Math 2374 Final Exam FALL 2002 PRINT NAME____________________________

12/16/02 SIGNATURE_____________________________

Time Limit: Three Hours WORKSHOP INSTRUCTOR________________

This exam contains 9 problems on 11 pages, including this cover page. Check to see if any pages are missing. Enter all requested information on the top of this page, and put your initials on the top of every page, in case the pages become separated. You may not use your books and notes on this exam, but you may use your graphing calculator and one 8.5 by 11 inch crib sheet.

Do not give numerical approximations to quantities such as or or . Do simplify , e⁰ = 1, etc.

Show your work, in a reasonably neat and coherent way, in the space provided. All answers must be justified by valid mathematical reasoning, including the evaluation of definite and indefinite integrals. Using your calculator to find an indefinite integral is not "valid mathematical reasoning", even if you then check the answer by differentiation.

Mysterious or unsupported answers will not receive full credit. Your work should be mathematically correct and carefully and legibly written. A correct answer, unsupported by calculations, explanation, or algebraic work will receive no credit; an incorrect answer supported by substantially correct calculations and explanations might still receive partial credit. Full credit will be given only for work that is presented neatly and logically; work scattered all over the page without a clear ordering will receive very little credit.

Recall that .

If f(r ,q ,f ) = (r sinf cosq , r sinf sinq , r cosf ) is the transformation from spherical coordinates to rectangular coordinates, then the Jacobian determinant of this transformation is -r ² sinf . You can use this without doing the calculation.

1. Find all critical points of the function f(x,y)= x² + y² + xy² . Then classify each critical point as a local maximum, local minimum, or saddle point.
2. The temperature at the point (x,y,z) in space is given by T(x,y,z) = xyz³. T is measured in degrees Celsius (°C); x, y, and z are measured in kilometers (km). At a certain instant of time, a space ship is at the point (2,3,1) and headed toward the point (3,4,3) at a rate of 5 kilometers per second (5 km/s). At what rate is the temperature experienced by the spaceship increasing at that instant? Your answer should be in terms of degrees Celsius per second (°C/s).
3. (a) Let M be the surface parametrized by f(s,t) = (t,t²,s³). Find the equation of the tangent plane to this surface at the point (2,4,1) . Express your answer in the form Ax+By+Cz=D.
   (b) Find a parametrization for the plane in R³ with equation 4x - y = 4; that is, find an equation of the form x = x₀+sa + tb.
4. Let S be the solid bounded above by the sphere x² + y² + z² = 16 and below by the paraboloid z = (x² + y²)/6.
   (a) Parametrize the solid S using cylindrical coordinates.

(b) Let F(x,y,z) = (x+yz, sin(x⁹z⁶), cos (x⁷y⁸)). Use the Divergence Theorem to find the flux of F through the boundary of S. Use the outward pointing normal.

5. Let F(x,y,z) = (-y, x, sin(x⁷y⁸z⁹)) and let M be the surface given by
   z = x² + y² - 9, z < 0.
   (a) Give a parametrization of the boundary of M as a curve in 3-space.
   (b) Using any valid method, evaluate the integral of CURL F over the surface M, where M has the outward pointing normal.
6. Let M be the surface x² + y² + z² = 4, z > 0. Using the outward-pointing normal, find the flux through M for the vector field F(x,y,z) = (y,x,z).
7. Let f(x,y) = (y², x + y) and let g be a function from R² to R such that the Jacobian matrix of g is given by [sin(x²), cos (xy)]. Let h be the composite h = g ° f.

(a) Find the Jacobian matrix for h at the point (2,3).

(b) Assuming that h(2,3) = 4, use the result of part (a) to find the equation of the tangent plane to the graph of h at the point (2,3). Write your answer in the form z =Ax+By+C.

8. Let f(x,y) = (x+y)² and let R be the region in the plane bounded by the lines
   x + y = 0, x + y = 1, 2x – y = 0, 2x –y = 1. Use the change of variables
   u = x + y, v = 2x - y to evaluate the double integral of f over R.
9. Let M be the level surface g(x,y,z) = 3 where g(x,y,z) = x² + y²/4 + z²/9.
   (a) Find the equation of the tangent plane to the surface M at the point (1,2,3)

(b) Compute the line integral of the vector field F(x,y,z) = (2x, y/2, 2z/9) over the curve C, where C is parametrized by
x(t) = (cos⁴ t², sin⁴ t², t +1), 0 < t < (2p )^1/2