Math 223 Final Exam Review Packet Fall 2012: The Following Q

Math 223 Final Exam Review Packetfall 2012the Following Questions C

The following questions can be used as a review for Math 223. These questions are not actual samples of questions that will appear on the final exam, but they will provide additional practice for the material that will be covered on the final exam. When solving these problems keep the following in mind: Full credit for correct answers will only be awarded if all work is shown. Exact values must be given unless an approximation is required. Credit will not be given for an approximation when an exact value can be found by techniques covered in the course.

The answers, along with comments, are posted as a separate file on 1. A sonic boom carpet is a region on the ground where the sonic boom is heard directly from the airplane and not as a reflection. The width of the carpet, W, can be expressed as a function of the air temperature on the ground directly below the airplane, t, and the vertical temperature gradient at the airplane’s altitude, d. Suppose ( , ) t W t d k d = for some positive constant k. (a) If d is fixed, is the width of the carpet an increasing or decreasing function of t. (b) If t is fixed, is the width of the carpet an increasing or decreasing function of d. 2.

Describe the following sets of points in words, write an equation, and sketch a graph: (a) The set of points whose distance from the line L is five. The line L is the intersection of the plane 3y = and the xy-plane. (b) The set of points whose distance from the yz-plane is three. (c) The set of points whose distance from the z-axis and the xy-plane are equal. 3. By setting one variable constant, find a plane that intersects 2 2cos 3y x z+ = in a: (a) parabola (b) waves (related to cosine curves) (c) line(s) 4. Consider the function 2( , )f x y y x= − . (a) Plot the level curves of the function for 2, 1, 0,1, 2z = − − . (b) Imagine the surface whose height above any point ( , )x y is given by ( , )f x y .

Suppose you are standing on the surface at the point where 1, 2x y= = . (i) What is your height? (ii) If you start to move on the surface parallel to the y-axis in the direction of increasing y, does your height increase or decrease? (iii) Does your height increase or decrease if you start to move on the surface parallel to the x- axis in the direction of increasing x? 5. Describe the level surfaces of each: (a) 2 2( , , )f x y z x y z= − − (b) 2 2 21( , , ) x y zg x y z e − − −= 2.5 3.0 3..... The figure at the right shows the level curves of the temperature T in degrees Celsius as a function of t hours and depth h in centimeters beneath the surface of the ground from midnight ( 0t = ) one day to midnight ( 24t = ) the next. (a) Approximately what time did the sunrise?

When do you think the sun is directly overhead? (b) Sketch a graph of the temperature as a function of time at 20 centimeters. (c) Sketch a graph of the temperature as a function of the depth at noon. from S. J. Williamson, Fundamentals of Air Pollution, (Reading: Addison-Wesley, . Given the table of some values of a linear function, complete the table and find a formula for the function. 8.

Consider the planes: I. 3 5 2x y z− − = II. 5 3x y= + III. 5 3 2x y+ = IV. 3 5 2x y+ = V.

3 5 2x y z+ + = VI. 1 0y + = List all of the planes which: (a) Are parallel to the z-axis. (b) Are parallel to 3 5 7x y z= + + . (c) Contain the point (1, 1, 6)− . (d) Are normal to ( ) ( )2 3 3i k i k+ à— âˆ’ ï¶ ï¶ ï¶ ï¶ . (e) Could be the tangent plane to a surface ( , )z f x y= , where f is some function which has finite partial derivatives everywhere. 9. A portion of the graph of a linear function is shown. (a) Find an equation for the linear function. (b) Find a vector perpendicular to the plane. (c) Find the area of the shaded triangular region. x -.111 0.167 0.200 0.167 0..167 0.333 0.500 0.333 0.167 y 0 0.200 0.500 1.000 0.500 0..167 0.333 0.500 0.333 0..111 0.167 0.200 0.167 0.111 x -.00 -3.00 -4.00 -3.00 0.00 1 3.00 0.00 -1.00 0.00 3.00 y 0 4.00 1.00 0.00 1.00 4..00 0.00 -1.00 0.00 3..00 -3.00 -4.00 -3.00 0.00 x y x y x y x -.828 2.236 2.000 2.236 2..236 1.414 1.000 1.414 2.236 y 0 2.000 1.000 0.000 1.000 2..236 1.414 1.000 1.414 2..828 2.236 2.000 2.236 2..

Match each of the following functions (a) – (f), given by a formula, to the corresponding tables, graphs, and/or contour diagrams (i) – (ix). There may be more than one representation or no representations for a formula. (a) 2 2( , )f x y x y= − (b) ( , ) 6 2 3f x y x y= − + (c) 2 2( , ) 1f x y x y= − − (d) 2 2 1 ( , ) 1 f x y x y = + + (e) ( , ) 6 2 3f x y x y= − − (f) 2 2( , )f x y x y= + (i) Table 1 (ii) Table 2 (iii) Table 3 (iv) (v) (vi) (vii) (viii) (ix) 11. Let 3 2 2v i j k= + − ï¶ ï¶ ï¶ ï¶ and 4 3w i j k= − + ï¶ ï¶ ï¶ ï¶ . Find each of the following: (a) A vector of length 5 parallel to wï¶ . (b) A vector perpendicular to vï¶ but not perpendicular to wï¶ . (c) The angle between vï¶ and wï¶ . (d) The component of vï¶ in the direction of wï¶ . (e) A vector perpendicular to both vï¶ and wï¶ .

12. Consider the vectors 2 3u i j k= − + ï¶ ï¶ ï¶ ï¶ and 2v ai aj k= − + − ï¶ ï¶ ï¶ ï¶ . (a) For what value(s) of a are uï¶ and vï¶ perpendicular? (b) For what value(s) of a are uï¶ and vï¶ parallel? (c) Find an equation of the plane normal to uï¶ and containing the point (1, 2, 3)− . (d) Find a parameterization for the line parallel to uï¶ and containing the point (1, 2, 3)− . 13. Let 3 2u j k= − ï¶ ï¶ ï¶ . If vï¶ is a vector of length 12 in the yz-plane such that the angle between uï¶ and vï¶ is 3 Ï€ , find u và—ï¶ ï¶ .

14. Find the following: (a) ( ) ( )( )3 2 2ln 3 arctanx y x y x ∂ + − + ∂ (b) Hf if ( )3 2 ( , ) 5 H T f H T H + = − (c) 2 x y x y y x  ∂ + ∂ ∂ ï£ ï£¸ 15. Find an equation for the tangent plane to: (a) ( , ) xyf x y = at ( , ) (1, 2)x y = (b) x y z− + − + − = at (3, 3, . A ball is thrown from ground level with initial speed v (m/sec) and at an angle of α with the horizontal. It hits the ground at a distance 2 sin(2 ) ( , ) v s v g α α = where 10 g ≈ m/sec2. (a) Find the differential ds . (b) What does the sign of (20, 3)sα Ï€ tell you? (c) Use the linearization of s about (20, 3)Ï€ to estimate the change in α that is needed to get approximately the same distance if the initial speed changes to 19 m/sec.

17. The depth of a lake at the point ( , )x y is given by 2 2( , ) 2 3h x y x y= + feet. A boat is at (-1,2). (a) If the boat sails in the direction of the point (3, 3) , is the water getting deeper or shallower? (b) In which direction should the boat sail for the depth to remain constant? Give your answer as a vector. (c) If the boat moves on the curve ( ) ( )r t t i t j= − + + ï¶ ï¶ ï¶ for t in minutes, at what rate is the depth changing when 2t = ? 18.

Calculate the following: (a) yz grad x    +ï£ ï£¸ (b) ( ) ( ) ( )( )2 2 2curl x y z i y z j xz k+ + − + + ï¶ ï¶ ï¶ (c) ( ) ( ) ( )( )2 3cos sec zdiv x i x y j e k+ +  ∂ + ∂ ∂ ï£ ï£¸ 15. Find an equation for the tangent plane to: (a) ( , ) xyf x y = at ( , ) (1, 2)x y = (b) x y z− + − + − = at (3, 3, . A ball is thrown from ground level with initial speed v (m/sec) and at an angle of α with the horizontal. It hits the ground at a distance 2 sin(2 ) ( , ) v s v g α α = where 10 g ≈ m/sec2. (a) Find the differential ds . (b) What does the sign of (20, 3)sα Ï€ tell you? (c) Use the linearization of s about (20, 3)Ï€ to estimate the change in α that is needed to get approximately the same distance if the initial speed changes to 19 m/sec.

17. The depth of a lake at the point ( , )x y is given by 2 2( , ) 2 3h x y x y= + feet. A boat is at (-1,2). (a) If the boat sails in the direction of the point (3, 3) , is the water getting deeper or shallower? (b) In which direction should the boat sail for the depth to remain constant? Give your answer as a vector. (c) If the boat moves on the curve ( ) ( )r t t i t j= − + + ï¶ ï¶ ï¶ for t in minutes, at what rate is the depth changing when 2t = ? 18.

Calculate the following: (a) yz grad x    +ï£ ï£¸ (b) ( ) ( ) ( )( )2 2 2curl x y z i y z j xz k+ + − + + ï¶ ï¶ ï¶ (c) ( ) ( ) ( )( )2 3cos sec zdiv x i x y j e k+ +  ∂ + ∂ ∂ ï£ ï£¸ 15. Find an equation for the tangent plane to: (a) ( , ) xyf x y = at ( , ) (1, 2)x y = (b) x y z− + − + − = at (3, 3, . A ball is thrown from ground level with initial speed v (m/sec) and at an angle of α with the horizontal. It hits the ground at a distance 2 sin(2 ) ( , ) v s v g α α = where 10 g ≈ m/sec2. (a) Find the differential ds . (b) What does the sign of (20, 3)sα Ï€ tell you? (c) Use the linearization of s about (20, 3)Ï€ to estimate the change in α that is needed to get approximately the same distance if the initial speed changes to 19 m/sec.

17. The depth of a lake at the point ( , )x y is given by 2 2( , ) 2 3h x y x y= + feet. A boat is at (-1,2). (a) If the boat sails in the direction of the point (3, 3) , is the water getting deeper or shallower? (b) In which direction should the boat sail for the depth to remain constant? Give your answer as a vector. (c) If the boat moves on the curve ( ) ( )r t t i t j= − + + ï¶ ï¶ ï¶ for t in minutes, at what rate is the depth changing when 2t = ? 18.

Calculate the following: (a) yz grad x    +ï£ ï£¸ (b) ( ) ( ) ( )( )2 2 2curl x y z i y z j xz k+ + − + + ï¶ ï¶ ï¶ (c) ( ) ( ) ( )( )2 3cos sec zdiv x i x y j e k+ +  ∂ + ∂ ∂ ï£ ï£¸ 15. Find an equation for the tangent plane to: (a) ( , ) xyf x y = at ( , ) (1, 2)x y = (b) x y z− + − + − = at (3, 3, . A ball is thrown from ground level with initial speed v (m/sec) and at an angle of α with the horizontal. It hits the ground at a distance 2 sin(2 ) ( , ) v s v g α α = where 10 g ≈ m/sec2. (a) Find the differential ds . (b) What does the sign of (20, 3)sα Ï€ tell you? (c) Use the linearization of s about (20, 3)Ï€ to estimate the change in α that is needed to get approximately the same distance if the initial speed changes to 19 m/sec.

17. The depth of a lake at the point ( , )x y is given by 2 2( , ) 2 3h x y x y= + feet. A boat is at (-1,2). (a) If the boat sails in the direction of the point (3, 3) , is the water getting deeper or shallower? (b) In which direction should the boat sail for the depth to remain constant? Give your answer as a vector. (c) If the boat moves on the curve ( ) ( )r t t i t j= − + + ï¶ ï¶ ï¶ for t in minutes, at what rate is the depth changing when 2t = ? 18.

Calculate the following: (a) yz grad x    +ï£ ï£¸ (b) ( ) ( ) ( )( )2 2 2curl x y z i y z j xz k+ + − + + ï¶ ï¶ ï¶ (c) ( ) ( ) ( )( )2 3cos sec zdiv x i x y j e k+ +  ∂ + ∂ ∂ ï£ ï£¸ 15. Find an equation for the tangent plane to: (a) ( , ) xyf x y = at ( , ) (1, 2)x y = (b) x y z− + − + − = at (3, 3, . A ball is thrown from ground level with initial speed v (m/sec) and at an angle of α with the horizontal. It hits the ground at a distance