Due: Fri Sep :00 PM MDT Question

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1 Exam 1 Review ( ) Due: Fri Sep :00 PM MDT Question Description This is a collection of problems that can be used as an exam review. It is worth no credit. This review is not complete! It is only a collection of problems from the book. Problem types on the exam maybe problems from previous assignments, the notes and the quizzes.

2 1. Question Details SCalcET [ ] Consider the point. (3, 5, 6) What is the projection of the point on the xyplane? (x, y, z) = What is the projection of the point on the yzplane? (x, y, z) = What is the projection of the point on the xzplane? (x, y, z) = Draw a rectangular box with the origin and (3, 5, 6) as opposite vertices and with its faces parallel to the coordinate planes. Label all vertices of the box. Find the length of the diagonal of the box.

3 2. Question Details SCalcET [ ] Find an equation of the sphere that passes through the origin and whose center is (4, 4, 2).

4 3. Question Details SCalcET [ ] Find the sum of the given vectors. a = 1, 0, 3, b = 0, 7, 0 a + b = Illustrate geometrically.

5 4. Question Details SCalcET [ ] Find a + b, 9a + 7b, a, and a b. a = 9i 8j + 7k, b = 7i 9k a + b = 9a + 7b = a = a b = 5. Question Details SCalcET [ ] Find a unit vector that has the same direction as the given vector. 4i j + 8k 6. Question Details SCalcET [ ] Find the vector that has the same direction as 3, 2, 6 but has length Question Details SCalcET [ ] If v lies in the first quadrant and makes an angle π/3 with the positive xaxis and v = 4, find v in component form. v = 8. Question Details SCalcET [ ] Find a b. a = p, p, 4p, b = 3q, q, q 9. Question Details SCalcET [ ] Find a b. a = 20, b = 90, the angle between a and b is 3π/4.

6 10. Question Details SCalcET [ ] Find, correct to the nearest degree, the three angles of the triangle with the given vertices. A(1, 0, 1), B(2, 5, 0), C(1, 4, 4) CAB = ABC = BCA = 11. Question Details SCalcET [ ] Determine whether the given vectors are orthogonal, parallel, or neither. (a) a = 9, 6, b = 4, 6 orthogonal parallel neither (b) a = 4, 5, 2, b = 3, 1, 5 orthogonal parallel neither (c) a = 6i + 3j + 9k, orthogonal parallel neither b = 4i 2j 6k (d) a = 2i j + 2k, orthogonal parallel neither b = 5i + 6j 2k 12. Question Details SCalcET [ ] Find the cross product a b. a = 3i + 3j 3k, b = 3i 3j + 3k Verify that it is orthogonal to both a and b. (a b) a = (a b) b =

7 13. Question Details SCalcET [ ] If a = 2, 1, 5 and b = 3, 2, 1, find the following. a b = b a = 14. Question Details SCalcET [ ] Find u v and determine whether u v is directed into the screen or out of the screen. u v = u v is directed into the screen. u v is directed out of the screen. 15. Question Details SCalcET [ ] Find the area of the parallelogram with vertices A( 3, 0), B( 1, 3), C(5, 2), and D(3, 1). 16. Question Details SCalcET [ ] Consider the points below. P(1, 0, 1), Q( 2, 1, 3), R(5, 2, 5) (a) Find a nonzero vector orthogonal to the plane through the points P, Q, and R. (b) Find the area of the triangle PQR.

8 17. Question Details SCalcET [ ] Find a vector equation and parametric equations for the line. (Use the parameter t.) The line through the point (2, 2.2, 3.1) and parallel to the vector 4i + 2j k r(t) = (x(t), y(t), z(t)) = 18. Question Details SCalcET [ ] Find a vector equation and parametric equations for the line. (Use the parameter t.) The line through the point (1, 0, 8) and perpendicular to the plane x + 3y + z = 5 r(t) = (x(t), y(t), z(t)) = 19. Question Details SCalcET [ ] Find an equation of the plane. The plane through the point (8, 9, 3) and with normal vector 5i + j k 20. Question Details SCalcET [ ] Find an equation of the plane. The plane through the points (4, 1, 4), (5, 8, 6), and ( 4, 5, 1)

9 21. Question Details SCalcET [ ] Sketch the curve with the given vector equation. Indicate with an arrow the direction in which t increases. r(t) = sin(8t), t

10 22. Question Details SCalcET XP. [ ] Match the parametric equations with the graphs. (I) (II) (III) (IV) (V) (VI) (a) x = e t cos(7t), y = e t sin(7t), z = e t Select 1 (b) x = t, y =, z = t t 2 Select (c) Select x = cos(t), y = sin(t), z = sin(5t) (d) Select x = cos(9t), y = t, z = sin(9t) (e) Select x = cos(t), y = sin(t), z = ln(t) (f) Select x = t, y = t 2, z = e t

11 23. Question Details SCalcET XP. [ ] Consider the given vector equation. r(t) = 3e t i + 2e t j (a) Find r'(t). r'(t) = (b) Sketch the plane curve together with the position vector r(t) and the tangent vector r'(t) for the given value of t = 0.

12 24. Question Details SCalcET XP. [ ] Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. x = t, y = t 5 t, z = t 5 + t; (11, 0, 2) x(t), y(t), z(t) =

13 25. Question Details SCalcET [ ] Find the velocity, acceleration, and speed of a particle with the given position function. r(t) = 7 cos(t)i + 6 sin(t)j v(t) = a(t) = v(t) = Sketch the path of the particle and draw the velocity and acceleration vectors for t =. π 3

14 26. Question Details SCalcET [ ] (a) Find the position vector of a particle that has the given acceleration and the specified initial velocity and position. a(t) = 11t i + e t j + e t k, v(0) = k, r(0) = j + k r(t) = (b) On your own using a computer, graph the path of the particle. 27. Question Details SCalcET [ ] A projectile is fired with an initial speed of 250 m/s and angle of elevation 60. The projectile is fired from a position 90 m above the ground. (Recall g 9.8 m/s 2. Round your answers to the nearest whole number.) (a) Find the range of the projectile. m (b) Find the maximum height reached. m (c) Find the speed at impact. m/s 28. Question Details SCalcET MI. [ ] A ball is thrown at an angle of 45 to the ground. If the ball lands 90 m away, what was the initial speed of the ball? (Round your answer to the nearest whole number. Use g 9.8 m/s 2.) v 0 = m/s 29. Question Details SCalcET MI. [ ] Find the length of the curve. r(t) = 9t, 3 cos(t), 3 sin(t), 4 t 4

15 30. Question Details SCalcET [ ] Match the function with its graph (labeled IVI). I II III IV V VI (a) f(x, y) =? x 2 + y 2 (b) f(x, y) =? x 2 y 2 (c) f(x, y) = ln(x 2 + y 2 )? (d) f(x, y) = cos x 2 + y 2? (e)? f(x, y) = xy (f)? f(x, y) = cos(xy)

16 31. Question Details SCalcET [ ] Consider the function below. z = sin(xy) (a) Match the function with its graph (labeled AF). A B C D E F (b) Match the function with its contour map (labeled IVI).

17 I II III IV V VI Give reasons for your choices. This function is Select in both x and y, and the function is Select when x is interchanged with y, so its graph is Select about the plane y =. In addition, the function is Select along the x and yaxes. These conditions are satisfied only by? and?. 32. Question Details SCalcET [ ] Find all the second partial derivatives. f(x, y) = x 4 y 3x 5 y 2 f xx (x, y) = f xy (x, y) = f yx (x, y) = f yy (x, y) = 33. Question Details SCalcET [ ] Find the indicated partial derivative. f xyz (x, y, z) =

18 34. Question Details SCalcET [ ] Determine the signs of the partial derivatives for the function f whose graph is shown below. (a) f xx ( x 0, y 0 ) positive negative (b) f yy ( x 0, y 0 ) positive negative 35. Question Details SCalcET [ ] Find the first partial derivatives of the function. z = x sin(xy) z x z y = = Assignment Details Name (AID): Exam 1 Review ( ) Submissions Allowed: 10 Category: Homework Code: Locked: Yes Author: Skriletz, Jaimos ( jaimosskriletz@boisestate.edu ) Last Saved: Sep 14, :09 PM MDT Permission: Protected Randomization: Person Feedback Settings Before due date Question Score Assignment Score Publish Essay Scores Question Part Score Mark Help/Hints Response

19 Which graded: Last Save Work After due date Question Score Assignment Score Publish Essay Scores Key Question Part Score Solution Mark Add Practice Button Help/Hints Response