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This is an open-book quiz You may (and should) refer to the text as you solve these problems Don t hurry! You ll find worked-out answers in App B The solutions in the appendix may not represent the only way a problem can be figured out If you think you can solve a particular problem in a quicker or better way than you see there, by all means try it! 1 Suppose that a plane contains the point (0,0,0), and the standard form of a vector normal to the plane is 4i + 4j 4k Find the plane s equation in standard form 2 Suppose that a plane contains the point (4,5,6), and the standard form of a vector normal to the plane is 2i + 0j + 0k Find the plane s equation in standard form 3 Consider a sphere whose equation is x2 + 2x + 1 + y2 2y + 1 + z2 + 8z + 16 = 64 What are the coordinates of the center of this sphere What s its radius 4 What s the equation of a sphere centered at the point (5,7, 3) and whose radius is equal to the positive square root of 23 5 Consider the equation 8(x 1)2 + 8(y + 2)2 + 6(z + 7)2 = 24 What sort of object does this equation describe Does the object have a center If so, what are the coordinates of the center point Does the object have axial radii If so, what are they 6 Consider the equation 400(x + 2)2 + 225(y 4)2 + 144z2 3\600 = 0 What sort of object does this equation describe Does the object have a center If so, what are the coordinates of the center point Does the object have axial radii If so, what are they.

7 Consider a surface whose equation is x2 + 2x + 1 + y2 2y + 1 z2 + 6z 9 = 36 What sort of object is this What are the coordinates of the center How is the axis oriented 8 Write down a generalized equation for an elliptic cone whose axis is parallel to the coordinate y axis, and whose vertex is at ( 2,3,4) 9 Suppose we slice the elliptic cone described in Problem 8 straight through with the coordinate xz plane The cone s surface intersects the xz plane in a curve Derive a generalized equation of that curve in the variables x and z What sort of curve is it Here s a hint: At every point in the xz plane, y = 0 10 Suppose we slice the elliptic cone described in Problem 8 straight through with the coordinate xy plane The cone s surface intersects the xy plane in a curve Derive a generalized equation of that curve in the variables x and y What sort of curve is it Here s a hint: At every point in the xy plane, z = 0

In Chap 16, we learned how parametric equations can define curves that are difficult to portray as conventional relations Parametric power becomes more apparent when we graduate to three dimensions

We now have a thorough, detailed list of tasks. We can do a walk-through with the customer, asking if there is anything else they want or need and are willing to pay for.

Finding an equation for a straight line in Cartesian three-space is harder than it is in the Cartesian plane The extra dimension makes expressing the line s location and orientation more complicated There are at least two ways we can do it: the symmetric method and the parametric method

Symmetric method A straight line in Cartesian xyz space can be represented by a three-part symmetric-form equation Suppose that (x0,y0,z0) are the coordinates of a known point on the line, and a, b, and c are nonzero real-number constants Given this information, we can represent the line as

The performance that a control syst,em is capable of giving is relative to the amount of intelligence built into it. It is possible to design a programmed system that will out-perform any conventional feedback system in response to command (i.e., set-point) inputs. But the program must embody more of the characteristics of the process than do conventional controllers. It has been found necessary to use an antiwindup feature in a controller with reset in order to prevent overshoot on large set-point changes. It has been further noted that the controlled variable will respond best if the reset mode is preloaded to the st eady-state conditions that are expected to prevail. In a sense, this is programmed control. An optimum program is one which will place the controlled variable at the designated set point in the minimum time (or with minimum energy or minimum cost). On-off control mill drive the process at maximum speed, but will cause overshoot with more than a single capacity in the loop. Overshoot means that some oscillation, however damped, is present, requiring a specified settling time after the set, point is crossed. This is scarcely consistent with minimum time.