Connecting Algebra to Calculus Inefinite Integrals Objective: Fin Antierivatives an use basic integral formulas to fin Inefinite Integrals an make connections to Algebra an Algebra. Stanars: Algebra.0, 0.0,.0/N-RN., N-RN., A-SSE., Calculus 5.0 Lesson: Suppose you were aske to fin a function F whose erivative is f ( )? Coul you go back an fin the original function whose erivative is? Talk to a neighbor, an then I will ask for a non-volunteer to answer. Since, then F( ). We can check by taking the erivative of F( ). Check: ( ) ( ) F F! Now, we can see that F! ( ) f ( ) We call F an antierivative of f. The function F is an antierivative of f. What if we trie saying that the original function is F erivative that we are looking for? What about F How many antierivatives are there for f equal to 0, there are infinitely many antierivatives for the function f can write F ( ) + 5? Does this give us the correct ( ) 5? ( )? Well, since the erivative of a constant is ( ). In general, we ( ) since any constant C will result in F ( ) f ( ). We can call F( ) a Family of Antierivatives. We can always check our work by taking the erivative F. Page of 5 MCC@WCCUSD 05/0/0
Eploration: Fining Antierivatives With your partner or group, for each erivative, fin the original function F. In other wors, fin the antierivative of f. Justify your work. a) f b) f c) f ) f e) f ( )! F ( ) ( ) F ( ) ( ) F ( )!! ( ) F ( ) ( ) F ( ) Stuents can justify their work by showing that F! ( ) f ( ). Definition of an Antierivative A function F is an antierivative of f on an interval I if F! Note: An antierivative is also calle an Inefinite Integral. ( ) f ( ) for all in I. Notation for Inefinite Integrals: Integran f ( ) F ( ) Constant of Integration Integral Sign Variable of Integration Antierivative of f Page of 5 MCC@WCCUSD 05/0/0
We can substitute to show the inverse nature of ifferentiation an integration. Since, F ( ) f ( ) an F ( ) F ( ), Then f ( ) F( ) So, if f ( ) F( ) Then [ f ( ) ] f ( ) Some Basic Integration Rules (There is a stuent half-sheet hanout at the en of the lesson.) Differentiation Formula Integration Formula [ C ] 0 0 C [ k ] k k k Constant Multiple Rule for Derivatives k f [ ( )] k f ( ) Sum an Difference Rule for Derivatives [ f ( ) ± g( ) ] f ( ) ± g ( ) Power Rule for Derivatives n [ ] n n Constant Multiple Rule for Integrals k f ( ) k f ( ) Sum an Difference Rule for Integrals [ f ( ) ± g( ) ] f Power Rule for Integrals n n+, n n + ( ) ± g( ) Note: Notice that we o not have a Prouct Rule or Quotient Rule for Integrals at this time. Page of 5 MCC@WCCUSD 05/0/0
Eample: Fin the inefinite integral using two ifferent methos. + Metho : Decomposition + + ( + ( ( + ( + ( + ( 4 + 6 ( + 9) Metho : Rewrite Quotient as a Prouct + + ( + ) + ) ( + * - *, / + +. -, / +. 4 + 6 ( + 9) Check the result by ifferentiation:! ( + 9)! 4 + 6 4 + 6 + 0 + + ), +.+ * - + + which is our integran! Page 4 of 5 MCC@WCCUSD 05/0/0
Class Group Activity In your group, fin the inefinite integral using two methos. Check your result by ifferentiation. Your group will be aske to isplay your work. ) + ) + 4 ) + + 4) 5) + 6) 4 + 5 7 Page 5 of 5 MCC@WCCUSD 05/0/0
Solutions to the Class Group Activity Problem Metho : Decomposition + + ( ( + ) + + + ( ( 6 6 Metho : Rewrite Quotient as a Prouct + ( + ) ( + ) + + + ( * ) ( * ) 6 6 Check the result by ifferentiation: 6 6 + 0 + + ), +.+ * - + which is our integran! Page 6 of 5 MCC@WCCUSD 05/0/0
Problem Metho : Decomposition + 4 + 4 4 4 + 4 4 4 + 4 + 4 + + Metho : Rewrite Quotient as a Prouct + 4 4 ( ) 4 ( ) + + + 4 + + 4+ + + + 4+ + + + + C + C Check the result by ifferentiation: C + + + + C ( ) ( ) ( ) + + 0 + 4 ( ) 4 + + 4 which is our integran! Page 7 of 5 MCC@WCCUSD 05/0/0
Problem : Metho : Decomposition + + + + + + + + + + 5 5 + + 5 ( + 5 +5) Metho : Rewrite Quotient as a Prouct + + + + + + + + + + 5 5 + + 5 ( + 5 +5) Check the result by ifferentiation: ( ) 5 + 5 +5 5 5 + + 5 5 5 + + + 0 + + + i + + + + + which is our integran! Page 8 of 5 MCC@WCCUSD 05/0/0
Problem 4: Metho : Decomposition ( + ) + + + ( + ) Metho : Rewrite Quotient as a Prouct ( ) + 4 ( ) ( ( + ), +. * - + +C ( ) +C ( ) + + +C + +C Checking the result by ifferentiation is shown on the net page. Page 9 of 5 MCC@WCCUSD 05/0/0
Check the result by ifferentiation: ( )! + +C! ( + ) +C! + +C ( + * ), - + ( + ( * ) - *,) ( + * - * - ), ( * * ) + - -, + - + 0, which is our integran! Page 0 of 5 MCC@WCCUSD 05/0/0
Problem 5: Metho : Decomposition + + + + + + + ) +, +. +C * - + +C ( + ) +C Check the result by ifferentiation: Metho : Rewrite Quotient as a Prouct + + ( + ) + + + + + + C + ( + )! +! + +C ( + * - + ( ), * ) + + ( ) +C + - + 0, ( + * - ) +, ( * + ), - + + which is our integran! Page of 5 MCC@WCCUSD 05/0/0
Problem 6: Metho : Decomposition 4 +5 7 4 + 5 7 ( ( 5 +5 7 ( ( 5 +5 7 4 + +5 5 8 + ( ( 7 + ( ( +C 4 4 + 5 8 8 +C ( ) +C 56 4 4 + 5 96 Check the result by ifferentiation: Metho : Rewrite Quotient as a Prouct 4 +5 7 ( ) 4 +5 7 5 +5 7 ( 5 +5 7 4 + + 5 5 ( + 7 ( 8 4 4 + 5 8 8 +C ( ) +C 56 4 4 + 5 96 + +C 56 4 4 + 5 96 ( ) 4 4 + 5 8 8 ) + + 8 * 5 8 5 + 5 7 5 + 5 7 ), ) 5 +. +. + 5 + + * - * 4 + 5 7 4 + 5 7 5,. ) -, +. + 0 * -,.. 7 - which is our integran! Page of 5 MCC@WCCUSD 05/0/0
Differentiation Formula Integration Formula [ C ] 0 0 C [ k ] k k k Constant Multiple Rule for Derivatives k f [ ( )] k f ( ) Sum an Difference Rule for Derivatives [ f ( ) ± g( ) ] f ( ) ± g ( ) Power Rule for Derivatives n [ ] n n Constant Multiple Rule for Integrals k f ( ) k f ( ) Sum an Difference Rule for Integrals [ f ( ) ± g( ) ] f Power Rule for Integrals n n+, n n + ( ) ± g( ) Differentiation Formula Integration Formula [ C ] 0 0 C [ k ] k k k Constant Multiple Rule for Derivatives k f [ ( )] k f ( ) Sum an Difference Rule for Derivatives [ f ( ) ± g( ) ] f ( ) ± g ( ) Power Rule for Derivatives n [ ] n n Constant Multiple Rule for Integrals k f ( ) k f ( ) Sum an Difference Rule for Integrals [ f ( ) ± g( ) ] f Power Rule for Integrals n n+, n n + ( ) ± g( ) Page of 5 MCC@WCCUSD 05/0/0
Warm Up CCSS: Calculus 4.4 Review: Calculus 4.0 Which of the following are true for this polynomial A. ( ) ( ) f + 6 + 8+? f 6 + + 8 B. ( ) C. f 6 + + 8 True True False False f ( ) + 8 True False D. f!( ) 6( +) ( ) True False E. f ( ) 6( )( + ) True y False Power Rule for Derivatives: Given: n n Fin the erivative: a) b) n 5 c) 4 Recall: Power Rule for Derivatives: n ( ) n ( ) f f n Page 4 of 5 MCC@WCCUSD 05/0/0
Solutions to Warm-Up Quarant I a) 5 5 4 b) c) 4 4 4 4 4 5 4 5 Quarant II ( ) + 6 +8 +!( ) 6 + +8 This makes choice B true ( ) 6( )!( ) 6( +) ( ) This makes choice D true f f f f Choices A, C an E are all false Page 5 of 5 MCC@WCCUSD 05/0/0