Rules for gradients Use the definition of the gradient (in two or three dimensions), assume that f and g are differentiable functions on ¡ 2 or ¡ 3 , and let c be a constant. Prove the following gradient rules. a. Constants Rule: ▿ ( cf ) = c ▿ f b. Sum Rule : ▿ ( f + g ) = ▿ f + ▿ g c. Product Rule: ▿ ( fg ) = (▿ f ) g + f ▿ g d. Quotient Rule : ∇ ( f g ) = g ∇ f − f ∇ g g 2 e . Chain Rule: ∇ ( f ∘ g ) = f ’ ( g ) ∇ g , where f is a function of one variable
Rules for gradients Use the definition of the gradient (in two or three dimensions), assume that f and g are differentiable functions on ¡ 2 or ¡ 3 , and let c be a constant. Prove the following gradient rules. a. Constants Rule: ▿ ( cf ) = c ▿ f b. Sum Rule : ▿ ( f + g ) = ▿ f + ▿ g c. Product Rule: ▿ ( fg ) = (▿ f ) g + f ▿ g d. Quotient Rule : ∇ ( f g ) = g ∇ f − f ∇ g g 2 e . Chain Rule: ∇ ( f ∘ g ) = f ’ ( g ) ∇ g , where f is a function of one variable
Solution Summary: The author explains that the constant rule nabla is differentiable at the point (x,y,z).
Rules for gradients Use the definition of the gradient (in two or three dimensions), assume that f and g are differentiable functions on ¡2 or ¡3, and let c be a constant. Prove the following gradient rules.
a. Constants Rule: ▿ (cf) = c▿f
b. Sum Rule: ▿ (f + g) = ▿f + ▿g
c. Product Rule: ▿ (fg) = (▿f)g + f▿g
d. Quotient Rule:
∇
(
f
g
)
=
g
∇
f
−
f
∇
g
g
2
e. Chain Rule:
∇
(
f
∘
g
)
=
f
’
(
g
)
∇
g
, where f is a function of one variable
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.
Let f(x, y)
=
xex²-y and P = (3,9).
Calculate ||Vfpll.
(Express numbers in exact form. Use symbolic notation and fractions where needed.)
||Vfp|| =
Find the rate of change of f in the direction Vfp.
(Express numbers in exact form. Use symbolic notation and fractions where needed.)
the rate of change
Use the contour diagram of f to decide if the specified directional derivative is positive, negative, or approximately zero.
?
✓1. At the point (1, 0) in the direction of -3.
✓ 2. At the point (0, -2) in the direction of (2-23)/√5,
✓3. At the point (0, 2) in the direction of 3,
4. At the point (-1, 1) in the direction of (-7+3)/√2,
?
?
?
?
?
V
5. At the point (-1, 1) in the direction of (-7-3)/√2,
✓6. At the point (-2, 2) In the direction of 7,
V
>
2.4
1.6
0.8
0
0.8
-1.6-
-2.4
12.0
12,0
10.0
10.0
-2.4
6.0
-1.6 -0.8
0
X
0.8
4.0
(Click graph to enlarge)
1.6
12.0
10.0
8.0
10.0
12.0
2.4
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