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Published byGregory Reed Modified over 8 years ago
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Clicker Question 1 Suppose the acceleration (in feet/second) 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
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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
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Integration (4/1/11) 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?)
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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.
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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]?
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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!
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Assignment for Monday Work on Hand-in #3 (due Tuesday 4:30) 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] using a. 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, ] using a. 2 trapezoids b. 4 trapezoids (Exact is 2)
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