MAT 220 Sect 5.3.pptx Fundamental Theorem

Theorem of Calculus

1. 5.3 The Fundamental Theorem of Calculus NOTES: The Fundamental Theorem of Calculus: If f is continuous on [ a , b ] and F is an antiderivative of f on [ a , b ], then   F ( b ) – F ( a ) 1.) Act like the integral is an indefinite integral (antiderivative) and find the answer like you usually would, leave out the + C as it is no longer needed. 2.) (Evaluate at the top number) – (Evaluate at the bottom number)

2. EX # 1: Evaluate . a.)   b.)  

3. c.)   EX # 1: Evaluate . d.)  

4. EX # 1: Evaluate . e.)   f.)   No need for absolute value bars.

5. EX #2: Find the area of the region bounded by the graph of , the x - axis, and the vertical lines x = 0 and x = 2.  

6. Differentiation and Integration as Inverse Processes: We end this section by bringing together the two parts of the Fundamental Theorem. We noted that Part 1 can be rewritten as which says that if f is integrated and then the result is differentiated, we arrive back at the original function f . Since F  ( x ) = f ( x ), Part 2 can be rewritten as This version says that if we take a function F , first differentiate it, and then integrate the result, we arrive back at the original function F , but in the form F ( b ) – F ( a ). Taken together, the two parts of the Fundamental Theorem of Calculus say that differentiation and integration are inverse processes. Each undoes what the other does.

7. EX # 3: Evaluate . a.)   b.) Find F ’( x ).   c.)   d.)  

8. EX #4: Find the derivative of   Other than “x”. Chain Rule. By FTC1.

9. EX #5: Find F ’( x ).   Chain Rule and FTC1. Find F ’( x ).  

10. EX # 6: Find F ’( x ).  

11. EX #7: (a) g (0) = g (2) = g (4) = g (6) = g (8) = (b) g ’ ( x ) = f ( t ) g’ ( x ) > 0, increase (0, 4) g ’ ( x ) < 0, decrease (4, 8) (c) Relative max at x = 4, (4, 9)