MTH 254 Final Exam Technology Free-access Exam. dy dx calculates the area of region(s). dx dy calculates the area of region(s).

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1 00604 Technolog Free-access Eam Name 1.. The circle + = 8 is shown in Figure 1. The region has been subdivided into 16 subregions. Each of the double integrals stated in questions (a) - (g) directl finds the area of one these subregions or a combination of two or more of the subregions. In each provided blank, state the subregion(s) whose area is directl calculated b the stated double integral. No work need be shown. (1 points) Figure 1 a. π /4 /cos( θ) rdrdθ calculates the area of region(s). π /4 0 b. 0 d d calculates the area of region(s). c. 0 8 d d calculates the area of region(s). d. 3 π / rdrdθ calculates the area of region(s). 5 π /4 /sin( θ) 0 e. d d calculates the area of region(s). 8 f. 5 π /4 rdrdθ calculates the area of region(s). 3 π /4 0 Page 1 of 10

2 . State a polar double integral that will find the area of the region described in the caption for Figure. Do not evaluate the integral. (6 points) Figure The region inside the rose curve r = 4sin ( θ ) and outside the circle r = z Write a brief sentence that clearl outlines the meaning/purpose of the smbols ou write. (7 points) 3. Find the maimum rate of change in the function f (,, z) = + at the point (,3, 1) 4. Page of 10

3 4. Find a triple integral that evaluates to the volume of the tetrahedron with vertices at the points ( 0,0,0), ( 3,0,0), ( 0,5,0), and ( 0, 0,15). Draw the solid onto Figure 3. Label Figure 3 in a manner that describes the significance of all 6 limits of integration. If ou like, ou ma draw a Figure 3a to illustrate the planar limits of integration. State the volume of the tetrahedron assuming that the linear unit is mm. Use our calculator to evaluate the triple integral. If ou cannot come up with the equation of the plane at the top of tetrahedron, come on up front and I ll sell it to ou for points. (1 points) z Figure 3: Tetrahedron Page 3 of 10

4 5. Consider the tangent plane to the surface z= 16 at the point (, 4, ). Eactl one of the lines given in options 1 5 lies on this tangent plane. Which one? To earn an credit for this problem, ou must clearl eplain how ou came to our determination. (10 points) r t t t t a. ( ) =,4 4, r t t t t c. ( ) = 4,4, 4 r t t t t b. ( ) = + 4, 4+, + 4 r t = + t t + t d. ( ), 4 4, Page 4 of 10

5 8. One equation for the curve shown in Figure 4 is = 0. Find the slope of the tangent line to this curve at the point ( 0,0 ). Show enough well-documented work so that our reasoning is clear. (8 points) Figure 4 9. Sketch onto Figure 5 the domain of the function f (, ) + ln( ) work so that our reasoning is clear. (10 points) =. Show enough Figure 5 Page 6 of 10

6 6. Suppose that over a certain region of space the electrical potential V is given b (,, z) = 5 3 z V + Find the instantaneous rate of change in the potential at the point (,4,5) iˆ + ˆj kˆ. (7 points) 3 in the direction 3 f, = Make sure 3 that ou show all relevant work in a well-organize, well-documented manner. 7. Find the critical points for the function ( ) Page 5 of 10

7 10. Prove that the unit normal vector to the function r ( t) = t, sin( t), cos( t) is parallel to the same coordinate plane for all values of t. State the plane to which the vectors are parallel. Write enough sentences so that our argument is clear. (Note: to prove this ou will have to find the () general formula for ˆ T t N() t =. ) Please work this problem without our calculator. T t () Page 7 of 10

8 11. All questions on this page refer to the function ( ) the level curve through (,1 ) is vertical. (10 points) f, shown in Figure 6. The tangent line to a. Is (,1) f positive, negative, or zero. b. Is (,1) f positive, negative, or zero. c. Is (,1) f positive, negative, or zero. d. Is ( 3,) f positive, negative, or zero. e. Is ( 3, ) f positive, negative, or zero z = z = 1 z = 0 z = 1 Figure 6: Level curves for = f (, ) Page 8 of 10

9 Etra Credit (Up to 8 points EC) 1 Rewrite d dz d in the order dd dz. The Region G is shown in Figure 7. G (Note I worked the problem before sending it off to the printer. This is the problem I mean to ask. :) Figure 6: G Page 9 of 10

10 Crimin Even more Etra Credit (Up to 4 points EC.) Erma was analzing a function, z f (, ) f (, ) = and that f ( ) =. In the process of her analsis, Erma determined that, = Without tring to recreate the formula for f, how can ou establish with 100% certaint that Erma mucked something up when taking her derivatives of f? Page 10 of 10