Slide 1: LIMITS AND DERIVATIVES
Slide 2: In mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value.[1] The concept of limit allows one to, in a complete space, define a new point from a Cauchy sequence of previously defined points.[2] Limits are essential to calculus (and mathematical analysis in general) and are used to define continuity, derivatives and integrals.
The concept of the limit of a function is further generalized to the concept of topological net, while the limit of a sequence is closely related to limit and direct limit in category theory.
In formulas, limit is usually abbreviated as lim as in lim(an) = a or represented by the right arrow (→) as in an → a. LIMITS
Limit of a function : Limit of a function Suppose f(x) is a real-valued function and c is a real number. The expression
means that f(x) can be made to be as close to L as desired by making x sufficiently close to c. In that case, we say that "the limit of f of x, as x approaches c, is L". Note that this statement can be true even if f(c) ≠ L. Indeed, the function f(x) need not even be defined at c.
For example, if
Limit of a sequence : Limit of a sequence Consider the following sequence: 1.79, 1.799, 1.7999,... We could observe that the numbers are "approaching" 1.8, the limit of the sequence.
Formally, suppose x1, x2, ... is a sequence of real numbers. We say that the real number L is the limit of this sequence and we write
to mean
For every real number ε > 0, there exists a natural number n0 such that for all n > n0, |xn − L| < ε.
Intuitively, this means that eventually all elements of the sequence get as close as we want to the limit, since the absolute value |xn − L| is the distance between xn and L. Not every sequence has a limit; if it does, we call it convergent, otherwise divergent. One can show that a convergent sequence has only one limit.
Slide 5: The limit of a sequence and the limit of a function are closely related. On one hand, the limit of a sequence is simply the limit at infinity of a function defined on natural numbers. On the other hand, a limit of a function f at x, if it exists, is the same as the limit of the sequence
xn = f(x + 1/n).
Convergence and fixed point : Convergence and fixed point A formal definition of convergence can be stated as follows. Suppose pn as n goes from 0 to is a sequence that converges to a fixed point p, with for all n. If positive constants λ and α exist with
then pn as n goes from 0 to converges to p of order α, with asymptotic error constant λ
Given a function f(x) = x with a fixed point p, there is a nice checklist for checking the convergence of p.
1) First check that p is indeed a fixed point:
f(p) = p
2) Check for linear convergence. Start by finding . If....
Slide 7: 3) If we find that we have something better than linear we should check for quadratic convergence. Start by finding If....
Limits for general functions : Limits for general functions
Limits of general functions : Limits of general functions
Notable special limits : Notable special limits
Simple functions : Simple functions
Logarithmic and exponential functions : Logarithmic and exponential functions
Trigonometric functions : Trigonometric functions
Near infinities : Near infinities
DERIVATIVES : DERIVATIVES In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's instantaneous velocity. Conversely, the integral of the object's velocity over time is how much the object's position changes from the time when the integral begins to the time when the integral ends.
Slide 16: The derivative of a function at a chosen input value describes the best linear approximation of the function near that input value. For a real-valued function of a single real variable, the derivative at a point equals the slope of the tangent line to the graph of the function at that point. In higher dimensions, the derivative of a function at a point is a linear transformation called the linearization. A closely related notion is the differential of a function
The process of finding a derivative is called differentiation. The reverse process is called antidifferentiation. The fundamental theorem of calculus states that antidifferentiation is the same as integration.
DIFFERENTIATION : DIFFERENTIATION Differentiation is a method to compute the rate at which a dependent output y changes with respect to the change in the independent input x. This rate of change is called the derivative of y with respect to x. In more precise language, the dependence of y upon x means that y is a function of x. This functional relationship is often denoted y = ƒ(x), where ƒ denotes the function. If x and y are real numbers, and if the graph of y is plotted against x, the derivative measures the slope of this graph at each point.
Slide 18: The simplest case is when y is a linear function of x, meaning that the graph of y against x is a straight line. In this case, y = ƒ(x) = m x + b, for real numbers m and b, and the slope m is given by
where the symbol Δ (the uppercase form of the Greek letter Delta) is an abbreviation for "change in." This formula is true because
y + Δy = ƒ(x+ Δx) = m (x + Δx) + b = m x + b + m Δx = y + mΔx.
It follows that Δy = m Δx.
Slide 19: At each point, the derivative of f(x)=1+xsin2x is the slope of a line that is tangent to the curve. The line is always tangent to the blue curve; its slope is the derivative. Note derivative is positive where green, negative where red, and zero where black.
Slide 20: This gives an exact value for the slope of a straight line. If the function ƒ is not linear (i.e. its graph is not a straight line), however, then the change in y divided by the change in x varies: differentiation is a method to find an exact value for this rate of change at any given value of x.
The idea, illustrated by Figures 1-3, is to compute the rate of change as the limiting value of the ratio of the differences Δy / Δx as Δx becomes infinitely small.
In Leibniz's notation, such an infinitesimal change in x is denoted by dx, and the derivative of y with respect to x is written
suggesting the ratio of two infinitesimal quantities. (The above expression is read as "the derivative of y with respect to x", "d y by d x", or "d y over d x". The oral form "d y d x" is often used conversationally, although it may lead to confusion.)
Definition via difference quotients : Definition via difference quotients Let ƒ be a real valued function. In classical geometry, the tangent line to the graph of the function ƒ at a real number a was the unique line through the point (a, ƒ(a)) that did not meet the graph of ƒ transversally, meaning that the line did not pass straight through the graph. The derivative of y with respect to x at a is, geometrically, the slope of the tangent line to the graph of ƒ at a. The slope of the tangent line is very close to the slope of the line through (a, ƒ(a)) and a nearby point on the graph, for example (a + h, ƒ(a + h)). These lines are called secant lines. A value of h close to zero gives a good approximation to the slope of the tangent line, and smaller values (in absolute value) of h will, in general, give better approximations. The slope m of the secant line is the difference between the y values of these points divided by the difference between the x values, that is,
Slide 22: This expression is Newton's difference quotient. The derivative is the value of the difference quotient as the secant lines approach the tangent line. Formally, the derivative of the function ƒ at a is the limit
of the difference quotient as h approaches zero, if this limit exists. If the limit exists, then ƒ is differentiable at a. Here ƒ′ (a) is one of several common notations for the derivative.
Equivalently, the derivative satisfies the property that
which has the intuitive interpretation (see Figure 1) that the tangent line to ƒ at a gives the best linear approximation
to ƒ near a (i.e., for small h). This interpretation is the easiest to generalize to other settings
DERIVATION FORMULA : DERIVATION FORMULA 1. Constant Rule:
If y = k, then y' = 0
The Derivative of a Constant is 0
If ƒ(x) = k for some constant k, then ƒ'(x) = 0
2. Power Rule
If y = x", then y' = nxn-1If ƒ is a differentiable function, and if ƒ(x) = x", then ƒ'(x) = nxn-1 for any real number n
3. Exponential Rule:
If y = ex, then y' = ex
4. Logarithm Rule:
If y = 1n|x|, then y' = 1/x
Slide 24: 5. Constant Times a Function Rule:
If y = kƒ, then y' = kf '
6. Sum Rule
If y = ƒ g, then y' = ƒ' g'
7. Product Rule
If y = ƒg, then y' = ƒg' ƒ' gIf ƒ and g are differentiable functions such that y = ƒ(x)g(x), then y' = ƒ(x)g' ƒ' (x)g(x)
8. Difference Rule
If y = ƒ - g, then y' = ƒ' - g‘
10. Chain Rule:
If y is a differentiable function of u and u is a differentiable function of x and
Slide 26: THANKING YOU FOR GIVING ME
YOUR PRECIOUS TIME