c Dr Oksana Shatalov, Spring 2013
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Spring 2013 Math 251
Week in Review 3
courtesy: Oksana Shatalov
(covering Sections 12.1-12.4 )
12.1: Functions of Several Variables
Key Points
• The level (contour) curves of a function of two variables z = f (x, y) are the curves with equations
f (x, y) = k, where k is a constant in the range of f .
• Note that for functions of three variables it is impossible to visualize its graph. However we can examine
them by their level surfaces: f (x, y, z) = k where k is a constant in the range of f .
1. Let f (x, y) = sin(xy) + π. Find f (x − y, x + y).
2. Let f (x, y, z) = x + yz. Find f (y 3 , x − y, x + y).
3. Find the domain and range of the functions:
(a) f (x, y) = ln (9 − x2 − y 2 )
−
(b) f (x, y, z) = e
1
x2 +y 2 +z 2
4. Sketch the graphs of the following functions.
(a) f (x, y) = x2 + y 2 − 3
r
(b) f (x, y) =
4 − x2 −
y2
25
(c) f (x, y) = y 2
(d) f (x, y) = y + 1
5. Find and sketch the domain of the given function and draw a contour map of the function
(i.e. draw several level curves).
2
(a) z = p
.
x2 − y
√
(b) f (x, y) = x − y
(c) f (x, y) = e−x
2 −y 2
6. Describe and sketch several level surfaces of the function f (x, y, z) = −x2 − y 2 − z 2 .
7. Let f (x, y) = yex . Find an equation of the level curve that passes through the point (1, −2).
c Dr Oksana Shatalov, Spring 2013
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8. Let f (x, y, z) = xyz 2 + 1. Find an equation of the level surface that passes through the point
(1, 2, −1).
9. An ant walks on a surface of a thin metal plate that occupies the first quadrant of the
xy-plane. The temperature at a point (x, y) of the plate is given by T (x, y) = x2 y. An ant,
initially at (1, 3), wants to walk on the plate so that the temperature along its path remains
constant. What path should the ant take and what is the temperature along the path?
12.3: Partial Derivatives
Key Points
• RULE FOR FINDING PARTIAL DERIVATIVES OF z = f (x, y):
1. To find fx , regard y as a constant and differentiate f (x, y) with respect to x.
2. To find fy , regard x as a constant and differentiate f (x, y) with respect to y.
• Clairaut’s Theorem. Suppose f is defined on a disk D that contains the point (a, b). If the functions fxy
and fyx are both continuous on D then fxy (a, b) = fyx (a, b).
10. Find the first partial derivatives of the functions
2
(a) w(x, y) = x4 ln(x + y) − e−xy
y
(b) g(x, y, z) = x cos
x+z
p
11. If f (x, y) = 3 x3 + y 3 find fx (0, 0).
12. Find the slope of the surface z =
13. If u = xy , show that
√
2x + 3y in the y-direction at the point (5, 2)
x ∂u
1 ∂u
+
= 2u.
y ∂x ln x ∂y
14. If f and g are twice differentiable functions of a single variable, show that the function
u(x, t) = f (x + at) + g(x − at) is a solution of the wave equation
utt = a2 uxx .
15. Find all partial derivative of the second order for the function z =
Z
16. Find fx and fy if f (x, y) =
y
x2
2
et dt.
√
y sin x.
c Dr Oksana Shatalov, Spring 2013
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12.4: Tangent Planes and Differentials
Key Points
• An equation of the tangent plane to the graph of the function z = f (x, y) at the point P (x0 , y0 , f (x0 , y0 )) is
z − f (x0 , y0 ) = fx (x0 , y0 )(x − x0 ) + fy (x0 , y0 )(y − y0 ).
• The line through the point P (x0 , y0 , f (x0 , y0 )) parallel to the vector n is perpendicular to the above tangent
plane. This line is called the normal line to the surface z = f (x, y) at P .
• The differential dz (or the total differential): dz =
∂z
∂z
dx +
dy.
∂x
∂y
• ∆z ≈ dz.
• f (a + ∆x, b + ∆y) ≈ f (a, b) + dz(a, b)
18. Sketch the circular paraboloid z = 7 − x2 − y 2 to see why the tangent plane at its vertex
should be horizontal. Confirm that by finding the equation of this tangent plane.
19. Find an equation of the tangent plane to the graph of the function z = ln(3x + 2y) at the
point (−1, 2, 0).
20. Find parametric equations for the normal line to the surface z = ex+y at the point P (1, −2, 1/e).
21. Find an equation of the tangent plane to the surface z = x2 + 3y 3 which is parallel to the
plane 2x + y − z = 1.
22. Where does the plane tangent to the surface z = ex−y at (1, 1, 1) meet the z -axis?
23. Find the differential of the function
v = y ln(cos(x2 + 3))
24. Find the differential of the function
f (x, y, z) =
p
x2 + y 2 + z 2
25. Use differentials to estimate
p
(4.01)2 + (3.98)2 + (2.02)2
26. The two legs of a right triangle are measured as 5 m and 12 m respectively, with a possible
error in measurement of at most 0.2 cm in each. Use differentials to estimate the maximum
error in the calculated value of
(a) the area of the triangle
(b) the length of the hypotenuse.