Question

Let $f(x, y) = xe^{x^{2} - y}$ and $P = (8, 64)$. (a) Calculate $||\nabla f_{P}||$. (b) Find the rate of change of $f$ in the direction $\nabla f_{P}$. (c) Find the rate of change of $f$ in the direction of a vector making an angle of $45^{\circ}$ with $\nabla f_{P}$. Answers: (a) (b) (c)

          Let $f(x, y) = xe^{x^{2} - y}$ and $P = (8, 64)$. 
(a) Calculate $||\nabla f_{P}||$.
(b) Find the rate of change of $f$ in the direction $\nabla f_{P}$.
(c) Find the rate of change of $f$ in the direction of a vector making an angle of $45^{\circ}$ with $\nabla f_{P}$.
Answers:
(a)
(b)
(c)

Let f(x, y) = xe^x^2 - y and P = (8, 64).
(a) Calculate ||∇ fP||.
(b) Find the rate of change of f in the direction ∇ fP.
(c) Find the rate of change of f in the direction of a vector making an angle of 45^∘ with ∇ fP.
Answers:
(a)
(b)
(c)

Added by Kenneth B.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Adi S

07/15/2023

Step 1

The gradient of f(x, y) is given by ∇f = (∂f/∂x, ∂f/∂y). Taking the partial derivatives of f(x, y) with respect to x and y, we get: ∂f/∂x = ex² + 2xex² ∂f/∂y = -1 Show more…

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Let f(x, y) = xex² − y and P = (8, 64). (a) Calculate ||∇fP||. (b) Find the rate of change of f in the direction ∇fP. (c) Find the rate of change of f in the direction of a vector making an angle of 45° with ∇fP. Answers: (a) (b) (c)