Mathematics, 10.07.2019 13:20 lilly198o

Find the gradient of the function at the given point. function point f(x, y, z) = x2 + y2 + z2 (3, 9, 8)

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Answer from: tareas7009

$\displaystyle \nabla f(3, 9, 8) = 6 \hat{\i} + 18 \hat{\j} + 16 \hat{\text{k}}$

General Formulas and Concepts:

Calculus

Differentiation

DerivativesDerivative Notation

Derivative Rule [Basic Power Rule]:

f(x) = cxⁿf’(x) = c·nxⁿ⁻¹

Multivariable Calculus

Differentiation

Partial DerivativesDerivative Notation

Gradient: $\displaystyle \nabla f(x, y, z) = \frac{\partial f}{\partial x} \hat{\i} + \frac{\partial f}{\partial y} \hat{\j} + \frac{\partial f}{\partial z} \hat{\text{k}}$

Gradient Property [Addition/Subtraction]: $\displaystyle \nabla \big[ f(x) + g(x) \big] = \nabla f(x) + \nabla g(x)$

Gradient Property [Multiplied Constant]: $\displaystyle \nabla \big[ \alpha f(x) \big] = \alpha \nabla f(x)$

Step-by-step explanation:

Step 1: Define

Identify.

$\displaystyle f(x, y, z) = x^2 + y^2 + z^2$

$\displaystyle P(3, 9, 8)$

Step 2: Find Gradient

[Function] Differentiate [Gradient]: $\displaystyle \nabla f = \frac{\partial}{\partial x} \Big( x^2 + y^2 + z^2 \Big) \hat{\i} + \frac{\partial}{\partial y} \Big( x^2 + y^2 + z^2 \Big) \hat{\j} + \frac{\partial}{\partial z} \Big( x^2 + y^2 + z^2 \Big) \hat{\text{k}}$ [Gradient] Rewrite [Gradient Property - Addition/Subtraction]: $\displaystyle \nabla f = \bigg[ \frac{\partial}{\partial x}(x^2) + \frac{\partial}{\partial x}(y^2) + \frac{\partial}{\partial x}(z^2) \bigg] \hat{\i} + \bigg[ \frac{\partial}{\partial y}(x^2) + \frac{\partial}{\partial y}(y^2) + \frac{\partial}{\partial y}(z^2) \bigg] \hat{\j} + \bigg[ \frac{\partial}{\partial z}(x^2) + \frac{\partial}{\partial z}(y^2) + \frac{\partial}{\partial z}(z^2) \bigg] \hat{\text{k}}$ [Gradient] Differentiate [Derivative Rule - Basic Power Rule]: $\displaystyle \nabla f = 2x \hat{\i} + 2y \hat{\j} + 2z \hat{\text{k}}$ [Gradient] Substitute in point: $\displaystyle \nabla f(3, 9, 8) = 2(3) \hat{\i} + 2(9) \hat{\j} + 2(8) \hat{\text{k}}$ [Gradient] Evaluate: $\displaystyle \nabla f(3, 9, 8) = 6 \hat{\i} + 18 \hat{\j} + 16 \hat{\text{k}}$

∴ the gradient of the function at the given point is <6, 18, 16>.

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Learn more about multivariable calculus: link

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Topic: Multivariable Calculus

Unit: Directional Derivatives

Answer from: Quest

the answer is 2/54

step-by-step explanation:

Answer from: Quest

The answer is 270 cars.you can do this by making a proportional relationship. for instance, 200 is the whole we are using, so on a “number line” 200 would be over 100%. now, since he sells 35% more, add 35% to 100% to get 135% on the other side of the “number line” on the bottom half, leaving the top half empty.cross multiply 200 by 135 then divide by 100 to get your answer. i hope this , if it did, , if you don’t mind, giving me brainiest! 200/100% } 200 • 135 /100 ? /135%

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Find the gradient of the function at the given point. function point f(x, y, z) = x2 + y2 + z2 (3, 9...

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