MATH 21A Midterm: MATH 21A Harvard 21a Spring 09Midterm1 Spr07

• School: Harvard University
• Department: Mathematics
• Course: MATH 21A
• Professor: All

Document Summary

You do not need to justify your answers in this part of the exam. 1 the vectors a = 2i + (t 1)j + 2k and b = j + tk are parallel when: (a) t = 0; (b) t = 1. 3 ; (c) t = 1 and t = 2; (d) all values of t; (e) no values of t. 2 t sin ti, 0 t , has length. 2 (cid:16)e (b) (c) q 2 (d) q 3 (e) given by a de nite integral which cannot be evaluated explicitly; (f) given by an inde nite integral. The questions in problems 4-5 refer to the marked points in the diagram below, which shows the level curves of a function f (x, y). 4 at which of the labeled points is the partial derivative f. 5 at the point y , what can one say about the values of f.