Connecting Algebra to Calculus Indefinite Integrals
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1 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
2 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
3 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
4 Eample: Fin the inefinite integral using two ifferent methos. + Metho : Decomposition + + ( + ( ( + ( + ( + ( ( + 9) Metho : Rewrite Quotient as a Prouct + + ( + ) + ) ( + * - *, / , / ( + 9) Check the result by ifferentiation:! ( + 9)! ), +.+ * which is our integran! Page 4 of 5 MCC@WCCUSD 05/0/0
5 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) Page 5 of 5 MCC@WCCUSD 05/0/0
6 Solutions to the Class Group Activity Problem Metho : Decomposition + + ( ( + ) ( ( 6 6 Metho : Rewrite Quotient as a Prouct + ( + ) ( + ) ( * ) ( * ) 6 6 Check the result by ifferentiation: ), +.+ * - + which is our integran! Page 6 of 5 MCC@WCCUSD 05/0/0
7 Problem Metho : Decomposition Metho : Rewrite Quotient as a Prouct ( ) 4 ( ) C + C Check the result by ifferentiation: C C ( ) ( ) ( ) ( ) which is our integran! Page 7 of 5 MCC@WCCUSD 05/0/0
8 Problem : Metho : Decomposition ( ) Metho : Rewrite Quotient as a Prouct ( ) Check the result by ifferentiation: ( ) i which is our integran! Page 8 of 5 MCC@WCCUSD 05/0/0
9 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
10 Check the result by ifferentiation: ( )! + +C! ( + ) +C! + +C ( + * ), - + ( + ( * ) - *,) ( + * - * - ), ( * * ) + - -, , which is our integran! Page 0 of 5 MCC@WCCUSD 05/0/0
11 Problem 5: Metho : Decomposition ) +, +. +C * - + +C ( + ) +C Check the result by ifferentiation: Metho : Rewrite Quotient as a Prouct + + ( + ) C + ( + )! +! + +C ( + * - + ( ), * ) + + ( ) +C , ( + * - ) +, ( * + ), which is our integran! Page of 5 MCC@WCCUSD 05/0/0
12 Problem 6: Metho : Decomposition ( ( ( ( ( ( 7 + ( ( +C C ( ) +C Check the result by ifferentiation: Metho : Rewrite Quotient as a Prouct ( ) ( ( + 7 ( C ( ) +C C ( ) ) * ), ) * - * ,. ) -, * -, which is our integran! Page of 5 MCC@WCCUSD 05/0/0
13 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
14 Warm Up CCSS: Calculus 4.4 Review: Calculus 4.0 Which of the following are true for this polynomial A. ( ) ( ) f ? f B. ( ) C. f 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
15 Solutions to Warm-Up Quarant I a) b) c) Quarant II ( ) !( ) 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
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