MATH 21A Midterm: MATH 21A Harvard 21a Fall 17Practice15
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Tth 11:30 sebastian vasey: start by printing your name in the above box and check your section in the box to the left, do not detach pages from this exam packet or unstaple the packet, please write neatly. Answers which are illeg- ible for the grader cannot be given credit: show your work. Except for problems 1-3, we need to see details of your computation: no notes, books, calculators, computers, or other electronic aids can be allowed, you have 180 minutes time to complete your work. The angle between the vectors h1, 2, 3i and h2, 1, 1i is . 6 . h4, 6, 8i is a normal vector for the plane 2x 3y 4z = 5. The plane tangent to the graph of f (x, y) = x2 + y2 at (3, 4, 25) is 6x + 8y = The directional derivative of a function f in the direction of f can never be negative.
