# Mathematics for Economists

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ANSWERS PAMPHLET 1 Chapter 2 2.1 i ) y 5 3 x 2 2 is increasing everywhere, and has no local maxima or minima. See ﬁgure.* ii ) y 52 2 x is decreasing everywhere, and has no local maxima or minima. See ﬁgure. iii ) y 5 x 2 1 1 has a global minimum of 1 at x 5 0. It is decreasing on ( 2‘ , 0) and increasing on (0 , ). See ﬁgure. iv ) y 5 x 3 1 x is increasing everywhere, and has no local maxima or minima. See ﬁgure. v ) y 5 x 3 2 x ha saloca lmax imumo f2 6 3 p 3a t 2 1 6 p 3, and a local minimum of 2 2 6 3 p t1 6 p 3, but no global maxima or minima. It increases on ( 2‘ , 2 1 6 p 3) and (1 6 p 3 , ) and decreases in between. See ﬁgure. vi ) y 5 | x | decreases on ( 2‘ , 0) and increases on (0 , ). It has a global minimum of 0 at x 5 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. 2.3 1, 5, 2 2, 0. 2.4 a ) x ± 1; b ) x . 1; c )a l l x ; d ) x ±6 1; e ) 2 1 # x #1 1; f ) 2 1 # x 1, x ± 0. 2.5 a ) x ± 1, b l l x , c ) x ±2 1 , 2 2, d l l x . 2.6 The most common functions students come up with all have the nonnegative real numbers for their domain. 2.8 a )1 , b ) 2 1, c )0 , d )3 . 2.8 a ) The general form of a linear function is f ( x ) 5 mx 1 b , where b is the y -intercept and m is the slope. Here m 5 2and b 5 3, so the formula is f ( x ) 5 2 x 1 3. b ) Here m 3and b 5 0, so the formula is f ( x ) 3 x . *All ﬁgures are included at the back of the pamphlet.

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2 MATHEMATICS FOR ECONOMISTS c ) We know m but need to compute b . Here m 5 4, so the function is of the form f ( x ) 5 4 x 1 b .When x 5 1, f ( x ) 5 1, so b has to solve the equation 1 5 4 ? 1 1 b . Thus, b 52 3and f ( x ) 5 4 x 2 3. d ) Here m 2, so the function is of the form f ( x ) 2 x 1 b x 5 2, f ( x ) 2, thus b has to solve the equation 2 2 2 ? 2 1 b ,so b 5 2and f ( x ) 2 x 1 2. e ) We need to compute m and b . Recall that given the value of f ( x )attwo points, m equals the change in f ( x ) divided by the change in x . Here m 5 (5 2 3) 6 (4 2 2) 5 1. Now b solves the equation 3 5 1 ? 2 1 b b 5 1and f ( x ) 5 x 1 1. f ) m 5 [3 2 ( 2 4)] 6 (0 2 2) 7 6 2, and we are given that b 5 3, so f ( x ) (7 6 2) x 1 3. 2.9 a ) The slope is the marginal revenue , that is, the rate at which revenue increases with output. 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. d ) The slope is the marginal propensity to consume , that is, the rate at which aggregate consumption increases with national income. e ) The slope is the marginal propensity to save , that is, the rate at which aggregate savings increases with national income. 2.10 a ) The slope of a secant line through points with x -values x and x 1 h is [ m ( x 1 h ) 2 mx ] 6 h 5 mh 6 h 5 m .
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answers1 - ANSWERS PAMPHLET 1 Chapter 2 2.1 i) y 3x 2 is...

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