(a) Vectors a, b and c are given as a = (2,−1, 1), b = (−1, 0, 2) and c = (0, 1, 0).
i. Find (a + b) · (b + c).
ii. Determine which of the vectors a, b and c are perpendicular to each other, if at
iii. Find the unit vector in the direction of a + 2b.
iv. Determine a × c.
(b) Use the scalar triple product to show that the points A(1, 0,−1), B(2, 1, 2), C(0, 1, 0)
and D(1, 2/3, 1/3) are coplanar, i.e., they belong to the same plane.
(c) Find the volume of the tetrahedron (or pyramid) with vertices O(0, 0, 0), A(2, 2, −1),
B(2, 0, 2), and C(0, −1, 1).
2. Consider the function
f(x) = |x + 1| + |x| .
(a) Find f′(x) at all points that it exists. Conclude on which intervals f is strictly
increasing or decreasing.
(b) Find all critical points.
(c) Find the y- and x-intercepts (if there are any).
(d) Locate and report all local minima and maxima (if they exist).
(e) Using the information obtained in parts (a)-(d), sketch the graph of f.
3. (a) Find one real solution of the equation
4c3 + 16c2 + 13c − 33 = 0
and then show there are no other real solutions.
(b) Demonstrate your understanding of the Mean Value Theorem by
i. checking its hypotheses (i.e., assumptions) and
ii. finding a point c,
corresponding to f(x) = x2 − 12√x + 3 on the interval [−3, 6].
4. Evaluate the following limits by first recognizing the indicated sum as a Riemann sum
associated with a regular partition of [0, 1] and then evaluating the corresponding integral.
√4 1 + √4 2 + √4 3 + · · · + √4 n
n √4 n
5. An object initially at the origin moves away with velocity
(t2 + 16)3/2 m/sec, t≥ 0.
Clearly the velocity is always positive and so the object moves to the right.