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A function f and a point P are given. Let correspond to the direction of the directional derivative. Complete parts a f(x,y)= In (1+6x²+5y²), P.- √5] a. Find the gradient and evaluate it at P. 24 80-5 211 211 b. Find the angles + (with respect to the positive x-axis) between 0 and 2x associated with the directions of maximum 10./5 The gradient at P is tan + 2x (Type your answer in radians. Type an exact answer in terms of Use a comma to separate answers as needed.) What angles are associated with the direction of maximum decrease? 10,5 tan +T

A function f and a point P are given. Let correspond to the direction of the directional derivative. Complete parts a f(x,y)= In (1+6x²+5y²), P.- √5] a. Find the gradient and evaluate it at P. 24 80-5 211 211 b. Find the angles + (with respect to the positive x-axis) between 0 and 2x associated with the directions of maximum 10./5 The gradient at P is tan + 2x (Type your answer in radians. Type an exact answer in terms of Use a comma to separate answers as needed.) What angles are associated with the direction of maximum decrease? 10,5 tan +T

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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A function f and a point P are given. Let correspond to the direction of the directional derivative. Complete parts a. through e. 1 f(x,y)= In (1+6x²+6y²), P-√6) 4 a. Find the gradient and evaluate it at P. 24 211 80 5 211 b. Find the angles (with respect to the positive x-axis) between 0 and 2x associated with the directions of maximum increase, maximum decrease, and zero change. What angles are associated with the direction of maximum increase? - (-10,5) -+27 3 (Type your answer in radians. Type an exact answer in terms of Use a comma to separate answers as needed.) What angles are associated with the direction of maximum decrease? The gradient at Pis tan tan 10,5 3 + T (Type your answer in radians. Type an exact answer in terms of Use a comma to separate answers as needed.) What angles are associated with the direction of zero change? - 10/5)+.tan - (-10,5)+370 1 tan 3 32 (Type your answer in radians. Type an exact answer in terms of c. Write the directional derivative at P as a function of ; call this function g(*). 24 80 √5 g(0) 211 cos- 211 sinė (Simplify your answer. Type an exact answer, using radicals as needed.) d. Find the value of between 0 and 2x that maximizes g(0) and find the maximum value. What value of maximizes g(+)? e= tan 3+2TT. (Type your answer in radians. Type an exact answer in terms of ) What is the maximum value? Use a comma to separate answers as needed.) 10√5 Yes O No (...) 8/509 g(*)= 211 (Type an exact answer, using radicals as needed.) e. Verify that the value of that maximizes g corresponds to the direction of the gradient. Verify that the maximum value of g equals the magnitude of the gradient. Are the values from part d consistent with the values from parts a and b?
Transcribed Image Text:A function f and a point P are given. Let correspond to the direction of the directional derivative. Complete parts a. through e. 1 f(x,y)= In (1+6x²+6y²), P-√6) 4 a. Find the gradient and evaluate it at P. 24 211 80 5 211 b. Find the angles (with respect to the positive x-axis) between 0 and 2x associated with the directions of maximum increase, maximum decrease, and zero change. What angles are associated with the direction of maximum increase? - (-10,5) -+27 3 (Type your answer in radians. Type an exact answer in terms of Use a comma to separate answers as needed.) What angles are associated with the direction of maximum decrease? The gradient at Pis tan tan 10,5 3 + T (Type your answer in radians. Type an exact answer in terms of Use a comma to separate answers as needed.) What angles are associated with the direction of zero change? - 10/5)+.tan - (-10,5)+370 1 tan 3 32 (Type your answer in radians. Type an exact answer in terms of c. Write the directional derivative at P as a function of ; call this function g(*). 24 80 √5 g(0) 211 cos- 211 sinė (Simplify your answer. Type an exact answer, using radicals as needed.) d. Find the value of between 0 and 2x that maximizes g(0) and find the maximum value. What value of maximizes g(+)? e= tan 3+2TT. (Type your answer in radians. Type an exact answer in terms of ) What is the maximum value? Use a comma to separate answers as needed.) 10√5 Yes O No (...) 8/509 g(*)= 211 (Type an exact answer, using radicals as needed.) e. Verify that the value of that maximizes g corresponds to the direction of the gradient. Verify that the maximum value of g equals the magnitude of the gradient. Are the values from part d consistent with the values from parts a and b?
Expert Solution
Step 1

Given the function fx,y=ln1+6x2+5y2 and point P as P14,-5.

# As per the guidelines we are entitled to solve one question at a time, please resubmit the other question 

(a) The gradient of the function is given by ∇f=∂f∂xi+∂f∂yj

Therefore,

fx,y=ln1+6x2+5y2∂f∂x=11+6x2+5y212x∂f∂y=11+6x2+5y210y

Find the value at P14,-5.

 

 

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