Nonuniform straight-line motion Consider the motion of an object given by the position function r ( t ) = f ( t ) 〈 a , b , c 〉 + 〈 x 0 , y 0 , z 0 〉 , for t ≥ 0 , where a, b, c , x 0 , y 0 , and z 0 are constants, and f is a differentiable scalar function, for t ≥ 0. a. Explain why this function describes motion along a line. b. Find the velocity function. In general, is the velocity constant in magnitude or direction along the path?
Nonuniform straight-line motion Consider the motion of an object given by the position function r ( t ) = f ( t ) 〈 a , b , c 〉 + 〈 x 0 , y 0 , z 0 〉 , for t ≥ 0 , where a, b, c , x 0 , y 0 , and z 0 are constants, and f is a differentiable scalar function, for t ≥ 0. a. Explain why this function describes motion along a line. b. Find the velocity function. In general, is the velocity constant in magnitude or direction along the path?
Solution Summary: The author explains why the function describes motion along a line.
Nonuniform straight-line motion Consider the motion of an object given by the position function
r
(
t
)
=
f
(
t
)
〈
a
,
b
,
c
〉
+
〈
x
0
,
y
0
,
z
0
〉
,
for
t
≥
0
,
where a, b, c, x0, y0, and z0 are constants, and f is a differentiable scalar function, for t ≥ 0.
a. Explain why this function describes motion along a line.
b. Find the velocity function. In general, is the velocity constant in magnitude or direction along the path?
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.
Nonuniform straight-line motion Consider the motion of an object given by the position function r(t) = ƒ(t)⟨a, b, c⟩ + ⟨x0, y0, z0⟩, for t ≥ 0,where a, b, c, x0, y0, and z0 are constants, and ƒ is a differentiable scalar function, for t ≥ 0.a. Explain why r describes motion along a line.b. Find the velocity function. In general, is the velocity constant in magnitude or direction along the path?
Calculus
Assume that the level surface equation x3+y3+z3+6xyz = 1 defines z implicitly as a function of x and y. Find zx(0, −1) and zy(0, −1). Use that information to find the equation of the plane tangent to the given level surface at the point corresponding to x = 0 and y = −1−1.
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