MAT1320 - Midterm - 2011 Fall

1. 0/6 points LarCalc11 13.1.010. [3864 VIA Evaluate the function at the given values of the independent variables. Simplify the results. fx, y) -4 -x2 - 4y2 (a) ro, 0) for (c) f2, 3) (e) x, o Cal 2. 0/2 points LarCalc11 13.1.019.MI. [386 Evaluate the function at the given values of the independent variables. Simplify the results. x, y)- 6x y2 (a) fix + ax, y)-f(x, y) Ay

3. 1 points LarCalc11 13.2 Find the indicated limit by using the limits limfx, y) 2 and im g(x, y)--4 lim 4. O/1 points LarCalci1 13 2 Find the indicated limit by using the limits lim x, y) 3 and lim (x, y) g(x, y)--5 (a, b) lim x, Y) (x, y) (a, b) g(x, y) 5. y1 points LarCalc11 13.3 Explain whether or not the Quotient Rule should be used to find the partial derivative. Do not differentiate. ay x2 51 O Yes, the function is a quotient. O No, y only occurs in the numerator. 6. 0/2 points LarCalc11 13.3 Find both first partial derivatives oz 7. 0/2 points LarCalc11 13.3 Find both first partial derivatives. hx, y) ex+y) h(x, y)

8. /2 points LarCalc11 13.3.031 Find both first partial derivatives. az dx az ay 9. /1 points LarCalc11 13.4.003. Find the total differential. z 3xSy10 dz 10. 0/2 points LarCalc11 13.5.012 Consider the following. (a) Find dw/dt using the appropriate Chain Rule. dw dt (b) Find dw/dt by converting w to a function of t before differentiating. dw dt 11. 0/1 points LarCalc11 13.6.008. Find the directional derivative of the function at P in the direction of v. rx, y) = x3-y3, Rio, 9), v-V2 (1+j)

12. 0/2 points LarCalc11 13.7.017 (a) Find an equation of the tangent plane to the surface at the given point. x+y+z=9, (2, 5, 2) (b) Find a set of symmetric equations for the normal line to the surface at the given point. ox -2y -5-z-2 13. /3 points LarCalc11 13.8.025 Use a computer algebra system to graph the surface and locate any relative extrema and saddle points. (If an an not exist, enter DNE.) -8x x2 y21 relative minimum (x, y, z) - relative maximum (x, y, z) saddle point 14. 0/1 points LarCalci1 13.10.027 Use Lagrange multipliers to find the minimum distance from the curve or surface to the indicated point. Surface Point Plane: x yz 1 (7,1,1)

Mnct 60. fx)-3x3x 61, f(x) = x.(x-z? Applications of Differentiation 212 CHAPTER 3 Find all critial hunber 35. 40, g(θ)-48-tane 39. F(x)-x""(x-4): 41. f)-2 cos8 + sino 63. Between 0°C and 30°C, the volume V Gn of I kg of water at a temperature T is 43-44 A formala for the derivartive of a function f'is given. How many critical numbers does f have? V-999.87-0.06426T0.00850437 Find the temperature at which water has its m APTE - 6x + 10 64. An object with weight W is dragged along a h attached to the o by a force acting along a rope rope makes an angle θ with the plane, then the 45-56 Find the absolute maximum and absolute minimurn values the force is μ sin θ + cos θ of f on the given interval. 45. fx)-12+4x -. [0,S] 46. fix)-5+54x-2x. (0,4) 司f(x)-2x,-3x1-12x + 1, [-23] where μ is a positive constant called the coefficie and where 0 θ π/2. Show that F is minimize 65. The water level, measured in feet above mean sea l Lake Lanier in Georgia, USA, during 2012 can be im the function L()-0.01441,, 50. f)- 4). [-2,3] 51.)(02.4) 52. x)-+T -0.4177,2 + 2703t + 1060. where t is measured in months since January 1,201 when the water level was highest during 2012. 66. On May 7, 1992, the space shuttle Endeavour was laun on mission STS 49, the purpose of which was to install perigee kick motor in an Intelsat communications satellit The table gives the velocity data for the shuttle between l and the jettisoning of the solid rocket boosters. (a) Use a graphing calculator or computer to find the cubi S4. ft)- 0.2 pol ymomial that best models the velocity of the shuttle interval r e [o, 125). Then graph this polynom acceleration of the shuttle and usew ss fc)-2 cos+sin 2t, [o, -/2 56. f(t)-' + cot('/2), [π/4.7v/4] 57. If a and b are positive bumbers, find the maximum value 58. Use a graph ximum and minimum values of the to estimate the critical numbers of fx)-II+5xcorect i e decil place 59-62 (a) Use a graph to estimate the absolute (b) Use calculus to find the exact 0 maximurm and 0 185 319 447 values of the function to two decimal places. minimum maxirsum and minimum values 20 32