Mathematics for Economists by Simon Blume 习题解答 answers1
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2.3 1, 5, Ϫ2, 0.
2.4 a) x 1; b) x Ͼ 1; c) all x; d) x Ϯ1; e) Ϫ1 Յ x Յ ϩ1; f ) Ϫ1 Յ x Յ ϩ1, x 0.
2.5 a) x 1, b) all x, c) x Ϫ1, Ϫ2, d) all x.
2.6 The most common functions students come up with all have the nonnegative real numbers for their domain.
d) Here m ϭ Ϫ2, so the function is of the form f (x) ϭ Ϫ2x ϩ b. When x ϭ 2, f (x) ϭ Ϫ2, thus b has to solve the equation Ϫ2 ϭ Ϫ2 и 2 ϩ b, so b ϭ 2 and f (x) ϭ Ϫ2x ϩ 2.
iii) y ϭ x2 ϩ 1 has a global minimum of 1 at x ϭ 0. It is decreasing on (Ϫϱ, 0) and increasing on (0, ϱ). See figure.
iv) y ϭ x3 ϩ x is increasing everywhere, and has no local maxima or minima. See figure.
ANSWERS PAMPHLET 1
Chapter 2
2.1 i) y ϭ 3xϪ2 is increasing everywhere, and has no local maxima or minima. See figure.*
ii) y ϭ Ϫ2x is decreasing everywhere, and has no local maxima or minima. See figure.
2.8 a) 1, b) Ϫ1, c) 0, d) 3.
2.8 a) The general form of a linear function is f (x) ϭ mx ϩ b, where b is the y-intercept and m is the slope. Here m ϭ 2 and b ϭ 3, so the formula is f (x) ϭ 2x ϩ 3.
b) The slope is the marginal cost, that is, the rate at which the cost of purchasing x units increases with x.
c) The slope is the rate at which demand increases with price.
vi) y ϭ |x| decreases on (Ϫϱ, 0) and increases on (0, ϱ). It has a global minimum of 0 at x ϭ 0. See figure.
2.2 Increasing functions include production and supply functions. Decreasing functions include demand and marginal utility. Functions with global critical points include average cost functions when a fixed cost is present, and profit functions.
f ) m ϭ [3 Ϫ (Ϫ4)] (0 Ϫ 2) ϭ Ϫ7 2, and we are given that b ϭ 3, so f (x) ϭ Ϫ(7 2)x ϩ 3.
2.9 a) The slope is the marginal revenue, that is, the rate at which revenue increases with output.
e) We need to compute m and b. Recall that given the value of f (x) at two points, m equals the change in f (x) divided by the change in x. Here m ϭ (5 Ϫ 3) (4 Ϫ 2) ϭ 1. Now b solves the equation 3 ϭ 1 и 2 ϩ b, so b ϭ 1 and f (x) ϭ x ϩ 1.
d) The slope is the marginal propensity to consume, that is, the rate at which aggregate consumption increases with national income.
v) y ϭ x3 Ϫ x has a local maximum of 2 3 3 at Ϫ1 3, and a local minimum of Ϫ2 3 3 at 1 3, but no global maxima or minima. It increases on (Ϫϱ, Ϫ1 3) and (1 3, ϱ) and decreases in between. See figure.
b) Here m ϭ Ϫ3 and b ϭ 0, so the formula is f (x) ϭ Ϫ3x.
*All figures are included at the back of the pamphlet.
2 MATHEMATICS FOR ECONOMISБайду номын сангаасS
c) We know m but need to compute b. Here m ϭ 4, so the function is of the form f (x) ϭ 4x ϩ b. When x ϭ 1, f (x) ϭ 1, so b has to solve the equation 1 ϭ 4 и 1 ϩ b. Thus, b ϭ Ϫ3 and f (x) ϭ 4x Ϫ 3.
2.4 a) x 1; b) x Ͼ 1; c) all x; d) x Ϯ1; e) Ϫ1 Յ x Յ ϩ1; f ) Ϫ1 Յ x Յ ϩ1, x 0.
2.5 a) x 1, b) all x, c) x Ϫ1, Ϫ2, d) all x.
2.6 The most common functions students come up with all have the nonnegative real numbers for their domain.
d) Here m ϭ Ϫ2, so the function is of the form f (x) ϭ Ϫ2x ϩ b. When x ϭ 2, f (x) ϭ Ϫ2, thus b has to solve the equation Ϫ2 ϭ Ϫ2 и 2 ϩ b, so b ϭ 2 and f (x) ϭ Ϫ2x ϩ 2.
iii) y ϭ x2 ϩ 1 has a global minimum of 1 at x ϭ 0. It is decreasing on (Ϫϱ, 0) and increasing on (0, ϱ). See figure.
iv) y ϭ x3 ϩ x is increasing everywhere, and has no local maxima or minima. See figure.
ANSWERS PAMPHLET 1
Chapter 2
2.1 i) y ϭ 3xϪ2 is increasing everywhere, and has no local maxima or minima. See figure.*
ii) y ϭ Ϫ2x is decreasing everywhere, and has no local maxima or minima. See figure.
2.8 a) 1, b) Ϫ1, c) 0, d) 3.
2.8 a) The general form of a linear function is f (x) ϭ mx ϩ b, where b is the y-intercept and m is the slope. Here m ϭ 2 and b ϭ 3, so the formula is f (x) ϭ 2x ϩ 3.
b) The slope is the marginal cost, that is, the rate at which the cost of purchasing x units increases with x.
c) The slope is the rate at which demand increases with price.
vi) y ϭ |x| decreases on (Ϫϱ, 0) and increases on (0, ϱ). It has a global minimum of 0 at x ϭ 0. See figure.
2.2 Increasing functions include production and supply functions. Decreasing functions include demand and marginal utility. Functions with global critical points include average cost functions when a fixed cost is present, and profit functions.
f ) m ϭ [3 Ϫ (Ϫ4)] (0 Ϫ 2) ϭ Ϫ7 2, and we are given that b ϭ 3, so f (x) ϭ Ϫ(7 2)x ϩ 3.
2.9 a) The slope is the marginal revenue, that is, the rate at which revenue increases with output.
e) We need to compute m and b. Recall that given the value of f (x) at two points, m equals the change in f (x) divided by the change in x. Here m ϭ (5 Ϫ 3) (4 Ϫ 2) ϭ 1. Now b solves the equation 3 ϭ 1 и 2 ϩ b, so b ϭ 1 and f (x) ϭ x ϩ 1.
d) The slope is the marginal propensity to consume, that is, the rate at which aggregate consumption increases with national income.
v) y ϭ x3 Ϫ x has a local maximum of 2 3 3 at Ϫ1 3, and a local minimum of Ϫ2 3 3 at 1 3, but no global maxima or minima. It increases on (Ϫϱ, Ϫ1 3) and (1 3, ϱ) and decreases in between. See figure.
b) Here m ϭ Ϫ3 and b ϭ 0, so the formula is f (x) ϭ Ϫ3x.
*All figures are included at the back of the pamphlet.
2 MATHEMATICS FOR ECONOMISБайду номын сангаасS
c) We know m but need to compute b. Here m ϭ 4, so the function is of the form f (x) ϭ 4x ϩ b. When x ϭ 1, f (x) ϭ 1, so b has to solve the equation 1 ϭ 4 и 1 ϩ b. Thus, b ϭ Ϫ3 and f (x) ϭ 4x Ϫ 3.