A slope field of the form y ′ = f y is said to be autonomous . (a) Explain why the tangent segments along any horizontal line will be parallel for an autonomous slope field. (b) The word autonomous means "independent?' In what sense is an autonomous slope field independent? (c) Suppose that G y is an antiderivative of 1 / f y and that C is a constant. Explain why any differentiable function defined implicitly by G y − x = C will be a solution to the equation y ′ = f y .
A slope field of the form y ′ = f y is said to be autonomous . (a) Explain why the tangent segments along any horizontal line will be parallel for an autonomous slope field. (b) The word autonomous means "independent?' In what sense is an autonomous slope field independent? (c) Suppose that G y is an antiderivative of 1 / f y and that C is a constant. Explain why any differentiable function defined implicitly by G y − x = C will be a solution to the equation y ′ = f y .
A slope field of the form
y
′
=
f
y
is said to be autonomous.
(a) Explain why the tangent segments along any horizontal line will be parallel for an autonomous slope field.
(b) The word autonomous means "independent?' In what sense is an autonomous slope field independent?
(c) Suppose that
G
y
is an antiderivative of
1
/
f
y
and that C is a constant. Explain why any differentiable function defined implicitly by
G
y
−
x
=
C
will be a solution to the equation
y
′
=
f
y
.
Verify Stoke's Theorem for F =( x²)i +(xz)j + (xyz)k and C is the curve of intersection betweenz=1 and z² = x² + y²
Plot the slope field of y = x + y using slopes m = -3, -2,-1,0,1,2. Sketch integral curves for
y (0) -6,-4,-2,0, 2.
For reference: y = -2x-4+ (y(0)-4)*/2
The maximum value of directional derivative of f in the direction of b is given by ...
(a)Vf.6
(b) Vf x6
(c) Vf.vb
(d)f
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