Download Spring 1999 Math 126 Exam Series by Prof. Brick and more Exams Calculus in PDF only on Docsity!
Do the problems in order in your bluebook. Show your work.
- Find
6 x + 9 x^2 (x + 3)
dx.
- Determine whether or not
π^3
x^5 + 3 x^6 + 7x^2 + 4 01
dx converges.
(Work out any integral you use in your comparison.)
- Find
∫ (^) e
1
ln(x^3 ) x
dx.
- Using integrals and Riemann sums, find (^) n→lim+∞
( (^) n ∑
i=
i^2 n^3
- Find
0
xe−x^ dx.
- Find the area between the curves y = 4x and y = x^3.
- Find
1
x^2 + 1
dx.
Do the problems in order in your bluebook. Show your work.
- Find the volume of the solid obtained by rotating about the x-axis the region bounded by the curves y = x + ex, the x-axis, x = 0 and x = 1.
- Set up (but don’t evaluate) an integral that gives the surface area obtained by rotating the curve y = tan(x) from x = 0 to x = π about the x-axis.
- Find the average value of y = ln^2 (x)/x over the interval [e, e^2 ].
- A heavy rope, 50 ft long, weighs 0.5 lb/ft and hangs over the edge of a tall building (taller than the length of the rope). How much work is done in pulling the rope to the top of the building?
- Find the centroid of the quarter unit-circle y =
1 − x^2 for 0 ≤ x ≤ 1. (You may use symmetry and the formula for the area of a circle.)
- Find the mean of the probability density given by f (x) = 2 √^12 π e−x
2 for 0 ≤ x < +∞.
- Solve the differential equation y′^ =
ey 1 + x^2
- A tank contains 10 lbs of salt dissolved in 100 gallons of water. A fluid containing a salt solution with concentration of 102 lbs per gallon enters the tank at a rate of 3 gallons per minute. A well-stirred mixtures leaves the tank at the same rate. Set up (but do not solve) an initial value problem that describes the amount of salt in the tank as a function of time.
Spring 99 Math 126 Final Exam Prof. Brick
Do the problems in order in your bluebook. Show your work.
- Find the following:
(a)
dt t^2 − 1
(b)
x^3 ex
2 dx
- Determine whether or not the following converge (justify your argument):
(a)
3
x(1 + e−x) x^2 + 2π
dx (b)
5
x^2 +
8 + sin^2 (x^6 + 1)
x^4 + 1
dx
- Find the area between the curves y = 2x^2 + 4x + 1 and y = x^3 + x + 1.
- Find the volume obtained by rotating about the x-axis the region bounded by y = ex, the x-axis, x = 1 and x = 2.
- Find the average value of y = tan^3 (x) sec^2 (x) over the interval [0, π/4]. Draw a graph that shows the geometric significance of the average value.
- A cylindrical water tank has height 20 feet and radius 6 feet. If the tank is half full of water, find the work required to pump all the water over the top rim. (Note: a cubic foot of water weighs 62.4pounds.)
- Solve the differential equation y′^ − x^2 y^2 = x^2
- Determine whether or not the series
n=
n en^2
converges
- Find the MacLaurin series for y =
3 x^3 7 − x
. For what x does it converge?
- Find the equation of the plane that contains the points (1, − 1 , 2), (1, 0 , 1) and (0, 2 , 2).
