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Drill: Find dy / dx

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- 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

Definite Integrals and Antiderivatives

Lesson 5.3

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

- Order of Integration
- Zero
- Constant Multiple

- 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

- Domination
f(x) > g(x) on [a,b]

f(x) > 0 on [a, b]

Suppose

Find each of the following integrals, if possible.

Suppose

Find each of the following integrals, if possible.

Suppose

Find each of the following integrals, if possible.

Suppose

Find each of the following integrals, if possible.

Not possible; not enough information given.

Suppose

Find each of the following integrals, if possible.

Not possible; not enough information given.

Suppose

Find each of the following integrals, if possible.

Not possible; not enough information given.

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

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.

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

- y = ln (sec x + tan x)
- y = xln x –x
- y = xex

sec(x)

If f is integrable over the interval [a, b], then

where f is the derivative of F.

Integrals: (where k and C are constants)

Note: when we are evaluating at definite integrals, we do not need to + C.

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!)

Find each integral.

Find each integral.