Functions of several variables

1. (1 pt)Find the limit, if it exists, or type N if it does not exist.lim 5×2(x,y)->(0,0) 1X2 + 33,2Answer(s) submitted:(incorrect)2. (1 pt) In this problem we show that the function4%2 – y2x‘ if( a) x2+yZdoes not have a limit as (x,y) -> (0,0).(a) Suppose that we consider (x, y) -> (0,0) along the curvey = 2x. Find the limit in this case:lim -4x2_y2(x,2r)->(0,0) x2+¥2(b) Now consider (x, y) -> (0, 0) along the curve y = 3x. Findthe limit in this case:lim -4x2_y2(x,3x)->(0,0) xzfl‘z(c) Note that the results from (a) and (b) indicate that f hasno limit as (x, y) -> (0,0) (be sure you can explain why!)To show this more generally, consider (x, y) -> (0.0) along thecurve y = mx, for arbitrary m. Find the limit in this case:lim 7-74-94 2(x,mx)->(0,0) x +3,(Be sure that you can explain how this result also indicates thatf has no limit as (x, y) -> (0,0).Answer(s) submitted:(incorrect)3. (1 pt) For the function f (x,y) below, determine whetherthere is a value for 0 making the function continuous every-where. If so, find it.4. (1 pt)The largest set on which the function f (x, y) = l / (11 -x2yz) is continuous isA. The interior of the circle x2 + y2 = 11B. The interior of the circle x2 + yz = 11, plus the circleC. All of the xy-planeD. All of the xy-plane except the circle x2 + yz : 11E. The exterior of the circle x2 + yz : llAnswer(s) submitted:(incorrect)5. (1 pt)Use thel level curves of the function z = f (x, y) todetermine if each partial derivative at the point P is pos-itive, negative, or zero.1- fxx (P)I2 fix (P)I3- f (P)I f (P)5I f (P)(x,mx) -> (0,0) x“ +3″(Be sure that you can explain how this result also indicates thatf has no limit as (x, y) -> (0,0).Answer(s) submitted:(incorrect)3. (1 pt) For the function f (x,y) below, determine whetherthere is a value for 6 making the function continuous every-where. If so, find it.f( ) C +31 x S 2?.x’y5-y x>2(If there is no value of c that works, enter none, and be sure thatyou can explain why there is no such. value.)Answer(s) submitted:(incorrect)6. (2 pts) The temperature adjusted for wind chill is a temper-ature that tells you how cold it feels, as a result of the combina-tion of wind speed, w, in miles per hour, and actual temperature,T, as measured by a thermometer in degrees Fahrenheit. The ta-ble below shows the temperature adjusted for wind chill, f(w,Ti),as a function of w and T. The left column of the table gives thewind speed in miles per hour, the top row gives the actual tem-perature in degrees Fahrenheit, and the remaining entries givethe temperature adjusted for wind chill in degrees Fahrenheit asa function of wind speed and actual temperature.-nmamun-“-n(a) Estimate f“. (15.0) and give a practical interpretation.f“. (15,0) xIf the wind is blowing _ mph and the actual temperatureis _ degrees Fahrenheit, it will feel _ degrees Fahren-heit colder for each 1 mph increase in wind speed.(b) Estimate f7 (10, 10) and give a practical interpretation.fT(10,10)If the wind is blowing _ mph and the actual temperatureis _ degrees Fahrenheit, it will feel _ degrees Fahren-heit warmer for each 1 degree Fahrenheit increase in actualtemperature.A nswer(s ) submitted:8. (2 pts)Compute the partial derivatives:Z : 8x3x3yAnswer(s) submitted:(incorrect)9. (1 pt)Compute the partial derivatives:z = In (x6 +y4)3xa)»Answer(s) submitted:(incorrect)10. (2 pts) Your monthly car payment in dollars is P(POJ, r), where $Po is the amount you borrowed, t is the num-ter of months it takes to pay off the loan, and r percent is theaterest rate.(a) Is aP/at positive or negative?Luppose that your bank tells you that the magnitude of 8P/8t is0.Vhat are the units of this value?For this problem, write our your units in full, writing dollarsor $, months for months, percent for %, etc. Note that frac-ional units generally have a plural numerator and singular de-ominator.(b) Is dP/ar positive or negative?Luppose that your bank tells you that the magnitude of dP/dr is0.Vhat are the units of this value?For this problem, write our your units in full, writing dollars