MULTIVARIATE CALCULUS, CH 14 - STUDY GUIDE

1. Find the domain of the function f(x,y) = ____________

Sketch the region representing the domain on an xy-plane.

Expect the function to contain a log expression or a sqare root, possibly in the denominator of a fraction.

2. Compare the limits along the paths y=___ and y=___:

Does the limit exist? (Yes or No or Not enough data; Justify your answer).

3. Use the chain rule to find if w = [function of x, y and z], x = [function of s and t], y = [function of s and t], z = [function of s and t].

4. Find equations of the tangent plane and the normal line to the surface f(x,y) = ____, or to a surface defined implicitly (such as g(x,y,z) = constant), at the point ( , , ).

5. The two shortest sides of a right triangle [or other figure] are measured as ___ inches and ___ inches, with a possible error in measurement of ____ inch in each measurement. Use differentials to approximate the maximum error in the calculated length of the hypotenuse.

6. Suppose that the temperature at a point (x,y,z) is given by T(x,y,z) = __________.

· Find the instantaneous rate of change of T at the point P( , , ) in the direction from P toward the point Q ( , , )

· Find the maximum rate of change of T at P.

7. Suppose that the temperature at a point (x,y) is given by T(x,y) = __________.

Find the path of a heat-seeking particle if the path passes through the point ( , ).

8. For the function f(x,y) = _________:

· Find f_x, f_y, f_xx, f_yy and f_yx.

· Find all the critical points.

· Use the second partials test to determine, for each critical point, whether it gives the location of a relative maximum, a relative minimum, a saddle points, or a point requiring further analysis.

· What are the values for the relative extrema (i.e. the values of f(x,y))?

9. Find the point on the surface z = ______ that is closest to the point ( , , ). That is to say, minimize the distance between the point (x,y,z) and the point ( , , ) subject to the constraint z = _______. Expect the surface to be a plane or a quadric surface.

Return to: Merced College; Don Power Updated 03/30/04 by Don Power