Math 1210-1 Final Practice 3
1. Evaluate
4
X
1
i=2
2. Simplify
n
X
i
.
(2i + 1)(3i − 1) such that there is no sigma notation in your result.
i=1
3. Suppose
6
X
i=2
Z
4. Suppose
ai = 3 and
6
X
bi = 6. Evaluate
i=2
f (x) dx = 3 and
2
(2ai − bi + 1).
i=2
Z
6
6
X
Z
6
g(x) dx = 6. Evaluate
2
6
(2f (x) − g(x) + 1) dx.
2
5. Suppose f (x) = x5 − 4x3 − 3x + 1. Evaluate f (2) and f (3). What can you conclude about the
solution(s) of f (x) = 0.
√
3
6. Suppose f (x) = x2 defined on the interval [−1, 27/8]. Find the average slope of f (x) on [−1, 27/8].
Find all real numbers c such that f 0 (c) equals to the average slope. Why there is no c inside the
interval [−1, 27/8]?
√
3
7. Suppose f (x) = x2 defined on the interval [−1, 8]. Find the average value of f (x) on [−1, 8]. Find
a real number c between −1 and 8 such that f (c) equals to the average value.