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.
A 20 foot ladder rests on level ground; its head (top) is against a vertical wall. The bottom of the ladder begins by being 12 feet from the wall but begins moving away at the rate of 0.1 feet per second. At what rate is the top of the ladder slipping down the wall? You may use a calculator.
Explain the focus and reasons for establishment of 12.4.1(root test) and 12.4.2(ratio test)
use Integration by Parts to derive 12.6.1
Chapter 15 Solutions
MyLab Math with Pearson eText -- Standalone Access Card -- for Calculus: Early Transcendentals (3rd Edition)
Elementary Statistics: Picturing the World (7th Edition)
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