Warmup
This presentation is the property of its rightful owner.
Sponsored Links
1 / 30

warmup PowerPoint PPT Presentation


  • 65 Views
  • Uploaded on
  • Presentation posted in: General

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.

Download Presentation

warmup

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


Warmup

warmup

1)

2)

3)


Warmup

Group Problem: 15 min


Warmup

solution


6 1a integration rules for antiderivatives

6.1a: Integrationrules for Antiderivatives


Warmup

Find the antiderivative of f(x) = 5x4

Work backwards (from finding the derivative)

The antiderivative of f(x)=F’(x) is x5


Warmup

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


Warmup

There is a whole family of functions having 2x as an antiderivative

This family differs only by a constant


Warmup

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


Warmup

Notice there are no lower or upper limits


Warmup

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!


Warmup

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.


Finding the antiderivative

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


Rules for antiderivatives

Rules for Antiderivatives

Power Rule:

Ex:

Ex:

You can always check your answers by taking the derivative!


You do

You Do

1.

2.


Rules for finding antiderivatives

Rules for Finding Antiderivatives

Constant Multiple and Sum/Difference:


Examples

Examples


Example

Example


Recall

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:


Indefinite integrals of exponential functions

Indefinite Integrals of Exponential Functions


Examples1

Examples


Indefinite integral of x 1

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.


Example1

Example


Warmup

5x + C


Warmup

No Calculators


  • Login