c Dr Oksana Shatalov, Spring 2014
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Spring 2014 Math 251
Week in Review 5
courtesy: Oksana Shatalov
(covering Sections 12.6-12.7)
12.6: Directional Derivatives and the Gradient Vector
Key Points
• The gradient of f (x, y) is the vector function ∇f defined by ∇f (x, y) = hfx (x, y), fy (x, y)i =
• The gradient of f (x, y, z) is ∇f (x, y, z) = hfx (x, y, z), fy (x, y, z), fz (x, y, z)i =
∂f
∂x i
+
∂f
∂y j
+
∂f
∂x i
+
∂f
∂y j.
∂f
∂z k.
• If f has continuous partial derivatives, then f has a directional derivative in the direction of any unit vector
û and Du f = ∇f · û.
• The maximum value of the directional derivative (or maximum rate of change) Du f of function f (which has
continuous partial derivatives) is |∇f | and it occurs when u has the same direction as the gradient vector
∇f .
1. Consider four points A(1, 2), B(3, 2), C(1, 12), and D(12, 5). Given:
~ is 10.
• The directional derivative of f (x, y) at A in the direction of the vector AB
~ is 6.
• The directional derivative of f (x, y) at A in the direction of the vector AC
~
Find the directional derivative of f (x, y) at A in the direction of the vector AD.
2. Let f (x, y, z) = zexy .
(a) Find ∇f (0, 1, 2) , the gradient of the function at (0, 1, 2).
(b) Find the directional derivative of f at (0, 1, 2) in the direction from this point to the
point (4, 5, 6).
(c) Find the direction in which f (x, y, z) increases most rapidly at (0, 1, 2).
(d) What is the maximum rate of increase of this function at the point (0, 1, 2)?
(e) What is the largest rate of decrease of f at this point? In which direction does this
change occur?
(f) When the directional derivative at this point is equal to half of its maximum value?
3. Find an equation of the tangent plane to the surface ln(xyz) = x + y + z 2 + 1 at (−1, −1, 1)
4. A ladybug at the point (1, 1, 1) on the egg x2 + y 2 + 2z 2 = 4 flies off along the normal line.
Does it reach a flower at (5, 5, 4)?
c Dr Oksana Shatalov, Spring 2014
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5. On a certain mountain, the elevation z above a point (x, y) in the xy-plane at sea level is
z = 1000 − 0.01x2 − 0.02y 2 , where x, y and z are in meters. The positive x-axis points East,
and the positive y-axis North. A climber is at the point with coordinates (60, 100, 764).
(a) If the climber uses compass reading to walk due West, will he begin to ascend or
descend?
(b) If the climber uses compass reading to walk due NorthEast, will he begin to ascend or
descend? At what rate?
(c) In which direction should the climber proceed initially in order to reach the top of the
hill fastest?
(d) If the climber proceeds in that direction, at what angle above the horizontal will he be
climbing initially?
12.7: Maximum and minimum values
Key Points
• If f has a local extremum (that is, a local maximum or minimum) at (a, b) and the first-order partial
derivatives exist there, then fx (a, b) = fy (a, b) = 0 (or, equivalently,∇f (a, b) = 0.)
• A point (a, b) such that fx (a, b) = 0 and fy (a, b) = 0 , or one of this partial derivatives does not exist, is
called a critical point of f .
• At critical point, a function could have a local max or a local min, or neither.
• THE EXTREME VALUE THEOREM:If f is continuous on a closed bounded set D in R2 , then f attains
an absolute maximum value f (x1 , y1 ) and an absolute minimum value f (x2 , y2 ) at some points (x1 , y1 ) and
(x2 , y2 ) in D.
• Suppose that the second partial derivatives of f are continuous near (a, b) and ∇f (a, b) = 0 (i.e. (a, b) is a
f
xx fxy 2
critical point). Let D = D(a, b) = fxx (a, b)fyy (a, b) − [fxy (a, b)] (= )
fxy fyy – If D > 0 and fxx (a, b) < 0 then f (a, b) is a local maximum.
– If D > 0 and fxx (a, b) > 0 then f (a, b) is a local minimum.
– If D < 0 then f (a, b) is not a local extremum (saddle point).
– If D = 0 or does not exist, then the test gives no information. fails.
6. Find three positive numbers whose sum is 45 and such that their product is as large as
possible.
7. A lamina occupies the region D which is the region in the first quadrant bounded by the
parabola y = x2 and the line y = 4. The temperature at each point of the lamina is given
by T (x, y) = xy − x − y + 20. Find the hottest and coldest points of the lamina.
8. The surface z = 4xy − x4 − y 4 has two peaks and one pass. Find them.
c Dr Oksana Shatalov, Spring 2014
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9. Find the top of the mountain z = x2 y + xy 2 + y 2 − y + 2000 defined in a square domain D
in the xy-plane with vertices (0, 0), (0, 2), (2, 0) and (2, 2).
10. Find the absolute minimum and absolute maximum of f (x, y) = 2x2 + y 2 + y + 1 on the
region D = {(x, y)|x2 + y 2 ≤ 1, x ≥ 0, y ≥ 0} .
11. Given the function f (x, y) = (x2 + y 2 )ey + 2014.
(a) Locate the critical points.
(b) Classify the critical points of f (i.e. local maximum, local minimum or saddle).