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Mathematics for Economics

Michael Hoy; John Richard Livernois; C.J. McKenna

Chapter 11

Calculus for Functions of n Variables - all with Video Answers

Educators

Section 1

Partial Differentiation

Problem 1

Find the partial derivatives of the function

$$
y=3 x_1+5 x_2
$$

using definition 11.1 (see example 11.1).

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Problem 2

Find the partial derivatives of the function

$$
y=a x_1+b x_2
$$

where $a$ and $b$ are any constants, using definition 11.1 (see example 11.1).

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01:18

Problem 3

For the revenue function of example 11.1, $R\left(x_1, x_2\right)=p_1 x_1+p_2 x_2$, find the partial derivative $\partial R\left(x_1, x_2\right) / \partial x_2$ by using definition 11.1. Give an intuitive explanation of your result.

Lucas Finney
Lucas Finney
Numerade Educator
00:04

Problem 4

Discuss why it is the case that the partial derivatives in questions 1,2 , and 3 are constant functions.

Norman Atentar
Norman Atentar
Numerade Educator

Problem 5

For the function of example $11.2, y=x_1^2 x_2$, find the partial derivative $\partial y / \partial x_2$ by using definition 11.1.

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00:26

Problem 6

For the function

$$
y=x_1 x_2
$$

find the partial derivatives by using definition 11.1.

Linh Vu
Linh Vu
Numerade Educator
01:42

Problem 7

Find the marginal-product functions for the Cobb-Douglas production function

$$
y=10 x_1^{1 / 2} x_2^{1 / 3} x_3^{1 / 4}
$$

Lucas Finney
Lucas Finney
Numerade Educator
01:04

Problem 8

Find the marginal-product functions for the Cobb-Douglas production function

$$
y=A x_1^{\alpha_1} x_2^{\alpha_2} x_3^{\alpha_3} x_4^{\alpha_4}, \quad A>0,0<\alpha_i<1 \quad \text { for } i=1,2,3,4
$$

Carson Merrill
Carson Merrill
Numerade Educator

Problem 9

Find the marginal-product functions for the CES production function

$$
y=12\left[0.4 x_1^{-1 / 2}+0.6 x_2^{-1 / 2}\right]^{-2}
$$

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01:05

Problem 10

Find the marginal-product functions for the CES production function

$$
y=A\left[w_1 x_1^{-r}+w_2 x_2^{-r}+w_3 x_3^{-r}\right]^{-1 / r}
$$

$$
A>0, r>-1,0<w_i<1 \text { for } i=1,2,3 \text { and } w_1+w_2+w_3=1
$$

Adrian Co
Adrian Co
Numerade Educator

Problem 11

Suppose both that the amount of capital at time $t, K=K(t)$, and that the efficiency with which it is used affect gross national product according to the function

$$
Y=f(K, t)=0.2(1+t)^{1 / 2} K, \quad \text { where } K=K_0 e^{0.05 t}
$$

Find and give the economic intuition of the derivative

$$
\frac{d Y}{d t}=f_t+f_K \frac{d K}{d t}
$$

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01:00

Problem 12

In general, a function $f(\mathbf{x}), \mathbf{x} \in \mathbb{R}^n$, is said to be continuous at the point $\mathbf{x}=\mathbf{a}$ if there is some $\delta>0$ (possibly very small) such that $|f(\mathbf{x})-f(\mathbf{a})|<$ $\epsilon$ whenever $\|\mathbf{x}-\mathbf{a}\|<\delta$ for any $\epsilon>0$, where $\|$.$\| is the Euclidean$ distance (equivalently, $d(\mathbf{x}, \mathbf{a})$ ).

Show that according to this definition, the linear function $f\left(x_1, x_2\right)=$ $c_0+c_1 x_1+c_2 x_2, c_1, c_2 \neq 0$, is continuous at any point $\left(a_1, a_2\right) \in \mathbb{R}^2$.

Raj Bala
Raj Bala
Numerade Educator