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Points = Solve[conditions, {x, y, z}] 1 , y 2, z 3 , x 7 14 14 1 , y 2, z 3 x 7 14 14 f[x, y,z] /.Points //N {4.74166, 2.64166}
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Farthest point. Closest point.
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10.3 The Total Differential
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The command D, discussed in Section 10.1, gives the partial derivative of a function of several variables. All variables other than the variable of differentiation are considered as constants. If f is a function, say, of two variables, x and y, but y is a function of x, D will compute an incorrect derivative. Dt gives the total differential of a function.
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Dt[f[x, y]] returns the total differential of f[x, y]. Dt[f[x, y], x] returns the total derivative of f[x, y] with respect to x.
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Of course, f may be a function of more than two variables and the independent variable, listed as x in the above description, can be any of the variables defining f. f df but Dt[f[x, y], x] returns . D[f[x, y], x] returns dx x
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EXAMPLE 9
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Calculus equivalent f[x_, y_] = x y ; D[f[x, y], x] 2 x y3 D[f[x, y], y] 3 x2 y2 Dt[f[x, y]] 2 x y3 Dt[x] + 3 x2 y2 Dt[y] Dt[f[x, y], x] 2 x y3 + 3 x2 y2 Dt[y, x] Dt[f[x, y], y] 3 x2 y2 + 2 x y3 Dt[x, y] df = 3x 2 y 2 + 2 xy3 dx dy dy df dy = 2 xy3 + 3 x 2 y 2 dx dx df = 2 xy3dx + 3 x 2 y 2 dy f = 3x 2 y 2 y f = 2 xy3 x
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f ( x , y) = x 2 y 3
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Multivariate Calculus
EXAMPLE 10
Suppose f (x, y) = x4 y5 where y = x3. The following sequence gives an incorrect result for the
derivative dz/dx. z = x4 y5; D[z, x] /. y x3 4 x18
The partial derivative is computed, then y is replaced by x3.
In reality z = x19, so dz should be 19 x18.
dx Dt[z, x] /. y x3 19 x18
The total derivative is computed, then y is replaced by x3.
In some expressions, constants represented by letters might cause confusion. The option Constants can be used to instruct Mathematica to treat a particular symbol as a constant.
Constants { objectlist } causes all symbols in objectlist to be treated as constants.
EXAMPLE 11
Dt[xn, x] //Expand n x 1 + n + xn Dt[n, x] Log[x] Dt[xn, x, Constants {n}] n x 1 + n
SOLVED PROBLEMS
10.12 Let z = sin xy. Let x = 1, y = 2, dx = x = 0.03, dy = y = 0.02. Compute dz and compare it with the value of z.
SOLUTION
z = f[x_, y_] = Sin[x y]; Dz = f[x + D x, y + D y] f[x, y]; Dt[z] /. {x 1, y 2, Dt[x] 0.03, Dt[y] 0.02} 0.0332917 z /. {x 1, y 2, x 0.03, y 0.02} 0.0364571
10.13 Use differentials to approximate e0 .1 4.01 and determine the percentage error of the estimate.
SOLUTION
We take advantage of the fact that 0.1 is near 0 and 4.01 is near 4. f[x_, y_] = Exp[x] Sqrt[y]; approximation = f[0, 4] + Dt[f[x, y]] /. {x 0, y 4, Dt[x] 0.1, Dt[y] 0.01} 2.2025 exactvalue = f[0.1, 4.01] 2.2131 percenterror = Abs[approximation exactvalue]/exactvalue * 100; Print["The error is ", percenterror, "%"] The error is 0.479103 %
10.14 Use differentials to approximate the amount of metal in a tin can with height 30 cm and radius 10 cm if the thickness of the metal in the wall of the cylinder is 0.05 cm and the top and bottom are each 0.03 cm thick.
Multivariate Calculus
SOLUTION
v = r2 h; 113.097 The amount of metal is approximately 113 cm3.
The change in height is the sum of the thicknesses of the top and bottom.
Dt[v] /. {h 30, r 10, Dt[r] 0.05, Dt[h] 0.06}
10.15 If three resistors of resistance, R1, R2, and R3 ohms are connected in parallel, their effective resistance is
1 1 ohms. If a 20-ohm, 30-ohm, and 50-ohm resistor, each with maximum error of 1%, + R2 + R3 are connected in parallel, what range of resistance is possible from this combination 1 R1 1
SOLUTION
f[R1_, R2_, R3_] = f[20, 30, 50]//N 9.67742
1 R1
1 1 1 ; + R2 + R3
Dt[f[R1, R2, R3]] /. {R1 20, R2 30, R3 50, Dt[R1] 0.2, Dt[R2] 0.3, Dt[R3] 0.5} 0.0967742 The combined resistance is 9.67742 0.0967742 ohms.
10.4 Multiple Integrals
Multiple integrals, or more precisely iterated integrals, are invoked by the Integrate command and are an extension of the command for a function of one variable.