Numerical Analysis and Simulation

Completing the square is advantageous in several situations:

• usually, it requires fewer steps than the quadratic formula, reducing the potential for error.
• it allows you to determine the exact value of the answer instead of a decimal approximation.
• it allows you to determine imaginary solutions.

If you have a scientific calculator then it will have a radical key for finding the square root of a number. In a spreadsheet it is sqrt(x) = x^0.5 or x^(0.5).

If you do not have a calculator or computer, then there is no simple formula. If you want to find the square root of k, define f(x) = x^2 – k. Then finding the square root of k is equivalent to solving f(x) = 0.

Let f’(x) = 2x. This is the derivative of f(x) but you do not need to know that to use the N-R method.
Start with x0 as the first guess.
Then let xn+1 = xn - f(xn)/f’(xn) for n = 0, 1, 2, … Provided you made a reasonable choice for the starting point, the iteration will very quickly converge to the true answer. It works even if your first guess is not so good:
Suppose you start with x0 = 5 (a pretty poor choice since 5^2 is 25, which is nowhere near 7).
Even so, x3 = 2.2362512515, which is less than 0.01% from the true value. Finally, remember that the negative value is also a square root.

A derivative of a function tells us how fast the output variable, y, is changing compared to the input variable, x. For example, if y is increasing 2 times as fast as x (like with the line y = 2x + 7), then we say that the derivative of y to the respect to x equals 2, and we write dy/dx = 2, which is the same as dy/dx = 2/1. That means that we can say that the rate of change of y compared to x is 2:1, or that the line has a slope of 2/1. We can think of a derivative dy/dx as basically rise/run. So, the derivative is basically just a rate or a slope. Thus, to solve a problem, all we have to do is answer the question as if it asks us to determine a rate or a slope instead of a derivative. Any line of the form y = mx + b has a slope equal to m.

Two problems which greatly influenced the development of the differential calculus are:

1. finding the equation of the tangent line to a given curve at a given point on the curve, and

2. finding the instantaneous velocity of a particle moving along a straight line at a varying speed (a derivative is always a rate, and a rate is always a derivative, assuming we are talking about instantaneous rates).

So, one way to think about a derivative like dp/dt is that it tells us how much the position, p, changes when the time, t, increases by a specific time. For example, a driver starts at a time=0, and go 50 miles/hour in his car. The rate of 50 miles/hour means that his position changes 50 miles each time the number of hours of his trip goes up by 1.

Using the information from the problem above, we can write a function that gives us the driver's position as a function of time.

p(t) = 50t or p = 50t, where p is in miles and t is in hours.

p = 50t is a line, of course, in the form of y = mx + b (where b = 0). So,the slope is 50 and the derivative also is 50. And again we see that a derivative is a slope and a rate.

We noticed in a plane geometry that a straight line intersects a circle in two points, or is tangent to the circle, or fails to intersect the circle at all. This might tempt us to define a tangent to a circle as a line that intersects the circle in one and only one point.

But such a definition would not do for most other curves. For example, the tangent line to the graph of y = x^3 at the point (1, 1) intersects the curve again at the point (-2, -8). This indicate that a different approach is needed.

Since we can write the equation of a line through a given point if we know the slope of the line, our task is to formulate a definition of the slope of the tangent to a curve which will apply to all curves as well.

The difference quotient is a magnificent tool that gives us the slope of a curve at a single point. For example, if we have a parabola and pick a point on it, let say the point (2, 4). We can't get the slope of the parabola at (2, 4) with algebra slope formula, m = (y2 - y1)/(x2 - x1), because no matter what other point on parabola we use with (2, 4) in the formula, we will get a slope that is steeper or less steep than the precise slope at (2, 4).

But, if our second point on the parabola is extremelyclose to (2, 4), for example the point (2.001, 5.00299...), our line would be almost exactly as steep as the tangent line. The difference quotient gives the precise slope of the tangent line by sliding the second point closer and closer to (2, 4) until its distance from (2, 4) is infinitely small.

The definition of the derivative based on the difference quotient is:

f'(x) = lim h -->0 = [f(x + h) - f(x)]/h

[In the example above h = 2.001 - 2 = 0.001, that is "delta x" and its symbol is Δx. By the symbol Δx wemean by how much the x-coordinate will change (as we move from A(2, 5) to B(2.001, 5.00299...). And by the symbol Δy("delta y") we mean by how much the y-coordinate will change.

Given any number x for which the limit exists, we assign to x the number f'(x). So, we can regard f' as a new function, called the derivative of f and defined by the above formula. We know that the value of f' at x, f'(x), can be interpreted geometrically as the slope of the tangent line to the graph of f at the point (x, f(x)).

The function f' is called the derivative of fbecause it has been derived from by limiting operation in the equation (formula) above. The domain of f' is the set {x| f'(x) exists} and may be smaller than the domain of f.

Definition: If f is a function and P(c, f(c)) is a point on the graph of y = f(x) the slope of the tangent to the graph at P(c, f(c)) is

lim Δx -->0 Δy/Δx = lim Δx-->0 [f(c + Δx) - f(c)]/Δx

provided that this limit exists.

The above limit is called the value of the derivative of the function f at c.

If f is a position function which gives the coordinate s = f(t) at time t of a particle moving along a coordinate line (Fig. below),

―•―――•―――――――――•―――――→

0; f(c); f(c + Δt) then the displacement of a particle from time c to time (c + Δt) is f(c + Δt) - f(c) and the time consumed during this displacement is Δt. Thus the average velocity of the moving particle from time c to time (c + Δt) is

V(Δt)= Δs/Δt = [f(c + Δt) - f(c)]/Δt

Its instantaneous velocity at time c is given by the following definition.

Definition: If f is a function such that the coordinate of a particle moving along a coordinate line, at the end of t units of time is s = f(t), then the instantaneous velocity of the moving particle at the end of c units of time is

V(Δt)= lim Δt -->0 Δs/Δt = lim Δt -->0 [f(c + Δt) - f(c)]/Δt,

provides that this limit exists.

If we compare our definition of the slope of a tangent to a curve with our definition of the instantaneous velocity of a particle moving on a straight line, we will see that they are formally the same.

Definition: The derivative of a function fis another function f' whose value at any point c in the domain of f is

f'(c) = lim Δx -->0 Δy/Δx= lim Δx -->0 [f(c + Δx) - f(c)]/Δx,

provided this limit exists.

If this limit does exist, we say that f is differentiable at c. The domain of f' is a subset of the domain of f.

Finding the derivative of a function is called differentiation; it is the basic process in differential calculus.

Definition: A function f is differentiable at a if f'(a) exists. it is differentiable on an open interval (a, b) [or (a, ∞) or (-∞ , a) or (-∞ , ∞)] if it is differentiable at every number in the interval.

Both continuity and differentiability are desiderable properties for a function to have. The following theorem shows how these properties are related.

Theorem: If f is differentiable at a, then f is continuous at a.

The converse of this theorem is false; that is that there are functions that are continuous but not differentiable.

For example, f(x) = |x| is continuous at 0 because lim x -->0 f(x) = lim x-->0 |x| = 0, but f is differentiable at all x except 0.

If we look at the graph of the function f(x) = |x|, we see that it changes direction suddenly when x= 0. In general, if a graph of a function f, has a "corner" or "kink" in it, then the graph of f has not tangent at this point, and f is not differentiable there. If we try to compute f'(a), we find that the left and the right limits are different.

The theorem says that if f is not continuous at a, then f is not differentiable at a. So, at any discontinuity (a jump discontinuity) f fails to be differentiable.

When x = a, the curve has a vertical line, that is, f is not continuous at and

lim x -->a |f(x)| = ∞. This means that the tangent line becomes steeper and steeper as x-->a.

Differentiation is a big idea in calculus. Differentiation is the study of the derivative , or slope, of functions: where the slope is positive, negative, or 0; where the slope has a minimum or maximum value; whether the slope is increasing or decreasing; how the slope of one function is related to the slope of another; and so on. When we study calculus we get differentiation basic, differentiation rules and techniques for analyzing the shape of curves, and solving problems with the derivative.