Presentation on theme: "Question from Test 1 Liquid drains into a tank at the rate 21e-3t units per minute. If the tank starts empty and can hold 6 units, at what time will it."— Presentation transcript:
1 Question from Test 1Liquid drains into a tank at the rate 21e-3t units per minute. If the tank starts empty and can hold 6 units, at what time will it overflow?A. log(7)/3B. (1/3)log(13/7)C. 3 log (13/7)D. 3log(7)E. Never
2 Fundamental Theorem of Calculus (Part 1) (Chain Rule) Chapter 5.3 & 5.5February 6, 2007Fundamental Theorem of Calculus (Part 1) (Chain Rule)If f is continuous on [a, b], then the function defined byis continuous on [a, b] and differentiable on (a, b) and
4 Fundamental Theorem of Calculus (Part 2) If f is continuous on [a, b], then :Where F is any antiderivative of f.( )Helps us to more easily evaluate Definite Integrals in the same way we calculate the Indefinite!
14 The same substitution holds for the higher power! With
15 Our Original Example of a Definite Integral: To make the substitution complete for a Definite Integral:We make a change of bounds using:
16 Substitution Rule for Indefinite Integrals If u = g(x) is a differentiable function whose range is an interval I and f is continuous on I, thenSubstitution Rule for Definite IntegralsIf g’(x) is continuous on [a,b] and f is continuous on the range of u = g(x), then
17 In-Class AssignmentIntegrate using two different methods:1st by multiplying out and integrating2nd by u-substitutionDo you get the same result? (Don’t just assume or claim you do; multiply out your results to show it!)If you don’t get exactly the same answer, is it a problem? Why or why not?