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Math 3A Wksheet #11

# Math 3A Wksheet #11 - Jenese Ortiz/Math 3A...

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Jenese Ortiz/Math 3A 3-9&10-09/Week #10/Session 1 Math 3A Worksheet #11 Topics Covered: Antiderivatives (Sec. 5.8) Let’ s Look at the following examples: 1. ? ? = 3 ? 2 2. ? ? = cos ? 3. ? ? = ? ? 4. ? ? = 1 ? Definition : A function F(x) is called an antiderivative of a function f(x) on an interval 𝐼 if F’(x) = f(x) for all ? ∈ 𝐼 . If f(x) is continuous on the closed interval [ a,b ] and differentiable on the open interval (a,b) , with f’(x) =0 for all ? ∈ ( ? , ? ) , then f is a constant on [ a,b ]. If F(x) and G(x) are antiderivatives of the continuous function f(x) on an interval I , then there exists a constant C so that: ? ? = ? ? + 𝐶 for all ? ∈ 𝐼 o Proof: Indeed if F(x) = f’(x) then [ F(x)+C]’ = f’(x)+(C)’ and (C)’ =0, thus ? ? = ? ? + 𝐶

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Examples: 1. ? ? = 3 ? 5 2. ? ? = ? 2 + 2 ? − 1 3. ? ? = 3 ? 2 2 ? 2 4. f(x)= ? 2 ? 5. sec 2 (3 ? ) 6. ? ? = ? 2 ? 3 + cos 𝜋 x 5 ? + ? 3 ? List of general antiderivatives of elementary functions: Function Particular Antiderivative ?? ? ?? ? +C f(x)+g(x) F(x)+G(x)+C ? ? , ? ≠ − 1 1 ? +1 ? ? +1 +C 1 ? ?? ? +C ? ?? 1 ? ? ?? +C sin ?? 1 ? ??? ?? +C cos ? ? 1 ? ?𝑖? ? ? +C sec 2 ?? 1 ? ??? ? ? +C
7. ?? ?? = 3 ? 2 2 ? 2 , ? ≠ 0 8. ?? ?? = ? 2 1 − ? 2 , ? ≥ 0 Examples: Solve the following initial value problems:

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Math 3A Wksheet #11 - Jenese Ortiz/Math 3A...

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