(a) Use a CAS to graph f x , y = x 2 + 3 y 2 e − x 2 + y 2 . (b) At how many points do you think it is true that D u f x , y = 0 for all unit vectors u? (c) Use a CAS to find ∇ f . (d) Use a CAS to solve the equation ∇ f x , y = 0 for x and y . (e) Use the result in part (d) together with Theorem 13.6.5 to check your conjecture in part (b).
(a) Use a CAS to graph f x , y = x 2 + 3 y 2 e − x 2 + y 2 . (b) At how many points do you think it is true that D u f x , y = 0 for all unit vectors u? (c) Use a CAS to find ∇ f . (d) Use a CAS to solve the equation ∇ f x , y = 0 for x and y . (e) Use the result in part (d) together with Theorem 13.6.5 to check your conjecture in part (b).
(a) Use a CAS to graph
f
x
,
y
=
x
2
+
3
y
2
e
−
x
2
+
y
2
.
(b) At how many points do you think it is true that
D
u
f
x
,
y
=
0
for all unit vectors u?
(c) Use a CAS to find
∇
f
.
(d) Use a CAS to solve the equation
∇
f
x
,
y
=
0
for
x
and
y
.
(e) Use the result in part (d) together with Theorem 13.6.5 to check your conjecture in part (b).
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.
I'm still confused on the position vector. The question is Let r(t) be the position vector corresponding with any point P on C. Write the position vector in component form. So does the position vector for this would be MP=<a,b>?
Consider the function f(x.y) = 4x? - 5y? - 9 and the point (- 2, - 1).
a. Find the unit vectors that give the direction of steepest ascent and steepest descent at P.
b. Find a vector that points in a direction of no change in the function at P.
a. What is the unit vector in the direction of steepest ascent at P?
(Type exact answers, using radicals as needed.)
Consider the following scenario.
A pilot is steering a plane in the direction N 45° W at an air-speed (speed still in air) of 150 mi/h. A wind is blowing in the direction S 30° E at a speed of 34 mi/h.
Set up the coordinate axes so that north is the positive y-direction and west is the negative x-direction. With respect to the still air, write a vector that represents the velocity of the plane and a vector that represents the velocity of the wind.
University Calculus: Early Transcendentals (3rd Edition)
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