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Velocity, Distance and Area under the Curve

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7. Figure 5.4 shows the velocity, v, of an object (in meters/sec). Estimate the total distance the object traveled between t = 0 and t = 6.

We can estimate this using 1 second intervals. Since the velocity is increasing on the interval from t = 0 to t = 6, the lower estimate of the velocity on each subinterval will occur at the left endpoint, while the upper estimate will occur at the right endpoint.

An underestimate:
Distance

An overestimate:
Distance

These estimates can be averaged to give an approximate distance of ???? meters.

Section 5.3 p. 236: 11, 21, 27

11. Find the area between and for .
You should look at the graphs of the two functions on one grid to determine how to calculate the area. Show the graph! If you are using Excel, try going from 0 to 1 on the x and on the y with major units of 0.2. Then use a computer or a calculator to calculate it!

21. (a) Compute the definite integral .
(Use a calculator or computer to calculate the value)

(b) Interpret the result in terms of areas. (You will need to show the graph of the function and refer to it in your explanation! If you are using Excel, try 0 to 1 on the x-axis and -1 to 5 on the y-axis with major units of 1.)

27. Using Figure 5.34, decide whether each of the following definite integrals is positive or negative. Explain your answer!

(a) is positive/negative (Delete the incorrect answer!)

Why?

(b) is positive/negative

Why?

(c) is positive/negative

Why?

(d) is positive/negative

Why?

Section 7.1 p. 281: 5, 16

In problems 5 and 16, find an antiderivative.

5.

16.

Solution Summary

Velocity, distance and area under the curve are investigated. The expert estimates the total distance the object traveled is determined.

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Unit 5
Practice Problems Name:

Section 5.1 p. 224: 7

7. Figure 5.4 shows the velocity, v, of an object (in meters/sec). Estimate the total distance the object traveled between t = 0 and t = 6.

We can estimate this using 1 second intervals. Since the velocity is increasing on the interval from t = 0 to t = 6, the lower estimate of the velocity on each subinterval will occur at the left endpoint, while the upper estimate will occur at the right endpoint.
First find the velocities at each time point:
V(0) = 0, v(1) = 14, v(2) = 20, v(3) = 25, v(4) = 30, v(5) = 34, v(6) = 37.
When using the left endpoint, let's take a look at from t = 0 1to t = 1.
The time interval is t= 1 - 0 = 1
Distance = velocity * time interval = v(0) * 1 = 0 * 1 = 0
From t = 1 to t = 2, ...

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