By David Berlinski

In its biggest element, the calculus capabilities as a celestial measuring tape, capable of order the limitless expanse of the universe. Time and house are given names, issues, and boundaries; doubtless intractable difficulties of movement, progress, and shape are lowered to answerable questions. Calculus used to be humanity's first try to characterize the area and maybe its maximum meditation at the subject of continuity. Charts and graphs all through.

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**Example text**

3 Differentiability, the Derivative Matrix, and Tangent Planes 31 (x, y) is given by function f (x, This equation is also used to compute a linear approximation. Compare this equation to the equation of a tangent line and the linear approximation in the one-variable case. 2. Differentiability. 2 --+ ---+ lR. function f: lR. IR,2 IR, JR. is differentiable if the tangent plane approaches n --+ lR. m , then f(xo, Yo) as (x, y) approaches (xo, Yo). Now, if f: U c C IR,n IR,m, iff: lR. m, llf(x)Df(xxo)ll -_ 00 - f(xo)f(xo) - Df(x - xo)1I lim Ilf(x) Ilx Xoll - ,' llx -- Kali Xo II x-+xo X-+Xo n --+ lR.

If c = = 0, the level surface is the origin. If c > 0, there is no level surface since the right-hand side must be nonpositive. The sketch shows the level surfaces for c = = 0, -1, and -4. -4. X z 17. 21. \' X 25. )')' 17. The equation may maybe be thought of as z = iyl. Iyl. " We sketch the section in the yz-plane and translate the section parallel to the x axis. This gives us the sketch shown. 21. We note that z is missing from the equation, so we have a cylinder. cylinder. In the xy plane, the level curve is the ellipse centered at the origin, with x-intercepts at x = ±2 and y-intercepts at y = ±4.

When velocity function is 8u/8t = 1 and t = = 1/3, we have 8u/at 8uf8t = = -6cos(-1) + 6cos(3) = x = 6 cos (3) = -6 cos (1) + 6 cos (3). 2. 2 Test for Chapter 2 1. True or false. If false, explain why. F is a function of x andy and y such that y = = f(x), then dy/dx dyjdx = = (a) IfF If Fis (8Fj8x)f(8Fj8y). (8F/8x)/(8F/8y). (b) The gradient is a directional derivative. (c) 8(x,y,z)/8(r,s) can be represented by a matrix composed of three rows and two columns. (d) If lim(x,y)-+(O,O) lim(x,y)-+(0,0) f(x, y) = = 0, then limx-+o limx-+0 f(x, 0) = = 0.