Warm up
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Warm Up. Read the paragraph at the top of page 267 titled “Chapter 6 Overview” Then continue to read all of page 267. 6 .1 Estimating with Finite Sums. Goal: Calculate the distance traveled and use the RAM method. . Example.

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Warm Up

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Warm up

Warm Up

Read the paragraph at the top of page 267 titled “Chapter 6 Overview”

Then continue to read all of page 267


6 1 estimating with finite sums

6.1 Estimating with Finite Sums

  • Goal: Calculate the distance traveled and use the RAM method.


Example

Example

  • Consider an object moving at a constant rate of 3 ft/sec. How far did it go after 4 sec?

Since rate . time = distance:

After 4 seconds, the object has gone 12 feet.


Example1

Example

velocity

time

If we draw a graph of the velocity, the distance that the object travels is equal to the area under the line.


Warm up

If the velocity is not constant,

we might guess that the

distance traveled is still equal

to the area under the curve.

We could estimate the area under the curve by drawing rectangles under the curve.


Big idea

Big Idea!

Area under the velocity curve represents the total distance traveled by an object.


Three different ways to draw rectangles

Three different ways to draw rectangles

Right-hand Rectangular Approximation

Method .

Left-hand Rectangular Approximation

Method .

Midpoint Rectangular Approximation

Method .


Example using lram

Example using LRAM

Approximate area:

A particle moves at a velocity of

How far did the particle travel after 4 seconds.


Warm up

Approximate area:

Same Example using RRAM

1

2

3

4


Warm up

Example of MRAM

Very similar to the other two.

See example #1 on page 268.


Vocab

Vocab

Circumscribed rectangles are all above the curve:

Inscribed rectangles are all below the curve:


Example2

Example

  • A particle starts at x=0 and travels along the x axis with velocity v(t)=4. Where is the particle at t=7?

  • Answer: Find the area of the rectangle under the curve to see the particle is at x=28.


Homework

Homework!


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