Question

Let $f(x, y) = xe^{x^2 - y}$ and $P = (5, 25)$. (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 $P = (5, 25)$. 
(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 = (5, 25).
(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 Craig B.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Vincenzo Zaccaro

03/05/2022

Step 1

To find df/dx, we take the partial derivative of f with respect to x: df/dx = e^-y To find df/dy, we take the partial derivative of f with respect to y: df/dy = -x*e^-y So, ∇fP = (e^-25, -5*e^-25) Show more…

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11(3 Let f(x,y) = x*e^-y and P = (5,25). (a) Calculate ∇fP. (b) Find the rate of change of f in the direction of ∇fP. (c) Find the rate of change of f in the direction of a vector making an angle of 45° with ∇fP.