Math Problem Statement
Funksjonen T(x,y,z)=10⋅e−x2−y2−z2 gir temperaturen for punkter i et rom. Hvis man befinner seg i punktet (−2,4,2) , hvilken retning skal man bevege seg for å oppleve størst mulig økning i temperatur? videre, Hvis man beveger seg i denne retningen, hva er stigningstallet for temperaturen?
Solution
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Math Problem Analysis
Mathematical Concepts
Multivariable Calculus
Gradient
Vector Calculus
Formulas
T(x, y, z) = 10 * e^(-x^2 - y^2 - z^2)
Gradient: ∇T(x, y, z) = (-20x * e^(-x^2 - y^2 - z^2), -20y * e^(-x^2 - y^2 - z^2), -20z * e^(-x^2 - y^2 - z^2))
Rate of Change: |∇T| = e^(-24) * sqrt(40^2 + (-80)^2 + (-40)^2)
Theorems
Gradient points in the direction of steepest ascent in scalar fields.
Suitable Grade Level
Undergraduate Calculus (or Advanced High School Math)
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