Calculus 3 Practice Test 5

Show all work and jusify each step.

(1) Find the maximum and minimum values attained by the function



at the points of the triangular region in the plane with vertices at and .

(2) Find the value of the integral where



(3) Evaluate where is the region bounded by the parabolas and

(4) Find the volume of the solid bounded by the cylinder that lies above the planes and

(5) Find the surface area of the part of the sphere that lies above the plane

(6) Estimate the center of mass of a lamina described by the region bounded by with density function

(7) Compute the double integral .

(8) Change to the order

(9) The temperature at point in a region of space is given by the formula Find the lowest temperature on the plane

(10)  Assume that Use triple integration to find the volume of the ellipsoid



(11) Suppose measures the time (in minutes) that a student spends working out problems on a calculus test and measures the time (in minutes) that a student spends checking over the test before turning it in. If the joint probability density function for a certain section of calculus students is



then find the probability that a student will have at least 10 minutes to check over the test for an 80 minute test.

(12) Find parametric equations for the tangent line to the curve with given parametric equations



at the point

(13) The acceleration of a moving particle is Find the particle's position as a function of if and   

(14)  Find the points of maximum curvature for the graph of

(15) Find the domain and range for the function .

(16) Find the limit, if it exists, or show that the limit does not exist:  

(17) Determine the largest set on which the function is continuous.



(18) Find the length of the curve   for   .

(19)  Re-parametrize the curve with respect to arc length measured with base point where in the direction of increasing .   for

(20)  Given the vector-valued functions







find



(21) Verify that the function is a solution of the three-dimensional Laplace equation  

(22) Find and if is defined implicitly as a function of and by the equation

(23) Sketch some level curves for the function for (at least four of them.)

(24)  Consider the function

.

Find the value of which will make continuous at the origin.

(25)  Show that but does not exist, where



(26) Find where and and

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