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Drill: Find dy / dx

Drill: Find dy / dx. y = - cosx y = sin x y = ln (sec x) y = ln (sin x). d y / dx = sin x dy / dx = cos x dy / dx = (1/sec x)(tan x sec x) = tan x dy / dx = (1/sin x) ( cos x) = cot x. Definite Integrals and Antiderivatives. Lesson 5.3. Objectives.

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Drill: Find dy / dx

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  1. Drill: Find dy/dx • y = -cosx • y = sin x • y = ln (sec x) • y = ln (sin x) • dy/dx = sin x • dy/dx = cos x • dy/dx = (1/sec x)(tan x sec x) = tan x • dy/dx = (1/sin x) (cos x) = cot x

  2. Definite Integrals and Antiderivatives Lesson 5.3

  3. Objectives • Students will be able to • apply rules for definite integrals and find the average value of a function over a closed interval.

  4. Rules for Definite Integrals • Order of Integration • Zero • Constant Multiple

  5. Rules for Definite Integrals • Sum and Difference • Additivity • Max-Min Inequality: If max f and min f are the maximum and minimum values of f on [a, b], then

  6. Rules for Definite Integrals • Domination f(x) > g(x) on [a,b] f(x) > 0 on [a, b]

  7. Example 1 Using the Rules for Definite Integrals Suppose Find each of the following integrals, if possible.

  8. Example 1 Using the Rules for Definite Integrals Suppose Find each of the following integrals, if possible.

  9. Example 1 Using the Rules for Definite Integrals Suppose Find each of the following integrals, if possible.

  10. Example 1 Using the Rules for Definite Integrals

  11. Example 1 Using the Rules for Definite Integrals Suppose Find each of the following integrals, if possible. Not possible; not enough information given.

  12. Example 1 Using the Rules for Definite Integrals Suppose Find each of the following integrals, if possible. Not possible; not enough information given.

  13. Example 1 Using the Rules for Definite Integrals Suppose Find each of the following integrals, if possible. Not possible; not enough information given.

  14. Average (Mean) Value If f is integrable on the interval [a, b], the function’s average (mean) value on the interval is

  15. Example 2 Applying the Definition of Average (Mean) Value Find the average value of f (x) = 6 – x2 on [0, 5]. Where does f take on this value in the given interval? Since 2.887 lies in the interval, the function does assume its average value in the interval.

  16. Homework • day 1: Page 290-292: 1-5 odd, 11-14, 47-49 • day 2: p. 291: 19-30, 31-35 odd

  17. Drill: Find dy/dx • y = ln (sec x + tan x) • y = xln x –x • y = xex sec(x)

  18. Using Antiderivativesfor Definite Integrals If f is integrable over the interval [a, b], then where f is the derivative of F.

  19. Determining Integrals with Power Functions Integrals: (where k and C are constants) Note: when we are evaluating at definite integrals, we do not need to + C.

  20. You will need to remember your derivative rules in order to do your anti-derivatives (integrals) Example: If y = sin x, dy/dx = cos x Therefore, Example: if y = tan x, dy/dx = sec2x Therefore, I would strongly suggest that you dig out your derivatives’ sheet from chapter 3! (You may use it on your next quiz!)

  21. Example 3 Finding an Integral Using Antiderivatives Find each integral.

  22. Example 3 Finding an Integral Using Antiderivatives Find each integral.

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