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Unformatted text preview: Math 10C Practice Exam 1 I won’t be posting solutions to this practice exam. If you have questions, make sure to ask at office hours or discussion section. 1. Consider the points in R3 : P = (4, 1, 3) Q = (5, 1, 1) R = (6, −1, 2) − → (a) Find P Q . − → (b) Find P R . − → (c) Find QR . (d) Which points are farthest apart? P and Q or P and R or Q and R 2. Consider the function of two variables f (x, y) defined as follows: z = f (x, y) = 1 x2 + y 2 (a) Find and graph the x = 0 cross section. hint: y 2 = |y|. (b) Find and graph the y = 0 cross section. (c) Make a contour diagram with z = 1, z = 1/2, and z = 1/3, level sets. (d) Graph f (x, y). 3. (a) Compute: (b) Compute: 3ı + k · − k 3ı + k × − k (c) If you locate v = 3ı + k so that it begins at the point (1, 4, −2), then at what point does v end? √ √ 4. Find the smallest positive angle between ı + 3 and ı − 3 . 5. Consider the plane described by the following equation. 3(x − 2) − (y − 4) − 2(z + 4) = 0 (a) Find a normal vector to the plane. (b) Find the x-slope of the plane. (c) Find the y-slope of the plane. (d) Find the z-intercept of the plane. 6. Find all vectors v = a, b with length √ 20 that are perpendicular to −1, 2 . 7. Find the linear function f (x, y) that has x = 1 cross section given by f (1, y) = −2y + 8 and y = 2 cross section given by f (x, 2) = 3x + 1 8. Suppose there are vectors v and w that satisfy v·v = 2 w · w = 18 √ v · w = −3 3 (a) Find cos(θ), where θ denote the smallest positive angle between v and w. hint: The rule u 2 = u · u holds for any vector u. (b) Use your answer to part (a) to find θ. 9. Match each description of points in R3 with its plot. (a) x = 0 and y = 0 (b) x = 0 and z = 0 (c) y = 0 (d) z = 0 ...
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