Schaum's Outline of Mathematica by Eugene Don

Author:Eugene Don
Language: eng
Format: epub, pdf
Publisher: McGraw-Hill Education
Published: 2019-03-12T16:00:00+00:00

SOLVED PROBLEMS

6.12 Solve the equation 5 cos x = 4 − x3. Make sure you find all solutions.

SOLUTION

Since 5 cos x = 4 − x3 if and only if 5 cos x − 4 + x3 = 0, we introduce the function f(x) = 5 cos x − 4 + x3 and look for x-intercepts. (Although we could look for the intersection of two curves, it is easier to approximate where points intercept an axis.)

f[x_] = 5 Cos [x] – 4 + x3;

Plot[f[x], {x, –1, 2}]

It appears that there are three solutions, near –0.5, 0.8, and 1.6.

FindRoot[f[x], {x, –0.5}]

{x → –0.576574}

FindRoot[f[x], {x, 0.8}]

{x → 0.797323}

FindRoot[f[x], {x, 1.6}]

{x → 1.61805}

6.13 Find a solution of the equation sin x = 2. (This problem may be omitted by those unfamiliar with functions of a complex variable.)

SOLUTION

Since –1 ≤ sin x ≤ 1 for all real x, this problem has no real solutions. We can force FindRoot to search for a complex solution by using a complex initial guess.

FindRoot[Sin[x] == 2,{x, I}]

{x → 1.5708 + 1.31696 }

6.14 Find a 20 significant digit approximation to the equation x + | sin (x - 1) | = 5.

SOLUTION

First we plot the function f(x) = x + | sin (x – 1) | – 5.

f[x_] = x + Abs[Sin[x – 1]] – 5;

Plot[f[x], {x, –10, 10}]

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