Clicker Question 1 Suppose the acceleration (in feet/sec/sec) of a rocket which starts from rest is...
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Transcript of Clicker Question 1 Suppose the acceleration (in feet/sec/sec) of a rocket which starts from rest is...
![Page 1: Clicker Question 1 Suppose the acceleration (in feet/sec/sec) of a rocket which starts from rest is a (t ) = 36t 2. How far does it travel during the first.](https://reader036.fdocuments.net/reader036/viewer/2022072006/56649f4a5503460f94c6c2bc/html5/thumbnails/1.jpg)
Clicker Question 1 Suppose the acceleration (in feet/sec/sec)
of a rocket which starts from rest is a (t ) = 36t 2. How far does it travel during the first 5 seconds? A. 360 feet B. 1250 feet C. 1875 feet D. 1500 feet/sec E. 1525 feet
![Page 2: Clicker Question 1 Suppose the acceleration (in feet/sec/sec) of a rocket which starts from rest is a (t ) = 36t 2. How far does it travel during the first.](https://reader036.fdocuments.net/reader036/viewer/2022072006/56649f4a5503460f94c6c2bc/html5/thumbnails/2.jpg)
Clicker Question 2 A car travelling at a speed of 90 feet/sec
now decelerates at a constant rate of 30 ft/sec/sec. How far does it travel (from the time the deceleration begins) before it stops? A. 90 feet B. 120 feet C. 135 feet D. 196 feet E. 270 feet
![Page 3: Clicker Question 1 Suppose the acceleration (in feet/sec/sec) of a rocket which starts from rest is a (t ) = 36t 2. How far does it travel during the first.](https://reader036.fdocuments.net/reader036/viewer/2022072006/56649f4a5503460f94c6c2bc/html5/thumbnails/3.jpg)
Integration (3/28/12)
Question: How can we “add up” all the values of a function f (x ) on some interval [a, b ]?
This is called “integrating f (x ) on [a, b ]”
Doesn’t seem to make sense since most functions have infinitely many values on an interval. (Which ones don’t?)
![Page 4: Clicker Question 1 Suppose the acceleration (in feet/sec/sec) of a rocket which starts from rest is a (t ) = 36t 2. How far does it travel during the first.](https://reader036.fdocuments.net/reader036/viewer/2022072006/56649f4a5503460f94c6c2bc/html5/thumbnails/4.jpg)
Interpreting Integration as Area under the Curve
If we have a graph of f (x ), we can interpret “adding up all values” as finding the area under the graph on the interval [a, b ].
Just as slope of a curve is a graphical interpretation of the derivative, area under the curve is a graphical interpretation of the integral.
![Page 5: Clicker Question 1 Suppose the acceleration (in feet/sec/sec) of a rocket which starts from rest is a (t ) = 36t 2. How far does it travel during the first.](https://reader036.fdocuments.net/reader036/viewer/2022072006/56649f4a5503460f94c6c2bc/html5/thumbnails/5.jpg)
Finding Areas
Question: How can we compute the area under a given function on a given interval?
Answer: Not at all obvious!! An easy case: If f is linear. Example: What is the area under
f (x ) = x + 4 on [0, 3]?
![Page 6: Clicker Question 1 Suppose the acceleration (in feet/sec/sec) of a rocket which starts from rest is a (t ) = 36t 2. How far does it travel during the first.](https://reader036.fdocuments.net/reader036/viewer/2022072006/56649f4a5503460f94c6c2bc/html5/thumbnails/6.jpg)
But what if f is not linear?
What is the area under f (x ) = x 2 + 4 on [0, 3]?????
Let’s estimate it by using a single trapezoid.
It turns out the exact answer 21 sq. units. (We don’t know how to do this yet.)
Let’s estimate it using 3 trapezoids!
![Page 7: Clicker Question 1 Suppose the acceleration (in feet/sec/sec) of a rocket which starts from rest is a (t ) = 36t 2. How far does it travel during the first.](https://reader036.fdocuments.net/reader036/viewer/2022072006/56649f4a5503460f94c6c2bc/html5/thumbnails/7.jpg)
Assignment for Wednesday
Work on Hand-in #3 (due Thursday 4:45) Do the following 3 exercises on areas:
1. Find the exact area under f (x ) = 6 – x on the interval [0, 4].
2. Estimate (to two decimal places) the area under f (x ) = x on [0, 4] usinga. 1 trapezoid b. 2 trapezoids c. 4 trapezoids(Note: Exact answer is 5 1/3)
3. Estimate (to two decimal places) the area under f (x ) = sin(x ) on [0, ] usinga. 2 trapezoids b. 4 trapezoids (Exact is 2)