Understanding Part 2 of The Fundamental Theorem of Calculus

Transcription

1 Understanding Part of The Fundamental Theorem of Calculus

2 Worksheet 8: The Graph of F () What is an Anti-Derivative? Give an eample that is algebraic: and an eample that is graphical: eample : Below is a graph of rate of snowfall, measured in cm/hour, after midnight. The graph shown is f '(, not f a. In words, what does 6 4 f '( stand for? b. Calculate the answer to part a. c. Find f ( 6) f (). Include units. d. Describe in words the meaning of f ( 6) f (). e. If ( ) 3 f find f () f. Eplain in words what problem e above is telling us about the snow. At midnight, and at AM, eample : The graph of f is shown below. Let F' f Without sketching a graph of F, answer the following questions. a. On what intervals is F increasing? Why? b. On what intervals is F concave up? Why? c. For what points does F have an inflection point? Why? d. Where does F have a local ma? Why?

3 eample three: Given the graph of f '( to the right. Below, make a table of values of the anti-derivative, f ( t f ( b. Use your table of values to make a sketch of f on the aes below. Label the coordinates of all local ma, min, and inflection points on f, f ', and the table. Graph b: Graph c: c. Do this problem over again, but this time let f ( ) 5. t f ( 5 Draw the new graph on the aes above and label all local ma, min, and inflection points.

4 Homework Worksheet 8 Graphing F (). If we say that we have two graphs f and F, and we say that F' f in plain language?, what does that mean. Each graph below is a graph of f (), where '( ) f ( ) -where does () - where does F() have inflection points? F. For each, answer: F have a local ma? -where does F() have a local min? 3. Given the sketch of f '( ) a. Let f ( ). Make a table of values of f t f( b. Sketch a graph of your function f below 4. Given the sketch of f '( ) a. Let f ( ) 3. Make a table of values of f t f( b. Sketch a graph of your function f above

5 5. The graph below shows the derivative '( ) - g of a function. It is given that ( ) 5 g. The graph of g'() is piece-wise linear, and goes through the points (-,), (-5,), (,5), (5,), (,), (5,), (,), (5,-), (3,-5) - a. Make a table of values of g t g( b. Sketch a graph of g, labeling all inflection points 6. The graph below is of p (. Assume that P( is the Anti-derivative of p ( and that P ( ) a. Make a table of values of P ( t P( b. Sketch a graph of P (, labeling all inflection points.

6 Worksheet 9: Intro the Area Function eample one: Let f ( be the function graphed below. the function f ( a. What is f (3)? b. f ( = c. d. If = what is f (? 3 f ( = eample : Let f ( be the function graphed above. Let A( ) f (. a. What does stand for? b. What does t stand for? c. What is A ()? A ()? (3) A? d. Is A() an anti-derivative of f? Why? Fact: We define the area function, A ( ) f ( to be the area under the curve of f (positive for f >, negative for f < ) from n (which is a constan to (which we decide the value of) n

7 eample three: Let f ( be the function graphed below. Let B ) the graph of f ( f (. a. Make a table of values of B () from - to 4. b. In the space below left, sketch a graph of B B() The Graph of B The Graph of C Let C( ) f ( c. Make a table of values of C () from - to 4 d. In the space above right, make a graph of C C() In Summary:

8 Homework Worksheet 9: Intro to the Area Function. Let f ( be the function defined at the right. and Let A( ) f ( a. What is f (3)? b. Make a table of values of A from A(). Let G ) ( f (, where f is the function graphed above. a. What is f (3)? b. Make a table of values of G from G() c. Fill in the blank: G ( ) A( ) 3. Let f ( t a. Sketch a graph of f on [ -3, 3 ] to the right. b. Let A ) ( f (. Make a table of values of A() from A() c Let B( ) f (. Make a table of values of B () from B() d. Fill in the blanks: B ( ) A( ) e. Graph A () and B () to the right Label the concavity of each portion of the graphs.

9 4. Let A( ) f ( a. Eplain why A is a function of. b. What does t stand for in this problem? c. Why is Aan ( ) anti-derivative of f? (Hint: Use the FTC ) d. What is the value of A ()? 5. Let C ( ) and B ( ) be two different anti-derivatives of some function gt () a. Write possible formulas for C ( ) and B ( ), using #4 above as a guide. b. What do C ( ) and B ( ) both have in common? c. What things are different C ( ) and B ( )? 6. What is an anti-derivative?

10 Worksheet : Properties of the Area Function We have used area to construct a new function based on another function. Today we will try to find rules relating f and A. We will also find properties relating one function A with others with different starting points. eample one: Let f ( t and let A ( ) f ( a. Draw the graph of f b. Make a table of values of A from 4 to 4 c. Draw the graph of A e. Where is f positive? What is true about A on this interval? f. Where is f negative? What is true about A on these intervals? g. Where does f change sign? What is true about A at these points? h. Where is f increasing? What is true about A on these intervals? i. Where is f decreasing? What is true about A on these intervals? j. Where does f have a local min or ma? What is true about A at this point? k. B( ) f ( Make a table of values for B from -4 to 4 and sketch the graph below B() How are A() and B() related to each other?

11 eample two: Let Q ( ) f ( and let K( ) f ( graph of f( -5 5 a. Q ( ) b. K ( ) c. Q ( ) K() d. Q( ) Q() e. K ( ) K() f. Q ( ) g. K ( ) h. Q ( ) K( ) b j. If Q ( 3) K(3) f ( then a = and b = a eample three: Let w( be a continuous function. Let A( ) w( and let B( ) w(. Fill in the upper and lower bounds on these definite integrals. 3 a. A ( 5) w( b. B ( 5) w( c. B ( 5) A(5) w( d. A ( ) w( e. B ( ) w( f. B ( ) A( ) w( g. B ( 7) A(7) w( h. A ( 5) B(5) w( What did we learn? The difference between two area functions is

12 Homework Worksheet : Properties of the Area Function. Let f ( be a continuous function defined for all numbers t. A possible graph of f is below. Let H( ) f ( and let J ) ( f (. Draw a possible picture of the area defined by each of the following: a. H (3) b. J (3) c. J( 3) H(3) d. H (). Using the equations in problem # above, write a definite integral which stands for each quantity a. H (3) b. J (3) c. J( 3) H(3) d. H () 3. The graph of f ( at the right is made up of a semi-circle and two lines. Let G( ) f ( and let H ( ) f ( the graph of f ( a. Make a table of values for () G and for H () - 4 G () H () b. Where is G increasing? c. Where does H have a local ma? g. What qualities do H and G have in common? d. Where is G concave up? e. Where does H have an inflection point? f. G ( ) H( ) h. What are the differences between H and G?

13 Worksheet : Notation for Anti-Derivatives eample one: for each picture below, fill in the limits of integration to make each equation true 4 f ( f ( f ( f ( f ( f ( f f ( 4 4 ( f ( What did you notice? eample two: Let A( ) f ( w) dw. Write a definite integral which stands for..a. A (5) b. A (6) c. A(4) d. A( 7) A(3) e. A ( 5) A( 3) eample three: Suppose that 3 a. G( ) g(. Write epressions in terms of G. g ( b. g ( c. g ( d. g ( e. g ( Summary: Let A() = f( a constants. and B() = f( b where f is a continuous function, and a and b are Then

14 Homework Worksheet : Notation for Anti-Derivatives Fill in the blanks to summarize what we have learned so far in this chapter:. The graph of y ' tells us a lot about y. We call the process of going from y' to y.. (If f is continuous) We can "build" an anti-derivative of f, Numerically, Graphically, Verbally, and Algebraically. Usually this anti-derivative has the form: A () 3. Every function has anti-derivatives. 4. All the of a function are vertical shifts of each other. 5. If F() and G() are both anti-derivatives of f then F() and G() differ by 6. If ) F ( f ( and G ( ) f ( then F ( ) G( ) a b 7. An anti-derivative A() is increasing when f( is. 8. An anti-derivative A() is concave down when f( is. 9. An anti-derivative A() has an inflection point when f( has a.. An anti-derivative A() has a local min when f( changes.. Refer to eample one and complete: For any integers p,q, and r, q ( f ( p f f (. If K( ) p(, write an integral epression for the following 5 K () K () K ( ) K(9) 3. If J( ) h( w) dw, write an epression in terms of J for the following: 7 h ( w) dw h ( w) dw h ( w) dw 3 8 4

15 Worksheet : The Fundamental Theorem of Calculus Part II During the first half of this unit we studied the function A ) f ( (. Now its time to use our tools of Calculus to analyze the function. In particular, can we find the derivative of this function? The answer is yes, and we will call our result The Fundamental Theorem of Calculus part Two. The FTC part II says that if ( ) f ( t then A( ) A ) In words, b d d b. Here is a proof: A'( ) A( h) lim h h A( ) What does it mean? It means that the slope of the line tangent to the curve of A () can be found from the graph of f evaluated at. eample one: Let right. a. Find () K( ) g( w) dw where g is graphed to the K K '() K ''() b. Find lim h K( h) K() h c. Where is K decreasing? What is true about g at these points? d. Where is K concave up? What is true about g at these points? e. Where does K have a local minimum? What is true about g at these points?

16 eample two: Let F( ) r(. The graph of r is shown at the right. a. Find F '(3) F ''(3) b. Find the -coordinate of each inflection point of F. What is true about r at these points? c. Find the Absolute Ma and Min values of F on [ -, 4 ] endpoint F Absolute Ma: critical point critical point Absolute Min: endpoint d. Find an equation of the line tangent to F at 3. Summary: According to Dan Kennedy (in his calculus te d It is difficult to overestimate the power of the equation d a It says: f ( f ( )

17 Homework Worksheet : The FTC part II. Let H ) ( f (. A graph of f is below. a. What is H () H '( ) H ''( ) b. Where is H increasing? What is true about f on these intervals? c. Where is H concave down? What is true about f on these intervals? d. Where does H have any inflection points? What is true about f at these points? e. Find an equation of the line tangent to H () at f. What are the critical numbers of H ()? What is true about f at these points? g. What are the global maimum and minimum values of H () on the interval [ -4, 4 ]? Show your calculus reasoning. h. Find the value c that satisfies the Mean Value Theorem for H () on [ -4, ] i. What does the Intermediate Value Theorem say about H () on [ -4, 4 ]? Write a complete sentence.

18 Worksheet 3: The Fundamental Theorem of Calculus with Equations First, let s set the stage for today s lesson: State the FTC part one using symbols State the FTC part two: We can use the FTC with functions defined by equations. Remember that the function defined by F( ) f ( is an antiderivative of f and that if f is continuous on the interval [, ] 3 eample one: Let F( ) t 7 a. What is F ()? d ab, then f t a d a ( ) f ( ) where is between a and b. b. What is '( ) F? '() F? c. Write an equation of the line tangent to F () at. 3 eample two: Let G( ) (4t 8t a. Find all critical numbers of G. ) b. Write an algebraic rule for G ''( ) c. Are the critical points in part a.) local min, local ma, or neither? Use the Second Derivative Test to find out. Justify your answers.

19 Homework Worksheet 3: The FTC with Equations. Let A( ), find the equation of the tangent line to A at t. Let B ) t ( cos. a. What is B ()? b. What is ' 3 B? c. Find lim h B( h) B( ) h t 3. Let F( ) e ( t ) a. Find all critical numbers of F (). b. Use the product rule to find F ''( ) c. Are the critical numbers above local min, local ma, or neither? Use the Second Derivative Test to find out: d. What does the MVT say about F () on [, ]? 4. Let G( ) sin t. What are the absolute maimum and minimum values of G () on the interval,? Endpoint Critical number Endpoint G () Conclusions: The Absolute Ma value is and the Absolute Minimum value is

20 Worksheet 4: The Chain Rule with the FTC A. Review of the Chain Rule B. Key Idea: We could have a function act as our upper limit of a definite integral: 5 4 eample one: Let B( ) 3 t t a. Find: '( ) 4 B B '() b. Let H( ) B(sin ) Write an integral epression for: H ( ) for H () c. Write a rule for H' ( ) two different ways cos 3 eample two: Suppose that F( ) t. Find F ' dy eample three: Use the chain rule to determine d 3 if y t

21 Homework Worksheet 4: The FTC with the Chain Rule. a. In words, what is the Chain Rule? b. If ( ) cos( ) 4 4 w, what is w (3 )? c. What is '(3 ). Let F( ) a. Find '( ) 3 t w? F b. Find F '(4) c. Suppose that K( ) F( ). Write an integral epression for K (). d d. What is d e. What is K '( )?? f. Find K '(4). dy 3. Use the chain rule to determine d a) y sin t b) y sin t 4. Find K '( ) if K ( ) tan 7t The graph of g is shown at the right. Let 4 A ( ) g( a. Find A (). b. Find A (). c. Write a rule for A '( ). d. Find A '(). e. Find an equation of the line tangent to A at =.

22 Review of Worksheets 8-4: Constructing Anti-Derivatives.. Let G( ) t. Sketch a graph of y t. Then determine whether the following statements are true or false. a. G( 3) G() b. G '( ) 3 c. The graph of G has a horizontal tangent at =. d. The graph of G has an inflection point at = 3. Let J( ) g( Write an epression in terms of J which stands for: 4 a g ( w) dw b. g ( c. g ( u) du 4 5

23 5. Given the picture of f ( below. Let H( ) f ( a. H ( 3) b. H ( ) c. H ( ) d. H ( ) e. Where does H have a Local Ma? Justify your answer. f. Where is H Concave Down? Justify your answer. 6. Let f ( be the continuous function (ignore the holes) graphed below. Let Q ( ) f ( where a is some constant between -3 and. a a. Write a definite integral that stands for Q(4) b. Calculate Q(7) - Q(5) f ( t ) c. Where does Q have a relative Ma? 5 d. What theorem helps you justify your answer? e. Where does Q have inflection points? f. How do you know? g. What is Q '(7)? h. If K() = f ( then K(6) Q(6) f (

24 7. Fill in the blanks: If a, b, and c, are positive constants, then f ( f ( c b a f ( 8. Let f ( be a continuous function. Let W ( ) f ( and let Q ) integral epressions in the blanks. ( f (. Write a. W ( 5) b. Q ( ) c. W ( 5) Q(5) 5 d. ( 9) Q(9) W e. Q ( 3) W(3) 5 9. Let P( ) (t 8t 3 ) a. What is P ()? b. What is P '( )? c. What is ''( ) P? d. What is the slope of the line tangent to P at? e. What is lim h P( h) P() h f. If J( ) P( e ) then what is J '() dy. Use the Chain Rule to determine if d a. y ( t ) b. y (sin c. y (tan d. y t 3. Find an equation of the line tangent to K( ) at 3 t 3

25 . Let H ) ( f (. A graph of f is below. a. What is H (3) H '( 3) H ''( 3) b. Where is H decreasing? What is true about f on these intervals? c. Where is H concave up? What is true about f on these intervals? d. Where does H have any inflection points? What is true about f at these points? e. What are the critical numbers of H ()? What is true about f at these points? f. What are the global maimum and minimum values of H () on the interval [ -4, 4 ]? endpoint H () critical point critical point critical point endpoint Conclusions: Global Maimum is and Global Minimum is g. Using the same graph of f ( above, let 5 K ( ) f (. Find an equation of the tangent line to () K at = 3