Calculus/Basics

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Expert: - 1/11/2010


Question

Calculus questions ima
As the Questions are written with the help of Equation editor, so image of questions  file is attached.

Answer
Questioner: vampire
Country: Pakistan
Category: Calculus
Private: No
Subject: Calculus
........................
1. Extrema of  f(x) = 3x^3 - 9x  on [0,2]

Here is what you do:

A. find f'(x) = 9x^2 - 9 and set it = 0.
B. Solve that, you get   x = +- 1, but only x = +1 is in your interval.
C. Compute  f(0), f(2) [endpoints] and f(1).
D. Pick your max and min.

2. cos x = x^2

Set  f(x) = x^2 - cos x.

Pick x0 = 1/2  (as good as any, I suppose)

Now:
 x1 = x0 - f(x0)/f'(x0)
 x2 = x1 - f(x1)/f'(x1)

etc.  

See:  en.wikipedia.org/wiki/Newton's_method

Excel is good for this stuff.

..................................
3. Integral:

{
| cos^3(x)  sin x dx
}

u = cos x looks like a good choice, and  du = - sin x dx

You can handle the rest.

Paul Klarreich

Expertise

All topics in first-year calculus including infinite series, max-min and related rate problems. Also trigonometry and complex numbers, theory of equations, exponential and logarithmic functions. I can also try (but not guarantee) to answer questions on Analysis -- sequences, limits, continuity.

Experience

I taught all mathematics subjects from elementary algebra to differential equations at a two-year college in New York City for 25 years.

Education/Credentials
(See above.)

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