Review Sheet for MATH 223 at UA
Review Sheet for MATH 223 at UA
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This 4 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Arizona taught by a professor in Fall. Since its upload, it has received 17 views.
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Date Created: 02/06/15
Math 223 Review 1 CHAPTER 12 Are the statements below TRUE or FALSE Eccplain lfthe statement is false make corrections 1 The plane 7x 2y 7 z 2 passes through the origin FALSE because the equation is satis ed with z 0y 07 z 0 2 The point 003 is the lowest point on the graph of z 3 x2y2 TR UE this can be seen from the graph orand contour diagram 3 The cross sections of a linear function are always lines TR UE we have seen this in class 4 Two contours with different heights can itersect FALSE we also pointed this out in class 5 A function Fpyz cannot have a level surface that is a sphere and a level surface that is a cylinder FALSE the level sets may not be of the same type Solve the following problems 1 Draw a contour diagram for the functions 2 Mp2 y2 and z x2 yz One is a cone the other one is an elliptical paraboloid How can you tell which one is which just from the contours This is straightforward the contours of the cone are evenly spaced 2 Represent the plane with intercepts z 27 y 37 z 4 as a graph of a function fpy and as the level set of a function Fpyz c The intercepts give us that P 200Q 030R 004 Now you can use the any method that we have studied or guess that 6x 4y 32 12 works Solve for z to nd the formula for fzy Also Fpyz 6x 4y 32 with c 12 3 Problem 17 and 19 from section 124 There are answers in the back of the teptbooh for these two 4 Find the level sets of the function Gzy7 z p2 22 They are cylinders along the y accis 1 2 CHAPTER 13 Are the statements below TRUE or FALSE Emplam lfthe statement is false make corrections 17 gtlt u cost9 FALSE H 2 13 17 is equal to the magnitude of 17 in the direction of 12 TRUE 3 H17 and 13 are any vectors than FALSE 4 The length of 7216 is twice the length of LU TRUE 5 lf171U0then1700r 4470 FALSE Solve the following problems 1 A plane is drawn through the points A 2107 B 013 and C 101 Find a Two vectors lying in the plane b A vector perpendicular to the plane c The equation of the plane 2 Given the points P 1237 Q 357 and R 2537 nd a A unit vector perpendicular to the plane containing P7 Q and R Answer 7121354 413 313k b The area of the triangle PQR 132 c The distance from R to the line through P and Q 13x29 3 Find a vector normal to 4W 7 1 62 3 12 42 6 4 Find a unit vector from the point P 12 and toward Q 46 Find a vector of length 10 in the opposite direction Draw the points in 3 space 3 CHAPTER 14 Are the statements below TRUE or FALSE Eccplairi lfthe statement is false make corrections 1 If fpy is differentiable then fwab is a scalar TRUE 2 If fpy is differentiable then fyab fab TRUE 3 If fpy is differentiable then f m7 n is perpendicular to gradfm7 FALSE 4 The directional derivative could be found using the formula fwab grad fab 11 for any vector LU FALSE only ifiU is a uriit vector 5 grad fmn is a scalar that could take positive values only FALSE 6 The directional derivative could be found using the formula fwa7 b ngad fa7 b cos where o is the angle between grad fa7 b and LU TRUE 7 The local linearization of fpy x2 y2 at A 11 gives an overestimate to the value of fpy at B 104095 FALSE cart be seeri from the graph Solve the following problems 8 Find the equation of the plane tangent to the surface xyxzyz 11 at A 115 Answer 0 6p 71 6y 71 22 7 5 9 Find the quadratic Taylor polynomial for og z13y2 near the origin 07 0 Use it to approximate the value of f027 701 Use the formula from the book 10 Show that the function og arctanyz satis es the equa tion 62 62 4 1 4 3x2 3y Just calculate the respective secorid order partials 0 11 Show that the function 2 hx2 yz satis es the equation Use the chain rule by letting t 2 y2 then compute the left hand side 12 A function ay has linear approximation fxy 735x7 3y near A 23 and fxy 73z2y near B 12 Say whether each of the following statements is TRUE7 FALSE7 or IMPOSSIBLE TO DETERMINE Explain you answer a 3 TRUE b Af0921 m 15 FALSE Af0921 5 01 i 3 01 c f1928 m 719 FALSE f19 28 m 73 5 19 i 3 28 d The equation of the tangent plane at B 17 2 is z 745x7 1 7 3y 7 2 FALSE 2 73x 2y is the equation e The differential of ay at A 23 is df 73 de 7 3dy FALSE df 5dz i 3dy 13 Find a normal vector to 3 2yz2 zay 12 at P 105 This is the gradient note that P is not on the graph use P 12137 07 5 instead
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