5. [0.51/1 Points] DETAILS PREVIOUS ANSWERS SCALCET9 14.4.016. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER Explain why the function is differentiable at the given point. f (x, y) = y tan(x), 6 The partial derivatives are f (x, y) = y sec (x) and f (x, y) = tan (x) , sof x 4 X and f fyl 6 = -1 X . Using n as an arbitrary integer, both f and f are continuous functions for 4 , so f is differentiable at (#, 6) Find the linearization L(x, y) of the function at (* , 6). L(x, y) = 4x - y - 1 X Need Help? Read It16. [011 Points] DETAILS PREVIOUS ANSWERS SCALCETQ 14.3.093.EP. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER Consider the following. 2 The ellipsoid 2x2 + 3y2 + z = 75 intersects the plane y = 4 in an ellipse. Find the slope of the tangent line to this ellipse at the point (3, 4, 3). :J Find parametric equations for the tangent line to this ellipse at the point (3, 4, 3). 'x=r+3y=4,z=33r' x Need Help? _'_ 10. [011 Points] DETAILS PREVIOUS ANSWERS SCALCET9 14.5.039.M|. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER The temperature at a point (x, y) is T(x, y), measured in degrees Celsius. A bug crawls so that its position after t seconds is given by x = V 2 + . V = 1 + it, where x and y are measured in centimeters. The temperature function satises Tx(2, 2) = 3 and 2 Ty(2, 2) = 8. How fast is the temperature rising on the bug's path after 2 seconds? (Round your answer to two decimal places.) 5.24 x C/s Need Help?| In\" H when: H mu I 11. [011 Points] DETAILS PREVIOUS ANSWERS SCALCET9 14.5.043. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER The length 1', width w, and height h of a box change with time. At a certain instant the dimensions are! : 5 m and w = h = 4 m, and f and w are increasing at a rate of 6 m/s while h is decreasing at a rate of 9 m/s. At that instant find the rates at which the following quantities are changing. (a) the volume (in m3/s) -x (b) the surface area {in mZ/s} -x me (c) the length of a diagonal (in m/s) (Round your answer to two decimal places.) -x Need Hem? __ 4. [0.79\" Points] DETAILS PREVIOUS ANSWERS SCALCET9 14.1.019. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER In this example we considered the function W: f(T v) where W is the wind- chill index, T' Is the actual temperature, and v is the wind speed A numerical representation is given in this table. (a) What is the value of f(15, 60)? What is its meaning? We get {(15, 60} = [-29.5 \\ X , which means that if the temperature is 15C and the wind speed is 60 km/h, then 2 the air would feel equivalent to approximately K C without wind. (b) Describe in words the meaning of the question "For what value of v is I(20, v) = 30?" The question is asking: when the temperature is -20 up C, what wind speed gives a wind-chill index of [30 ' y C? J Answer the question. 1/ = ['20 "l 9 km/h (c) Describe in words the meaning of the question "For what value of Tis f(T, 20) = ~37?\" The question is asking: when the wind speed is [20 J y kmfh, what temperature gives a wind-chill index of | -3:r y C? Answer the question. T=|-10 lit 0:2 6. [0.6/1 Points] DETAILS PREVIOUS ANSWERS SCALCET9 14.4.027.EP. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER Consider the following function. f ( x, y, z) = V x2 + y2 + z2 Find f (x, y, z), f( x, y, z), and fz ( x, y, z). X fx ( x, y, Z ) = Vx2+ 12+2 2 V fy ( x, y, z ) = Vx 2 + 12 + 2 2 Z f z ( x, y, Z ) = Vx2 + 12+ 2 2 Find the linear approximation of the function f(x, y, z) = v x2 + y2 + z2 at (4, 4, 7). f ( x, y, Z) = Use the linear approximation to approximate the number v 4.032 + 3.982 + 6.992. ( Round your answer to four decimal places.) f(4.03, 3.98, 6.99) = 8.9954 X Need Help? Read It Watch It6. [0/1 Points] DETAILS PREVIOUS ANSWERS SCALCET9 14.5.018. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER Let R(s, t) = G(u(s, t), v(s, t)), where G, u, and v are differentiable, and the following applies. u(-5, -1) = 3 v(-5, -1) = 4 Us (-5, -1) = -4 V s ( - 5, - 1) = -3 U. (-5, -1) = 7 v . ( - 5 , - 1 ) = -9 Gu (3, 4 ) = 1 GV ( 3 , 4 ) = -7 Find R (-5, -1) and R.(-5, -1). Rs ( -5, -1) = 16 X R . (-5, -1) = 56 X Need Help? Read It 7. [0.51/1 Points] DETAILS PREVIOUS ANSWERS SCALCET9 14.5.025.EP. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER Consider the following. z = x + x y, x = s+ 2t - u, y = stu2 Find the following partial derivatives. az as tu-x 2+ 4x" + 2xy az at su-x2 + 2( 4x3 + 2 xy) az au 2stux- - 4x3 - 2xy Find- az az dZ when s = 3, t = 5, u = 2. as' at du az 28052 as az 46072 at az 3584 X au Need Help? Read It Watch It 8. [-/1 Points] DETAILS SCALCET9 14.5.032. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER Use the equation OF dy = ax Fx dx aF Fy ay to find dy dx cos(xy) = 4 + sin(y) dy dy Need Help? Read It1. [0.67/1 Points] DETAILS PREVIOUS ANSWERS SCALCET9 14.1.004. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER Let h(x, y) = e Vy - x 2 (a) Evaluate h(-1, 10). e3 (b) Find and sketch the domain of h. Graph Layers Clear All Vertical Parabola 1 Delete Fill . -10 -9 -8 -7 -6 -5 4 -3 -2 -1 2 3 6 8 10 No Solution WebAssign. Graphing Tool + Submission Data (c) Find the range of h. (Enter your answer using interval notation.) [1,00 )2. [0.66/1 Points] DETAILS PREVIOUS ANSWERS SCALCET9 14.4.003.EP. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER Consider the following surface. z = 2x + y2 - 5y Let z = f(x, y). Find f (x, y) and f (x, y). f ( x, y ) = 4x f ( x, y ) = 2y - 5 Find an equation of the tangent plane to the given surface at the point (1, 2, -4). z = 4x -y - 9 X Need Help? Read It9. [0/1 Points] PREVIOUS ANSWERS SCALCET9 14.4.039. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER lfz = 3x2 + y2 and (x, y) changes from (1, 3) to (1.05, 3.1), compare the values ofAz and dz. dz:-x A2 = 0.1325 x Need Help? _' 10. [0/1 Points] DETAILS PREVIOUS ANSWERS SCALCET9 14.4.041.M|. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER The length and width of a rectangle are measured as 42 cm and 38 cm, respectively, with an error in measurement of at most 0.1 cm in each. Use differentials to estimate the maximum error in the calculated area of the rectangle. [3.8 x cm2 Need Help?| III-nun II when H MI