Could someone help me with my calculus review?

Question

The test for my calc class is the final one in the class and it is on Wednesday of this upcoming week. The teacher gave us a sample test to go over and I need some help. All of the stuff in this section has confused the h**l out of me and I'm not sure how to do them. If someone can show me how to do them then that would be awesome. I don't even need answers. Just a simple explanation.

Here's the review sheet:

http://img.photobucket.com/albums/v660/clarence2222/HPIM3966.jpg

Thanks for any help.

Answer 1

1.) i) Newton's method is a powerful way of finding roots. It works as follows: consider the function f. we are looking for x such that f(x)=0. We want to approximate x. So we take an educated guess which we denote y. In this case x0=1.  From this we will make a new guess, x1. This guess would be the intercept if the function was linear. Henceforth it is an approximation. We would then repeat the process again (with x1,x2,x3 ect). The formula for determining your new guess (x1)is:

x(1)=x0 - [f(x0)] / [f ' (x0)]

ii) to calculate the error, find the difference between the actual root (from foiling) and this estimate.

2.) The tangent line approximation is a method used to estimate less well known points from well known points of a function. In this case we know that the cubed root of eight is three. Over the small interval we are going to pretend that this function is linear to estimate the cubed root of 8.03. The general form of this is: given a function f(x), well known at point a (both the derivative and the value) then for values of x near a the tangent line approximation states that:

f(x) approxmately = f(a)+f ' (a)(x-a)

ii) find the difference of this approximation to your calculator's approximation.

3.) I can't quite see the numbers. But to evaluate a limit such as this remember that within a limit you can get rid of factors common to the top and bottom such as (x-0) if x is approaching 0. The limit never quite gets there. Also remember that the limit of a sum is the sum of the limits (lim(a+b)=lim(a)+lim(b)) where they approach the same thing. The same is true for quotients and products (assuming the denominator of a quotient does not = 0).

4.) THe method for determinating the absolute extrema of a function is quite simple if the interval is closed. You must check both ends of the interval and all the critical points inbetween. A critical point or value is a point at which the derivative is either undefined or 0. So determine the derivative, work out where it would be undefined or 0 (if anywhere) and check the end points. This method works because if a point's derivative is not undefined or 0, and it is not a end point, the point on eitherside will inevitably be greater and less than it. This only works for contiuous funtions.

5.) i) Critical points, values or numbers are points where the derivative is 0 or undefined. Determine the derivative, then work these out.

    ii) If the derivative changes from + to - at a point then the point is a local maximum. If it changes from - to + at a point then the point is a local minimum. If it doesn't change it is neither. I don't know what inflection points have got to do with any of this. Inflection points are points where curvature changes from bending up to bending down or vice versa. These can be determined by finding the places where the second derivative is 0 or undefined.

iii) Once you have the maximums and minimums, this should be easy. Places between a maximum and a minimum are decreacing, places between a minimum and maximum are increasing. At the end intervals, think it through.

iv) Concave up means that the second derivative is positive, concave down means that the second derivative is negative. Once you have the points of inflection (see ii) this should not be to hard to work out.

v) see iii

6, 7, & 8) THese are all minimize / maximize problems. The fist thing you should do is to find what you are trying to minimize or maximize, and on what interval. The minimum or maximum must either occur on a critical point, and end point, or neither occur because the funtion increases or decreases indefinately (which often happens on open intervals).