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Question #1: The base of a solid is the region in the first and second quadrants bounded by the graph of y = 1 – x2 and the x-axis. If cross-sections of the solid perpendicular to the x-axis are squares, what is the volume of the solid?
(a) 1.333
(b) 1.269
(c) 1.066
(d) .933
(e) 1.121-
Solution: The points of intersection between the x axis and
y = 1 – x2 are given by the equation 1 – x2 = 0.
x1 = -1 and x2 = 1
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Solution: The points of intersection between the x axis and
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Question #2:
t(hours) 0 .5 1 1.5 2 2.5 3 v(miles/hour) 32 30 16 22 20 24 26
A speedboat travels on a river. Its speed v, in miles per hour, at certain times is given in the table above. Using a left Riemann sum, what is the approximation of the total distance traveled by the speedboat from t = .5 to t = 3?(a) 85
(b) 56
(c) 86
(d) 78
(e) 66-
Answer:
The left Riemann sum is:
d = 30(1-.5) + 16(1.5-1) + 22(2-1.5) + 20(2.5-2) + 24(3-2.5).
d = 56
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Answer:
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Question #3:
At what value of x are the the tangent lines to the graphs of f(x) = ln(x) and g(x) = 6x parallel?(a) -1
(b) .5
(c) 1.2
(d) .32
(e) .43-
Answer:
If the tangent lines are parallel,
Using the graphing calculator, x = .32.
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Question #4: What is the average value of the function f(x) = sin2(3x) + x on the interval [0, ¶]?
(a) 2.07
(b) 1.05
(c) 3.3
(d) 1.23
(e) 1.9
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Answer:
The average value of a function f from x = a to x = b is the integral:
In our case,
Question #5: The base of a solid is the region in the first quadrant bounded by the graph of y = -x2 + 5x – 4 and the x-axis. If cross-sections of the solid perpendicular to the x-axis are equilateral triangles, what is the volume of the solid?
(a) 1.871
(b) 2.32
(c) 1.555
(d) 3.507
(e) 2
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Answer:
We need to find the area of an equilateral triangle function of its side, y.
The altitude of the triangle is y(√3)/2 so the area of the triangle is:
The volume of the solid is:
The limits of the integral are determined by the roots of the equation -x2 + 5x – 4 = 0