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on the interval [0, 2]. f[b] f[a] ; b a Plot[f'[x] m, {x, 0, 2}, PlotRange { 8, 8}]] f[x_]= x + Sin[2 x] a = 0; b = 2; m = ;
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We estimate the zeros of the function f '(x) m to determine the approximate locations of c. There appear to be four values: 0.3, 0.7, 1.3, and 1.7 (approximately).
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FindRoot[f'[c] m, {c, {.3, .7, 1.3, 1.7}}]
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{0.257071, 0.753319, 1.24344, 1.75836}}
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c1 = .257071; c2= .753319; c3 = 1.24344; c4 = 1.75836; l1[x_] f[c1] + f'[c1](x c1) /; c1 .25 x c1 + .25 l2[x_] f[c2] + f'[c2](x c2) /; c2 .25 x c2 + .25 l3[x_] f[c3] + f'[c3](x c3) /; c3 .25 x c3 + .25 l4[x_] f[c4] + f'[c4](x c4) /; c4 .25 x c4 + .25 l[x_] f[a] + m (x a) Plot[{f[x], l[x], l1[x], l2[x], l3[x], l4[x]}, {x, a, b}]
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The tangent lines are parallel to the secant connecting the endpoints of the curve.
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8.9 Compute the 3rd derivative of tan x.
SOLUTION
Tan'''[x] 2 Sec[x]4 + 4 Sec[x]2 Tan[x]2
8.10 Compute the values of the first ten derivatives of f ( x ) = e x at x = 0. Put the results in tabular form.
SOLUTION
f[x_] = Exp[x2] derivtable = Table[{n, D[f[x], {x, n}] /. x 0}, {n, 1, 10}]; TableForm[derivtable, TableAlignments Right, TableSpacing {1,5}, TableHeadings {None,{"n", "f(n) (0)"}}]
n 1 2 3 4 5 6 7 8 9 10 f(n) (0) 0 2 0 12 0 120 0 1680 0 30 240
8.11 Sketch the graph of f(x) = x4 50 x2 + 300 and its derivative, on one set of axes, for 10 x 10.
SOLUTION
PlotLegends` f[x_] = x4 50 x2 + 300; Plot[{f[x], f'[x]}, {x, 10, 10}, PlotRange { 1000, 1000}, PlotStyle {GrayLevel[0], Dashing[{.015}]}, PlotLegend {"f(x)", "f'(x)"}]
Observe that f '(x) = 0 precisely where f ( x) has a relative (local) maximum or minimum.
5 500
f(x) f'(x)
1000
Differential Calculus
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.
SOLUTION
f[x_] = Sin[x]; a = o/3; l[x_] = f[a] + f'[a](x a); Plot[{f[x] l[x]}, {x, 0, 2 o}] ,
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
1 1
8.13 Use Manipulate to show the tangent line at various positions along the curve y = sin x, 0 x 2 .
SOLUTION
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}]
This guarantees that the tangent line will have a constant length of 2. s n
Move the slider to change the location of the tangent line.
Differential Calculus
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].
SOLUTION
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}]
8.15 Verify the Mean Value Theorem for the function f ( x ) = x + sin 2 x on the interval [0, ].
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