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warmup

warmup. 1). 2). 3). Group Problem: 15 min. solution. 6.1a: Integration rules for Antiderivatives. Find the antiderivative of f(x) = 5x 4 Work backwards (from finding the derivative) The antiderivative of f(x)=F’(x) is x 5.

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warmup

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  1. warmup 1) 2) 3)

  2. Group Problem: 15 min

  3. solution

  4. 6.1a: Integrationrules for Antiderivatives

  5. Find the antiderivative of f(x) = 5x4 Work backwards (from finding the derivative) The antiderivative of f(x)=F’(x) is x5

  6. In the example we just did, we know that F(x) = x2 is not the only function whose derivative is f(x) = 2x G(x) = x2 + 2 has 2x as the derivative H(x) = x2 – 7 has 2x as the derivative For any real number, C, the function F(x)=x2 + C has f(x) as an antiderivative

  7. There is a whole family of functions having 2x as an antiderivative This family differs only by a constant

  8. Since the functions G(x) = x2 F(x) = x2 + 2 H(x) = x2 – 7 differ only by a constant, the slope of the tangent line remains the same The family of antiderivatives can be represented by F(x) + C

  9. Notice there are no lower or upper limits

  10. Integrals such as are called definite integrals because we can find a definite value for the answer. The constant always cancels when finding a definite integral, so we leave it out!

  11. When finding indefinite integrals, we always include the “plus C”. Integrals such as are called indefinite integrals because we can not find a definite value for the answer.

  12. Finding the Antiderivative Finding the antiderivative is the reverse of finding the derivative. Therefore, the rules for derivatives leads to a rule for antiderivatives Example: So

  13. Rules for Antiderivatives Power Rule: Ex: Ex: You can always check your answers by taking the derivative!

  14. You Do 1. 2.

  15. Rules for Finding Antiderivatives Constant Multiple and Sum/Difference:

  16. Examples

  17. Example

  18. Recall Previous learning: If f(x) = ex then f’(x) = ex If f(x) = ax then f’(x) = (ln a)ax If f(x) = ekx then f’(x) = kekx If f(x) = akx then f’(x) = k(ln a)akx This leads to the following formulas:

  19. Indefinite Integrals of Exponential Functions

  20. Examples

  21. Indefinite Integral of x-1 Note: if x takes on a negative value, then lnx will be undefined. The absolute value sign keeps that from happening.

  22. Example

  23. 5x + C

  24. No Calculators

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