Final of Calculus 1/15/2007
1. Show your work to get credits!
2. No Calculator!
3. No Cheating!
(1) Find the derivatives (a) f (x) = ln(x√
x2+ 1), (b) f (x) = 3x2+x. (2) Find the indefinite integral
(a) ∫ (√
x +√1 x
)
dx (b) ∫
(2x + 1)5dx (c)∫ 1
x ln xdx (d) ∫ x
√3x+1dx.
(3) Find the definite integral (a) ∫1
0 xex2dx (b) ∫2 1
1+ln x x dx.
(4) Evaluate ∫3
0 |x − 1|dx
(5) Find the area of the region bounded by the two graphs of func- tions f (x) = (x− 1)3 and g(x) = x− 1.
(6) Find the consumer and producer surpluses if the demand func- tion is given by p1(x) = 100− x2 and the supply function is given by p2(x) = 70 + x.
(7) The upper half of the ellipse
16x2+ 25y2 = 400
is revolved about the x-axis to form a football like spheroid.
Find the volume of the spheroid.
(8) The probability of recall in an experiment is found to be
P (a≤ x ≤ b) =
∫ b
a
105 16 x2√
1− xdx,
where x represents the percent of recall. (0≤ x ≤ 1)
Find the probablity that a randomly chosen individual will re- call 80% of the material.
(9) Sketch the graph of the function
f (x) = x3 x3− 1.
Find the intercepts, relative extrema, points of inflection, and asymptotes if they exist.
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