Multivariate Calculus Review 1

    This topic is a collection of problems and concepts that might help someone understand their working knowledge of multivariate calculus

(1) Find a vector function whose graph is the curve of intersection of the hemisphere and the parabolic cylinder

(2) Find a vector function whose graph is the line of intersection of the planes and

(3)  Determine the function given by



where and

(4)  Find the limit of the following vector-valued function, if it exists:

(5) Show that if is a vector function that is continuous at then is continuous at   

(6) Given determine so that

(7) Given determine so that

(8) Determine so that the parametric equations for the tangent line to the graph of the vector function at the point corresponding to are

   and    .

(9) Find a value of and such that



and

(10) The acceleration of a moving particle is Find the vectors and so that the particle's position as a function of is

(11) Find the unit tangent vector and the unit normal vector for the curve with

(12) Find the length of the curve over the interval

(13) Find the curvature of the plane curve at the point .

(14) If is a smooth curve in described by the parametric equations and then the curvature is given by



Use this formula to find the curvature of the curve described by the parametric equations   and

(15) Find the point(s) where the ellipse has maximum curvature.

(16)  Let Find and

(17) Find the domain and range for the multivariate function

(18) Find the domain and range for the multivariate function   

(19) Find the domain and range for the multivariate function    

(20) Find the domain and range for the multivariate function    

(21) Sketch some level curves for the multivariate function for

(22) Sketch some level curves for the multivariate function for   

(23) Sketch some level curves for the multivariate function for     

(24) Sketch some level curves for the multivariate function for     

(25) Evaluate the limit, if the limit exists.

(26) Evaluate the limit, if the limit exists.

(27) Evaluate the limit, if the limit exists.

(28) Evaluate the limit, assuming the limit exists.

(29) Given that the function

.

is continuous at the point what is the value of Explain.

(30) Determine and given

      and       .  

(31) Determine and by differentiating implicitly given   

     and     

(32) The heat equation is , and the wave equation is show which of the following satisfies these equations:

(33) The Cauchy-Riemann equations are   and where and are functions of and Show which of the following satisfies these equations:  

    (a)     (b)

(34) Find the tangent plane to the given surfaces at the given point.

    (i) Determine the standard-form equation for the tangent plane to the surface at .

    (ii) Determine the standard-form equation for the tangent plane to the surface at .

(35) Determine the total differential of the functions

.

(36) Use linear approximation (differentials) to find approximate values for

and      

(37) Show that the functions and are differentiable.

(38) Show that if and are sufficiently close to zero and is differentiable at then  

;

and use this approximation for the expressions and around

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