( ) ( ) Mat 241 Homework Set 5 Due Professor David Schultz. x y. 9 4 The domain is the interior of the hyperbola.
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1 Mat 4 Homework Set 5 Due Professor David Scultz Directions: Sow all algebraic steps neatly and concisely using proper matematical symbolism. Wen graps and tecnology are to be implemented, do so appropriately. Mecanics: #. Determine & Sketc te domains of te following functions. f (, y) = + y A. 0 & y 0 Te domain is te all non-negative pairs. B. 5 g(, y) = y y > y > y > 6 y so, < 9 4 Te domain is te interior of te yperbola. 8 y
2 C. (, ) = + + ln( 4 ) f y y y + y 0 and 4 y > 0 + y and + y Te domain is te annulus formed by te two concentric circles. - #. Describe te domain of te following function: f = + y + z 4 y z (, y, z) 4 y z 0 + y + z 4 Te domain is te set of all ordered triples wic do not lie on te spere of radius centered at te origin.
3 #. Find te it or sow it does not eist. State your reasons for noneistence. A. + sin y ; (, y) ( 0, + y + sin + 0 (, ( 0, (, ( 0, ( 0, y) ( 0, ( 0, y) ( 0, ( 0, y) ( 0, ( 0, y) ( 0, ( ) 0 = = ( y), 0 0, 0 0, 0, sin y sin y sin y sin y = = = = = 0 + y y y y Since f, y f, y, (, y) (, ) B. 0 0 (, y) ( 0, + sin + y y does not eist (, ( 0, (, ( 0, ( 0, y) ( 0, ( 0, y) ( 0, (, y) ( 0, y + y y y = = 0 + y y ( ) ( y), 0 0, 0 0, 0, 0 y + y = = Since f, y f, y, does not eist. A Grap B Grap
4 C. Use polar coordinates to compute tis one: ( = r cos θ; y = r sinθ (, y) (, ) 0 0 iff r 0 ). (, ) (, ) ( + ) sin r ( cos θ + sin θ ) + r ( cos θ + sin θ ) ( r ) 6r sin y sin r = y r y 0 0 r 0 r 0 cos = = r 0 r #4. Grap te level surfaces of were K = 4, 0, and - 4. F, y, z = + y z z y = + z = + y z y = + + F, y, z = + y z for te values K, 4 /* paraboloid sifted 4 down /*paraboloid 4 /* paraboloid sifted 4 up
5 f, y = cos + y. #5. Draw a contour map for: f, y = cos + y cos ( ) + = + = cos y k y k k coose k =, 0, + y = cos = π π + y = cos ( = π + y = cos = 4 Te contour plot is concentric circles.
6 #6. Consider te surface: f (, y) = 0 y A. Determine any restrictions upon te allowed values of K for wic te function as level curves. K is positive because: ( y) y 0 > 0, B. Pick tree convenient values of K and grap te corresponding level curves. Make sure your values sample all possibilities. f (, y) = 0 y Coose : k =, 0, 00 0 y y = 0 log0 log0 ( 0 ) y 0 = = 0 so y =, /* yperbola transverse orizontal log 0 = 0 log 0 = 0 = y so y y y 0 0 =, /* yperbola transvers log y y 0 0 e vertical, /* yperbola transverse vertical 00 = 0 log 00 = 0 = y so y =
7 Concept Development: #7. A tin metal rectangular plate, located in te y-plane, as temperature at point (, y ) on te plate given by T (, y) 5 y = + +. Te level curves of T are called isotermals because at all points on an isotermal te temperature is te same. Sketc several isotermals for tis plate and be accurate. (, ) = T y y coose : k = 9,, 4 y 4 y 8 6 y y = 9 + y = 4 + = y = + y = 6 + = y = 4 + y = 6 + = nested ellipses #8. In te clean and jerk weigtlifting competition a weigtlifter in te eavyweigt class wo weigs 0 kg lifts 0 kg. In te flyweigt class, a person wo weigs 50 kg lifts 0 kg. How can we compare tese feats of strengt? Wo actually is te superior lifter? Several different formulas ave been developed for andicapping lifts. Let wl be te lifted weigt (in kg) and wb be te body weigt of te lifter (in kg). Consider te following 4 proposed andicapping formulas.
8 A. Used in ABC s Superstars competition: = wl wb w = w HW FW B. Austin formula: l 4 b = 0 0 = 00 = 0 50 = 80 HW FW w = w C. Classical formula; D. O Carroll formula: = l b HW FW 0 = = = = w ( w 5) HW FW b l 0 = 5. ( 65) ( 5) 0 = 5. 7 Use tese formulas to determine weter te eavyweigt or te flyweigt is te superior lifter. In eac case te fly-weigt as a greater andicap so e/se is te superior lifter.
9 Computer Grapics & Computations. #9. Discuss te continuity of te function and evaluate te it along te suggested pats. Does te it eist? Wy or wy not? Include wit your analysis a computer generated grap. A. y pats y = 0 & y = (, y) ( 0, + y y 0 = = y (, ( 0, (, ( 0, = = (, ) ( 0, 0 ) + (, ) ( 0, y Since, (, ( 0, + y (, ) ( 0, + y does not eist + y (, y) ( 0, B. y pats y = 0 & y = (, y) ( 0, + y (, ( 0, (, ( 0, (, ) ( 0, (, ) ( 0, (, y) ( 0, = ± + + = ( ) ( ), 0 0, 0, 0, 0 y does not eist + y = Since f, y f, y,
10 Special Project Wen my fater studied to become a tool & die maker in te fifties people used clay to visualize ow planes would intersect cylinders and oter various sapes. Tey could cut te clay at angles and ten see te resulting cross sections. Te tool tey used for cutting te clay can still be bougt today and simply consists of a wire strung between two rigid rods. Two types available appear below. Today, we ave te advantage of powerful tree dimensional software packages but te visualization process for students is still etremely important. Building realistic models bot in a concrete or virtual world is a skill to be mastered by tose serious about teir matematical and engineering pursuits. Consider a simple slicing. A cylinder cut by a plane making a 45 degree angle wit te orizontal. Complete te following.. Build a pysical model of te scenario. An old toilet paper roll or some wood stock would do fine. I ope to get some great models!. Construct a virtual model by doing te following Determine te equation for a cylinder wit radius. Determine te equation of te plane wic cuts te cylinder at a 45 and contains te point:,, 0 and as a nice traces in octant ( +,y + ). Grap te cylinder, slanted plane, and te y-plane all on one grap. Determine te space curve wic represents te intersection of te plane and te cylinder. Determine te lengt of one trace of te space curve. EC. Determine te volume and surface area (entire solid) of te wedge cut out by te y-plane, te slanted plane, and te cylinder. Come by my office for a visual.
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