Solution:

At x = 0, the slope of the tangent line is -4. Therefore, the equation of the line is y = -4x+4. By setting this equal to the function y, we get two solutions: x = 0 and x = 1. Hence M is 1. Now we have to take the integral of y minus the line's equation from 0 to 1. Whatever the absolute value of that value equals to is the area of the question. Mathematically, it is integral of
x³-x²-4x+4 - (-4+4) or x³-x² from 0 to 1. The answer is 1/12. It is a very tiny area, isn't it?

Topic:

Derivatives and Integrals

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02/18/02) Find the limit of f(x) as x approaches infinity, if f(x) is the square root of x²+4x+10 minus the square root of x²-6x+17.

Solution:

As complicated as it sounds, a simple idea underlines this problem. As x approaches infinity the whole numbers do not count as much. Therefore, we can simply change 10 to 4 and change 17 to 9. This would make each of the terms under the square root a perfect square. Hence, the whole thing can be written as (x+2)-(x-3) which equals 5. Indeed, 5 is the limit of this function as x approaches infinity. Wasn't that sweet?

Topic:

Limits

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02/11/02) Find the derivative of the following function with respect to x.

Solution:

Derivative and Integral cancel each other, so we plug each limit of integration into the function and by chain rule, we multiply each term by its derivative. Hence, we would obtain:
(1/sin(x))*cos(x)-(1/cos(x))*-sin(x) which simplifies to cot(x)+tan(x). This can be simplified even more to 1/(cos(x).sin(x)) by using common denominator. However, either way is correct!

Topic:

Derivative of Integrals

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02/04/02) What is the slope of the line tangent to the point of inflection of f(x)=xⁿ + x ?

Solution:

First mission is to find the function's point of inflection, which is the place where its second derivative equals zero. The first derivative is n*x^n-1 +1 and the second derivative is (n-1)*x^n-2. The second derivative is zero only when x=0. Therefore, zero is its point of inflection. Now, since the slope of the tangent line or DERIVATIVE of that point is needed, we plug 0 into our first derivative function and we obtain 1. So the answer is 1, wasn't that easy?

Topic:

Derivatives

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