Find the local linear approximation L to the function f(x,y)=ln
xy at the point P(1,2). Compare the error in approximating f by
L at the point Q(1.01,2.01) with thw distance between P and Q
Answers
Answer:
Step-by-step explanation:
Linear approximation, also known as linearization, is a method for estimating the value of a function at a given point. Liner approximation is useful because determining the value of a function at a specific point can be difficult. A good example of this is square roots.
f(x,y) = ln(xy)
(x₀,y₀) = (1,2)
f(x₀,y₀) = ln2
⇒ fₓ(x₀,y₀) = 1/xy (y) = 1/x
fₓ(1,2) = 1
⇒ fy(x₀,y₀) = 1/xy (x) = 1/y
fy(1,2) = 1/2
L(x₀,y₀) = f(x₀,y₀) + fₓ(x₀,y₀) (x-x₀) + fy(x₀,y₀) (y-y₀)
= ln2 + 1/x(x-1) + 1/y (y-2)
⇒ x + 1/2(y) + ln(2-2)
L(1.01,2.01) = 1.01 + 1/2 * 2.01 + ln2 = 0.7081471806
F(1.01,2.01) = 0.7080850529
error = |F-L| = 0.00006212767585
Distance between (1.01.2.01) and (1,2)
= √(1.01-1)² + (2.01-2)²
= 0.1 √2
= 0.1414213562
Error/distance = 0.0004393003003 <= 1/2300
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