Calculus 3 Review 4

    This topic is a collection of problems and concepts that might help someone understand their working knowledge of Calculus 3.

(1) Find a vector-valued function whose graph is the curve of intersection of the plane and the plane

    Solution. One way to accomplish the task is by letting Then to find relations for and we will solve the system

.

The solution is and Therefore a vector-valued function for this intersection is   which has the following graph.

(2) Determine if the graph of the vector function  



is piecewise smooth on .

    Solution. The graph of the vector function is not piecewise smooth over because



and for where is an integer.

(3) Find the position vector and velocity vector given the acceleration vector function initial position vector   and initial velocity vector  

    Solution.  Given the velocity vector function is
     




We can find by using



Thus Therefore,



So the position vector function is



which is



We can find by using



Thus Therefore,

(4) Express the vector function in terms of arc length measured from the point corresponding to , in the direction of increasing :

    Solution. We have,
    


Solving for we have Thus

(5) Find the maximum curvature on the curve

    Solution. We have and



So,



Applying the first derivative test we have the maximum curvature at with curvature of   

(6) Evaluate the limit  



    Solution. We have along the curves :



But along the curve , we have



Therefore the limit does not exist.  

(7) On what set is the function continuous?

    Solution. Let and . Then



Now is continuous everywhere since it is a polynomial and is continuous on its domain . Thus, is continuous on its domain



which consists of all points outside the circle

(8) If , calculate and .

    Solution. Using the chain rule for functions of one variable, we have


and

(9) Find the tangent plane to the elliptic paraboloid at the point

    Solution. Let Then and Then the equation of the tangent plane is or

(10) Determine whether the function is differentiable at either or Explain why.

    Solution. The function is not cntinuou at because it is not defined at Moreover, the limit
    


does not exist because along we have



and along we have



Indeed, is not continuous at and thus is not differentiable at To show that is differentiable at we compute the partial derivatives,



Since these partial derivatives and are continuous on any open disk not containing   we conclude that is differentiable at any point except ; and in particular at

(11) If is differentiable where and then find   

  

    Solution. We compute,







Similarly,  







and


  

  

so  

(12) Find the absolute maximum and minimum values of the function  



over the rectangle  



    Solution. Since is a polynomial it is continuous on the closed bounded rectangle   therefore has both absolute maximum and minimum values. We first find the critical points by solving the system   



The only critical point is and the value of there is We look at the values of on the boundary of , which consists of four line segments as shown.



On we have and for This is an increasing function of so its minimum value is and its maximum value is On we have and on This is a decreasing function of so its maximum value is and its minimum value is On we have and on By observing that we see that the minimum value of this function is and the maximum value is Finally, on we have and on with maximum value and minimum value Thus on the boundary, the minimum value of is 0 and the maximum is 9.  We compare these values with the value at the critical point and conclude that the absolute maximum value of on is and the absolute minimum value is

(13) A cylindrical can is to hold of orange juice.  The cost per square inch of constructing the metal top and bottom is twice the cost per square inch of constructing the cardboard side. What are the dimensions of the least expensive can?

    Solution. Let and be the radius and height of the cylinder, respectively. We want to minimize the cost subject to the constraint where We have and Solving the system with the Lagrange multiplier







we find the radius in. and the height in.

(14)  Suppose the smooth curves and are given by



and



Evaluate

   and    

    Solution. Both and are smooth curves from to with the same trace which is the portion of the parabola for For we have    and therefore   

  

For we have    and therefore   

(15) Find the work done when an object moves in the force field once counterclockwise around the circular path

    Solution.  Using Gren's theorem we have,  








(16) Find the work done when an object moves in the force field once counterclockwise around the circular path  

    Solution. Using Green's theorem,

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Cite this as:
Calculus 3 Review 4
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/calculus-3-review-4.html