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# Drill: Find dy / dx PowerPoint PPT Presentation

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.

Drill: Find dy / dx

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### 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

## Definite Integrals and Antiderivatives

Lesson 5.3

### Objectives

• Students will be able to

• apply rules for definite integrals and find the average value of a function over a closed interval.

### Rules for Definite Integrals

• Order of Integration

• Zero

• Constant Multiple

### Rules for Definite Integrals

• Sum and Difference

• Max-Min Inequality: If max f and min f are the maximum and minimum values of f on [a, b], then

### Rules for Definite Integrals

• Domination

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

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

### Example 1 Using the Rules for Definite Integrals

Suppose

Find each of the following integrals, if possible.

### Example 1 Using the Rules for Definite Integrals

Suppose

Find each of the following integrals, if possible.

### Example 1 Using the Rules for Definite Integrals

Suppose

Find each of the following integrals, if possible.

### Example 1 Using the Rules for Definite Integrals

Suppose

Find each of the following integrals, if possible.

Not possible; not enough information given.

### Example 1 Using the Rules for Definite Integrals

Suppose

Find each of the following integrals, if possible.

Not possible; not enough information given.

### Example 1 Using the Rules for Definite Integrals

Suppose

Find each of the following integrals, if possible.

Not possible; not enough information given.

### Average (Mean) Value

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

### 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.

### Homework

• day 1: Page 290-292: 1-5 odd, 11-14, 47-49

• day 2: p. 291: 19-30, 31-35 odd

### Drill: Find dy/dx

• y = ln (sec x + tan x)

• y = xln x –x

• y = xex

sec(x)

### Using Antiderivativesfor Definite Integrals

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

where f is the derivative of F.

### 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.

### 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!)

### Example 3 Finding an Integral Using Antiderivatives

Find each integral.

### Example 3 Finding an Integral Using Antiderivatives

Find each integral.