MATH 101 SAMPLE MIDTERM 1
For full credit, please show all work. Time: 50 minutes.
Books, notes, calculators, or other aids are not allowed.
√
1. (6 marks) Write the upper and lower Riemann sum for the function f (x) = x on the
interval [1, 3] corresponding to the partition of [1, 3] into n intervals of equal length. (Do
not evaluate the sums!)
2. (6 marks) Let
(
f (x) =
4x − 3, 0 ≤ x ≤ 2
9 − x2 , 2 ≤ x ≤ 3.
Find the average value of f on the interval [0, 3].
3. (12 marks) Evaluate the following integrals:
Z 2
dx
(a)
,
2
1 x(1 + ln x)
Z
(b) cos(2x) cos x dx.
4. (8 marks) Find the area of the finite plane region bounded by the curve y = xex and
the line y = ex.
Z
2x−x2
5. (8 marks) Let F (x) =
√
2 + sin( 100 − t) dt for 0 ≤ x ≤ 4. Find the x where F
0
attains its maximum value on the interval [0, 4].
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