Section 4
Use of the Partial Derivatives
Practise Questions
For the following functions (1-4) find the
- marginal functions
- slope of the level set \(\frac{dy}{dx}\)
- \(MRS_{x,y}\)
1. \(f(x,y) = x^{0.5}y^{0.5} = 10\)
-
\(f_{x} = 0.5x^{-0.5}y^{0.5}\)
\(f_{y} = 0.5x^{0.5}y^{-0.5}\) - \(\frac{dy}{dx} = -\frac{f_{x}}{f_{y}} = -\frac{y}{x}\)
- \(MRS_{x,y} = -\frac{dy}{dx} = \frac{y}{x}\)
2. \(f(x,y) = x^{0.6}y^{0.4}\)
-
\(f_{x} = 0.6x^{-0.4}y^{0.4}\)
\(f_{y} = 0.4x^{0.6}y^{-0.6}\) - \(\frac{dy}{dx} = -\frac{f_{x}}{f_{y}} = -\frac{3y}{2x}\)
- \(MRS_{x,y} = -\frac{dy}{dx} = \frac{3y}{2x}\)
3. \(f(x,y) = 10x^{0.5}y^{0.5}\)
-
\(f_{x} = 5x^{-0.5}y^{0.5}\)
\(f_{y} = 5x^{0.5}y^{-0.5}\) - \(\frac{dy}{dx} = -\frac{f_{x}}{f_{y}} = -\frac{y}{x}\)
- \(MRS_{x,y} = -\frac{dy}{dx} = \frac{y}{x}\)
4. \(f(x,y) = Ax^{\alpha }y^{\beta }\)
-
\(f_{x} = A \alpha x^{\alpha - 1}y^{\beta }\)
\(f_{y} = A \beta x^{\alpha }y^{\beta - 1}\) - \(\frac{dy}{dx} = -\frac{f_{x}}{f_{y}} = -\frac{\alpha y}{\beta x}\)
- \(MRS_{x,y} = -\frac{dy}{dx} = \frac{\alpha y}{\beta x}\)
5. Find the critical points at which the function may be optimized:
\(f(x,y) = 5x^{2} - 1.5y^{2} - 30x - 4y + 5xy\)
Set first-order partial derivatives equal zero: $$f_{x} = 10x + 5y - 30 = 0 \quad \text{(1)}$$ $$f_{y} = 5x - 3y - 4 = 0 \qquad \text{(2)}$$ Multiply equation (2) by \(2\) and deduct from equation (1): $$11y = 22$$ $$y = 2$$ $$x = 2$$ The function may be optimized at \((2, 2)\).
6. Find the critical points at which the function may be optimized:
\(f(x,y) = 5x^{2} - 3y^{2} - 30x + 7y + 4xy\) [See Dowling P. 99]
Set first-order partial derivatives equal zero: $$f_{x} = 10x + 4y - 30 = 0 \quad \text{(1)}$$ $$f_{y} = 4x - 6y + 7 = 0 \qquad \text{(2)}$$ Multiply equation (2) by \(2.5\) and deduct from equation (1): $$19y = 47.5$$ $$y = 2.5$$ $$x = 2$$ The function may be optimized at \((2, 2.5)\).
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