GATE 2022 MA – Question 60

Calculus · Multivariable Differentiation and Optimization · 2 marks · Numerical answer

For f(x,y,z)=x+x cos z−y sin z+y, find the rate of change at P₀=(2,−1,0) toward P₁=(0,1,2).

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: 0 to 0

Explanation

The gradient is (1+cos z,1−sin z,−x sin z−y cos z), giving (2,1,1) at P₀. The displacement is (−2,2,2), with unit vector (−1,1,1)/√3. Its dot product with the gradient is (−2+1+1)/√3=**0**.

More Multivariable Differentiation and Optimization questions

GATE 2021 MA Q25 – Differentiable f:ℝ²→ℝ has directional derivatives at zero D u f=√5 and D v f=√2, where…
GATE 2021 MA Q47 – Define f(x,y)=√(x²+y²) sin(y²/x) for x≠0 and zero for x=0. P:f is continuous but not…
GATE 2021 MA Q49 – For f(x,y)=4xy−2x²−y⁴, which description holds?
GATE 2026 MA Q22 – f(x,y)=xx2+y2/ x for x ≠ 0 and 0 for x = 0. Which statements hold at (0,0)?
GATE 2025 MA Q18 – g(x,y)=f(x,y)e(2x+3y), where f is a nonzero C¹ homogeneous function of degree four on ℝ².…
GATE 2025 MA Q26 – For (x,y)≠(0,0), f=(x2-y2)/(x2+y2)+x[1/(x2+y2)]. S1: lim (x→0) lim (y→0) f exists; S2: lim…

All Multivariable Differentiation and Optimization questions →