GATE 2018 EC Q.52: solution and answer

GATE 2018 · ECQ.52 · 2 marks · NAT

Let $r = x^{2} + y - z$ and $z^{3} - x y + y z + y^{3} = 1$ . Assume that $x$ and $y$ are independent variables. At $(x, y, z) = (2, - 1, 1)$ , the value (correct to two decimal places) of $\frac{\partial r}{\partial x}$ is ________

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Answer and solution

ANSWER

$\frac{\partial r}{\partial x} = 4.50$ at $(2, - 1, 1)$ .

Official answer: 4.4 to 4.6Source: Official answer key, GATE 2018 EC (IIT Guwahati)

Since $x$ and $y$ are independent, the constraint $z^{3} - x y + y z + y^{3} = 1$ fixes $z$ as an implicit function $z (x, y)$ . So $r = x^{2} + y - z$ depends on $x$ in two ways: directly, and through $z$ . With $y$ held constant, $\frac{\partial r}{\partial x} = 2 x - \frac{\partial z}{\partial x}$
Check that the point lies on the surface: $1^{3} - (2) (- 1) + (- 1) (1) + (- 1)^{3} = 1 + 2 - 1 - 1 = 1$ . It does.
Differentiate the constraint implicitly with respect to $x$ , holding $y$ constant: $3 z^{2} \frac{\partial z}{\partial x} - y + y \frac{\partial z}{\partial x} = 0$
Solve for the derivative: $\frac{\partial z}{\partial x} = \frac{y}{3 z ^{2} + y}$ At the given point the denominator is $3 (1)^{2} + (- 1) = 2 \neq = 0$ , so $z (x, y)$ is well defined and differentiable there.
At $(2, - 1, 1)$ : $\frac{\partial z}{\partial x} = \frac{- 1}{3 ( 1 ) ^{2} + ( - 1 )} = \frac{- 1}{2} = - 0.5$ .
Substitute: $\frac{\partial r}{\partial x} = 2 (2) - (- 0.5) = 4 + 0.5 = 4.5$
Answer: $\frac{\partial r}{\partial x} = 4.50$ .

KEY IDEA

$z$ is not an independent variable. The constraint makes it a function of $x$ and $y$ , so $\frac{\partial r}{\partial x} = \frac{\partial r}{\partial x}_{direct} + \frac{\partial r}{\partial z} \frac{\partial z}{\partial x}$ . Find $\frac{\partial z}{\partial x}$ by implicit differentiation: $\frac{\partial z}{\partial x} = - \frac{F _{x}}{F _{z}}$ , where $F = z^{3} - x y + y z + y^{3} - 1$ .

WATCH OUT FOR

Treating $z$ as a constant gives $\frac{\partial r}{\partial x} = 2 x = 4$ , which is outside the key range. The other common slip is a sign error in $- \frac{F _{x}}{F _{z}}$ : here $F_{x} = - y$ , so $\frac{\partial z}{\partial x} = + \frac{y}{3 z ^{2} + y} = - 0.5$ . Dropping the minus sign gives $+ 0.5$ and a wrong answer of $3.5$ .

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GATE 2018 EC Q.52: solution and answer

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