(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 all. 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). 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. (a) lim n−→∞ √4 1 + √4 2 + √4 3 + · · · + √4 n n √4 n (b) lim n−→∞ !n i=1 1 n sin 5πi n 5. An object initially at the origin moves away with velocity v(t) = 120 t (t2 + 16)3/2 m/sec, t≥ 0. Clearly the velocity is always positive and so the object moves to the right.