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8.12 Given f(x) whose graph is C, the slope of the line tangent to C at a is f '(a). Let f(x) = sin x. Sketch the graph and its tangent line at a = /3.

f[x_] = Sin[x]; a = o/3; l[x_] = f[a] + f'[a](x a); Plot[{f[x] l[x]}, {x, 0, 2 o}] ,

Indicate possible exceptional points. DIFFERENTIALS 6.71. If z x3 xy 3y2 , compute (a) z and (b) dz where x 5, y 4, x 0:2, y 0:1. Explain why z and dz are approximately equal. (c) Find z and dz if x 5, y 4, x 2, y 1. Ans. (a) 11:658; b 12:3; c z 66; dz 123 q 5 Computer 3:8 2 2 2:1 3 approximately, using di erentials. Ans. 2.01 Find dF and dG if (a) F x; y x3 y 4xy2 8y3 ; b G x; y; z 8xy2 z3 3x2 yz, ln y=x . Ans. a 3x2 y 4y2 dx x3 8xy 24y2 dy b 8y2 z3 6xyz dx 16xyz3 3x2 z dy 24xy2 z2 3x2 y dz c f y2 ln y=x y2 g dx f2xy ln y=x xyg dy (c) F x; y xy2

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Recall that the equation of a line having slope m, passing through (x1, y1) is y y1 = m(x x1) or y = y1 + m(x x1) y = f ( a) + f '(a)(x a) Here, x1 = a, y1 = f (a), and m = f '(a) so

[-a] [-d] [-e] [-D debug level] [-n] \ [-r remote_machine] [-R name resolve order] [-m] \ [-j DOMAIN] [-U username] [-h] [-s] username smbpasswd program has several different functions, depending on whether or not it is run by the root user. When run as a normal user it allows the user to change the password for an SMB session on any machines that store SMB passwords.

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6.72. 6.73.

8.13 Use Manipulate to show the tangent line at various positions along the curve y = sin x, 0 x 2 .

The tangent line has equation y = f (a) + f '(a)( x a). f[x_] = Sin[x]; 644444444 44444444 7 8 1 1 l[x_, a_]:= f[a]+ f'[a](x a)/; a x a+ 1 + f'[a]2 1 + f'[a]2 Manipulate[Plot[{f[x], l[x, a]}, {x, 0, 2 o}, PlotRange { 1.5, 1.5}], {a, 0, 2 o}]

smbmnt mount-point [-u uid] [-g gid] [-f file mode] [-d dir mode] smbmnt is a helper application used by the f3 smbmount (8) program to do the actual mounting. Smbmnt is meant to be installed setuid root so that normal users can mount their smb shares. It checks whether the user has write permissions on the mount point and then mounts the directory. mount-point is a UNIX mount for normal users. With this program, normal users can mount smbfilesystems, provided that it is suid root.

8.14 Find the value(s) of c guaranteed by Rolle s Theorem for the function f(x) = 4x + 39x2 46 x 3 + 17 x 4 2 x 5 on the interval [0, 4].

6.74. 6.75.

Since f(x) is a polynomial, it is continuous and differentiable everywhere. First we verify that f(0) = f(4) = 0. f[x_]= 4 x + 39 x 2 46 x 3 + 17 x 4 2 x 5; f[0] 0 f[4] 0 Now we look to see where f '(c) = 0. Since f ' is a polynomial, we can use NSolve. NSolve[f'[c] 0]

0.0472411}, {c 1.05962}, {c 2.27466}, {c 3.51296}}

There are three values of c between 0 and 4 (Rolle s Theorem guarantees at least one). A plot of the graph confirms our result. Plot[f[x], {x, 1, 4}]

Prove that (a) d UV U dV V dU; b d U=V V dU U dV =V 2 ; c d ln U dU =U, 1 (d) d tan V dV = 1 v2 where U and V are di erentiable functions of two or more variables. Determine whether each of the following are exact di erentials of a function and if so, nd the function. a 2xy2 3y cos 3x dx 2x2 y sin 3x dy b 6xy y2 dx 2xey x2 dy c z3 3y dx 12y2 3x dy 3xz2 dz Ans. a x2 y2 y sin 3x c; b not exact, c xz2 4y3 3xy c

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