By Thao N. X.

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2. (a) Let w = 5t (so that dw/dt = 5), then y = ew and dy/dw = ew . Thus,by the chain rule, dy dt = dy dw dw dt = 5ew = 5e5t . (b) Let w = 3t , then y = 4ew and dy/dw = 4ew . Thus,we have dy dt = dy dw dw dt = 12e3t (c)Similarly to (b) above, dy/dt = −12e−2t . 3. 718)t . The value of t can be either positive, zero, or negative. 718)2 , still positive. Thus y 0 (t) and y 00 (t) are always positive, and the function y = et always increases at an increasing rate. 4. (a) The curve with a = −1 is the mirror image of the curve with a = 1 with reference to the horizontal axis.

Let ∇U be the row vector [U1 , . . , Un ], and dx be the column vector [dx1 , . . , dxn ]. Then dU = ∇U dx. 2. ¢ ¡ (a) dz = (6x + y) dx + x − 6y 2 dy (b) dU = (2 + 9x2 ) dx1 + (9x1 + 2x2 ) dx2 3. x1 x2 2 dx1 − 2 dx2 (x1 + x2 ) (x1 + x2 ) ³ ³ ´2 ´2 x1 2 (b) dy = 2 x1x+x dx + 2 dx2 1 x +x 2 1 2 (a) dy = 41 Chiang/Wainwright: Fundamental Methods of Mathematical Economics Instructor’s Manual 4. ∂Q ∂P dQ dR P 2bP 2 ; = Q a + bP 2 + R1/2 R 1 R1/2 ¢. = R−1/2 = ¡ 2 Q 2 a + bP 2 + R1/2 = 2bP, thus ²QP = 2bP = 1 −1/2 , thus ²QR R 2 5.

Only Q∗ = 11 gives a maximum profit. (e) Maximum profit = 111 1 3 b = 0. With its minimum at 3a zero output. The MC curve must be upward-sloping throughout. Since the increasing segment 4. If b=0, then the MC-minimizing output level becomes Q∗ = − of MC is associated with the convex segment of the C curve, b = 0 implies that the C curve will be convex throughout. 5. (a) The first assumption means π(0) < 0. Since π(0) = k, we need the restriction k < 0. (b) Strict concavity means π 00 (Q) < 0. Since π 00 (Q) = 2h, we should have h < 0.

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