 R x dx = 3 x (2) 1 1+1 x 1+1 = 3 x x 2 + C. (Rules 1, 2, 3, and 4) 25. R ( x 2 + x + x 3 ) dx = R x 2 dx + R x dx + R x 3 dx = 1 1+2 x 1+2 + 1 1+1 x 1+1 + 1 1+( 3) x 1+( 3) + C = 1 3 x 3 + 1 2 x 2 + 1 2 x 2 + C. (Rules 2 and 4) 27. R 4 e x dx = 4 R e x dx = 4 e x + C. (Rules 3 and 5) 29. R 1 + x + e x dx = R 1 dx + R x dx + R e x dx = x + 1 1+1 x 1+1 + e x + C = x + 1 2 x 2 + e x + C. (Rules 1, 2, 4, and 5) 31. R 4 x 3 2 x 2 1 dx = 4 R x 3 dx 2 R x 2 dx R 1 dx = (4) 1 1+3 x 1+3 (2) 1 1+( 2) x 1+( 2) x + C = x 4 + 2 x 1 x + C. (Rules 1, 2, 3, and 4) 33. R x 5 / 2 + 2 x 3 / 2 x dx = R x 5 / 2 dx + 2 R x 3 / 2 dx R x dx = 1 1+(5 / 2) x 1+(5 / 2) +(2) 1 1+(3 / 2) x 1+(3 / 2) 1 1+1 x 1+1 + C = (2 / 7) x 7 / 2 +(4 / 5) x 5 / 2 (1 / 2) x 2 + C. (Rules 2, 3, and 4) 35. R x + 3 x dx = R x 1 / 2 dx + 3 R x 1 / 2 dx = 1 1+(1 / 2) x 1+(1 / 2) + (3) 1 1+( 1 / 2) x 1+( 1 / 2) + C = (2 / 3) x 3 / 2 + 6 x 1 / 2 + C. (Rules 2, 3, and 4) 37. R u 3 +2 u 2 u 3 u du = R u 3 3 u + 2 u 2 3 u u 3 u du = R u 2 3 + 2 u 3 1 3 du = (1 / 3) R u 2 du +(2 / 3) R u du R 1 3 du = (1 / 3) 1 1+2 u 1+2 + (2 / 3) 1 1+1 u 1+1 (1 / 3) u + C = (1 / 9) u 3 + (1 / 3) u 2 (1 / 3) u + C....
View
Full
Document
This note was uploaded on 02/14/2012 for the course MAC 2233 taught by Professor Smith during the Spring '08 term at University of Florida.
 Spring '08
 Smith
 Calculus, Derivative

Click to edit the document details