Directional derivatives Consider the function f ( x.y ) = 8– x 2 /2 – y 2 . whose graph is a paraboloid (see figure). a. Fill in the table with the values of the directional derivative at the points ( a, b ) in the directions given by the unit vectors u . v . and w . ( a,b ) = ( 2,0 ) ( a , b ) = (0,2) ( a , b ) = (1,1) u = 〈 2 2 , 2 2 〉 v = 〈 − 2 2 , 2 2 〉 w = 〈 − 2 2 , − 2 2 〉 b. Interpret each of the directional derivatives computed in pan (a) at the point (2,0).
Directional derivatives Consider the function f ( x.y ) = 8– x 2 /2 – y 2 . whose graph is a paraboloid (see figure). a. Fill in the table with the values of the directional derivative at the points ( a, b ) in the directions given by the unit vectors u . v . and w . ( a,b ) = ( 2,0 ) ( a , b ) = (0,2) ( a , b ) = (1,1) u = 〈 2 2 , 2 2 〉 v = 〈 − 2 2 , 2 2 〉 w = 〈 − 2 2 , − 2 2 〉 b. Interpret each of the directional derivatives computed in pan (a) at the point (2,0).
f(x.y) = 8–x2/2 – y2. whose graph is a paraboloid (see figure).
a. Fill in the table with the values of the directional derivative at the points (a, b) in the directions given by the unit vectorsu. v. and w.
(a,b) = (2,0)
(a, b) = (0,2)
(a, b) = (1,1)
u
=
〈
2
2
,
2
2
〉
v
=
〈
−
2
2
,
2
2
〉
w
=
〈
−
2
2
,
−
2
2
〉
b. Interpret each of the directional derivatives computed in pan (a) at the point (2,0).
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
Interpreting directional derivatives Consider the functionƒ(x, y) = 3x2 - 2y2.a. Compute ∇ƒ(x, y) and ∇ƒ(2, 3).b. Let u = ⟨cos θ, sin θ⟩ be a unit vector. At (2, 3), for what values of θ (measured relative to the positive x-axis), with 0 ≤ θ < 2π, does the directional derivative have its maximum and minimum values? What are those values?
Both parts of this problem refer to the function f(x, y, z) = x2 + y3 + 5z.
(a) Find the directional derivative of f(x, y, z) at the point (1, 1, −2) in the direction of the vector <(1/√(3)), −(1/√(3)), (1/√(3))>(b) Find an equation of the tangent plane to the level surface of f for the function value −8 at the point (1, 1, −2).
VECTOR DIFFERENTIATION:
If R = e^(−t) i + ln(t^2+ 1) j - tant k. Find: (a) dR/dt, (b) d^2R/dt^2,(c) |dR/dt| ; (d) |d^2R/dt^2| at t = 0
Chapter 15 Solutions
MyLab Math with Pearson eText -- Standalone Access Card -- for Calculus: Early Transcendentals (3rd Edition)
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