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Section 4.8 - Antiderivatives

Section 4.8 - Antiderivatives. If the following functions represent the derivative of the original function, find the original function. Antiderivative – If F’(x) = f(x) on an interval, then F(x) is the antiderivative of f(x) for every value of x on the interval. .

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Section 4.8 - Antiderivatives

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  1. Section 4.8 - Antiderivatives If the following functions represent the derivative of the original function, find the original function. Antiderivative – If F’(x) = f(x) on an interval, then F(x) is the antiderivative of f(x) for every value of x on the interval.

  2. State the derivative of each function. Section 4.8 - Antiderivatives Theorem: – If F(x) is an antiderivative of f(x) on an interval I, then the general antiderivative of f(x) is:

  3. Antiderivative Formulas where k is a constant (from page 281 of the textbook) Section 4.8 - Antiderivatives

  4. Section 4.8 - Antiderivatives Write the general antiderivative of each of the following functions.

  5. Indefinite Integrals Section 4.8 - Antiderivatives

  6. Initial Value Problems Solve for the original equation if given and . Section 4.8 - Antiderivatives

  7. Estimating Area Under a Curve Approximate the area under the curve from to using 2 rectangles. . Section 5.1 – Area and Estimating Finite Sums Left-hand endpoints Right-hand endpoints Midpoints 2 1 2 1 1 2

  8. Estimating Area Under a Curve Approximate the area under the curve from to using 4 rectangles. . Section 5.1 – Area and Estimating Finite Sums Left-hand endpoints Right-hand endpoints Midpoints 2 1 1 2 1 2 LH RH Mid

  9. Average Value of an Integral Average Value: Given a closed interval for a continuous function, the average value is the function value that when multiplied by the length of the interval produces the same area as that under the curve. Section 5.1 – Area and Estimating Finite Sums AV AV AV

  10. Average Value of an Integral Estimate the average value for the function on the interval using four midpoint subintervals (rectangles) on equal width. . Section 5.1 – Area and Estimating Finite Sums 1 3 2 4

  11. Sigma Notation – A mathematical notation that represents the sum of many terms using a formula. Sequence – a function whose domain is positive integers. Sigma Notation Section 5.2 – Sigma Notation and Limits of Finite Sums g

  12. Sigma Notation Examples Section 5.2 – Sigma Notation and Limits of Finite Sums )

  13. Sigma Notation Express the sums in sigma notation. Section 5.2 – Sigma Notation and Limits of Finite Sums

  14. Sigma Notation Linearity of Sigma Section 5.2 – Sigma Notation and Limits of Finite Sums Example

  15. Summation Rules Section 5.2 – Sigma Notation and Limits of Finite Sums

  16. Summation Rules Examples Section 5.2 – Sigma Notation and Limits of Finite Sums

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