MATH 126 Study Guide - Midterm Guide: Electrical Network
MATH 126 FINAL EXAM December 17, 2012
INSTRUCTIONS
Answer all the questions. You must show your work to obtain full credit. Points may
be deducted if you do not justify your final answer. Please indicate clearly whenever you
continue your work on the back of the page. Calculators and other electronic devices are
NOT allowed. Books and lecture notes are NOT allowed. Turn off your cell phones.
You can use one self-prepared handwritten formula sheet (one piece of letter size paper,
both sides may be used).
1. [20 points] Determine whether or not the limit exists. If the limit exists find it, and
indicate clearly how you obtained your answer. If the limit does not exist give reasons why.
(i) lim
x→3
x5−243
x3−27 ,(ii) lim
x→∞ (xe1/x −x),(iii) lim
x→0+(cos x)1/x2.
2. [24 points] Evaluate the integrals.
(i) ∫x2cos(πx)dx, (ii) ∫5x−3
x3+ 3x2dx, (iii) ∫x2
(1 + x2)3/2dx.
3. [16 points] Determine whether the integral is convergent or divergent.
(i) ∫∞
1
2 + sin 3x
√1 + x+x3dx, (ii) ∫2
0
x
1−x2dx.
4. [20 points] Consider the region Rbetween the line y= 3xand the curve y=x2−2x.
(i) Sketch the region Rand find its area.
(ii) Set up but do not evaluate an integral giving the volume of the solid obtained by
rotating the region Rabout the y-axis.
(iii) Set up but do not evaluate an integral giving the length of the curved part of the
boundary of R.
Document Summary
You must show your work to obtain full credit. Points may be deducted if you do not justify your nal answer. Please indicate clearly whenever you continue your work on the back of the page. You can use one self-prepared handwritten formula sheet (one piece of letter size paper, both sides may be used). [20 points] determine whether or not the limit exists. If the limit exists nd it, and indicate clearly how you obtained your answer. 5x 3 x3 + 3x2 dx, (iii) x2 (1 + x2)3/2 dx: [16 points] determine whether the integral is convergent or divergent. (i) . 1 + x + x3 dx, (ii) 2. R = 3 meters is full of water. Find the work required to pump all of the water out over the rim of the tank. [you may leave your answer in terms of the density of water kg/m3 and the acceleration due to gravity g m/s2. ]
