MA271 SAMPLE PROBLEMS FOR FINAL EXAM
SET 1
1. Find the series’ radius and interval of convergence. For what values of xdoes
the series converge absolutely; conditionally?
∞
X
n=1
nxn
4n(n2+ 1)
2. Estimate the error of approximation of ln(1 + x) by its third order Taylor poly-
nomial in the interval |x|<0.1. Use Remainder Estimation Theorem.
3. Solve the initial value problem
y0−y=−x, y(0) = 2,
using power series.
4. Find the distance from the point (0,4,1) to the line in the parametric form
x= 2 + t, y = 2 + t, z =t
5. Let r(t) be the position of a moving particle at time t. Write the acceleration a
in the form aTT+aNNat t= 0 without finding Tand N.
r(t) = (2 + 3t+ 3t2)i+ (4t+ 4t2)j−(6 cos t)k
6. Find the length of the curve
r(t) = (3 cos t)i+ (3 sin t)j+ 2t3/2k,0≤t≤3
7. Find the extreme values of f(x, y) = x2+y2−3x−xy on the disk x2+y2≤9.
Hint. First find if the function has critical points in x2+y2<9, then use the
Lagrange multipliers method to find the critical points on x2+y2= 9.
8. Find the parametric equation for the line tangent to the curve of intersection of
the surfaces at the given point.
Surfaces: xyz = 1, x2+ 2y2+ 3z2= 6
Point: (1,1,1)
9. Sketch the region of integration, reverse the order of integration and evaluate
the integral
Z8
0Z2
3
√x
dy dx
y4+ 1.
10. Find the volume of the solid that is bounded above by the cylinder z= 4 −x2,
on the sides by the cylinder x2+y2= 4 and below by the xy-plane.
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