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exam1pracA

# exam1pracA - R in the plane bounded between the curves y =...

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Math 42 Practice Exam 1 2007 Notes: Skip problems 1(e), 2, 6(b), and 6(e); See other review resources for problems on “Work” (not covered below). 1. Evaluate each of the following integrals. (a) r r 2 - 4 dr (b) 1 r 2 - 4 dr (c) e 1 x ln x dx (d) sin 4 x dx (e) 1 0 1 5 x dx (f) π/ 2 π/ 4 sin 2 θ cos θ dθ (g) 1 0 1 ( x 2 + 9) 3 / 2 dx (h) x 3 x 3 - x 2 - x + 1 dx = x 3 ( x + 1)( x - 1) 2 dx (i) 4 1 ( at + b ) t dt ( a and b are constants.) 2. Determine whether the improper integral 3 ln x x dx converges or diverges. 3. The following chart gives the rate of poplulation growth (i.e., the number of births minus the number of deaths per year) of a certain small town in the given years. Year: 1980 1985 1990 1995 2000 Rate of growth: 1 - 1 0 1 2 Use Simpson’s Rule to estimate how much the population grew from 1980 to 2000. 4. Consider the integral 1 0 sin( x 2 ) dx . (a) Estimate the error made in approximating the value of this integral using n = 5 trapezoids. (b) How many trapezoids would be necessary to guarantee an error of at most 1 200 ?

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5. Consider the region
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Unformatted text preview: R in the plane bounded between the curves y = x and y = x 2 . (a) Find the area of R . (b) Find the volume of the solid obtained by revolving R about the x-axis. (c) Find the volume of the solid obtained by revolving R about the line x = 1. 6. True / False: (You do not need to justify your answer.) (a) Simpson’s Rule usually, but not always, gives a more accurate approximation for a deﬁnite integral than both the Midpoint Rule and the Trapezoid Rule. (b) If f ( x ) ≤ g ( x ) and ± ∞ g ( x ) dx diverges, then ± ∞ f ( x ) dx also diverges. (c) ± sin 2 x dx = 1 3 sin 3 x + C . (d) It is possible to antidiﬀerentiate every rational function in terms of ﬁnitely many familiar functions. (e) If ± ∞ a f ( x ) dx and ± ∞ a g ( x ) dx are both divergent, then ± ∞ a [ f ( x ) + g ( x )] dx is also divergent....
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exam1pracA - R in the plane bounded between the curves y =...

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