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If you trade the forex markets regularly, chances are that a lot of your trading is of the short-term variety; i. From my experience, there is one major flaw with this type of trading: h igh-speed computers and algorithms will spot these patterns faster than you ever will. When I initially started trading, my strategy was similar to that of many short-term traders. That is, analyze the technicals to decide on a long or short position or even no position in the absence of a clear trendand then wait for the all-important breakout, i. I can't tell you how many times I would open a position after a breakout, only for the price to move back in the opposite direction - with my stop loss closing me out of the trade. More often than not, the traders who make the money are those who are adept at anticipating such a breakout before it happens. # Graphical analysis of functions using derivatives in investing

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Similarly if the second derivative is negative, the graph is concave down. This is of particular interest at a critical point where the tangent line is flat and concavity tells us if we have a relative minimum or maximum. For the designated function and point, determine if the graph has a local minimum, local maximum, or non-extreme point, or if the second derivative test fails. However if we look a the graph, we can see the curve is concave up everywhere, and that this point is a local minimum.

However if we look a the graph, we can see this point is neither a local minimum or a local maximum. It is a place where the graph switches from being concave up to being concave down. This is called an inflection point. We will use the second derivative test for finding maximums and minimums in the next chapter. The mathematical first and second derivatives are used in pricing various financial products and options that are also called derivatives.

The first derivative is used to give a value to whether the underlying product has a price that goes up or down. It looks at the slope of the pricing curve. The second derivative is used to give a value to the volatility of the underlying product.

It looks at how much the pricing curve bends. We will return to these uses when we look at integrals. Find the critical points of the function. Classify each as a local minimum, a local maximum, neither, or not a local extremum. There are critical points when the derivative is undefined or 0. The function is concave down, where the second derivative is negative, which for our function is when the denominator is negative.

Note that local maximums of a function must correspond to zeroes of its derivative. The original function is F green. Its derivative is D Purple. The second derivative is E black. Looking at the middle segment of the curves, the green curve is a downward facing parabola, whose derivative is a line with a negative slope like the purples curve, whose derivative is a negative constant like the black line.

For the next set of problems, you need to know the formula for the price of a zero coupon bond given the face value, the interest rate, and the time to maturity. Consider this an expressway to Data Science with approachable methods and friendly concepts that will guide you to truly understanding the most important ideas in Differential Calculus. We will review some algebra basics, talk about what a derivative is, compute some simple derivatives and apply the basics of derivatives to graphing and maximizing functions.

Excellent introduction and review for those of us who took these courses some years ago! Really appreciated the clear teaching, which is so important for learrning math. The lectures are very clear and thorough. I really appreciate the effort and thoughtfulness put into this course. Use where derivatives are positive and negative to help graph a function. Using the Derivative to Graph Functions. Algebra and Differential Calculus for Data Science.

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What we have stated is that there may exist a second derivative, which is a derivative of a derivative. We will further define second derivative at a to refer to a derivation of a derivative only at the value a. This short section will first introduce some intriguing properties that differentiation has to offer.

Concepts here will help shed insight into the latter theorems. In the previous chapter , you have been introduced to the concept that being able to take the derivative of a function implies continuity at that point. This is important to remember as each theorem will use concepts that continuity offers to justify their proof. We will now introduce two new concepts about functions, both of which you may be familiar with.

We will then justify its existence by creating a theorem to go along with it. This proof will justify the minimum and maximum point definition by relating it to differentiation, namely through this statement. The proof is straightforward: it invokes the definition of differentiation to assert the result of the theorem. To prove the case for the minimum point, you simply reverse the initial inequality.

Note that the proof will still yield the same result, namely that the derivative of the minimum point is also 0. Likewise, we will now create two more complementary definitions related to functions. These are the convexity and concavity definitions, a method of describing functions based on how the function relates to some given reference point, in this case a line.

Convexity and concavity mirror the visual, physical description of convex and concave, although they are described in reference to how the line protrudes out from the function instead of the other way around, so that graphically the definitions appear reversed. Which can be expressed differently so that although it appears more foreign, it is more applicable to theorems. The first corollary that comes from this definition is a simple, yet often not explained facet of properties coming from negating a function.

Conceptually, finding the derivative means finding the slope of the tangent line to the function. Thus the derivative can be thought of as a linear, or first-order, approximation of the function. This unnamed theorem is the theorem that proves our definition earlier is not wrong. Using it, our understanding of what differentiation graphically represents; a tangent line, should become rigorously clear. It is so ingrained that just in the section above, we defined what a tangent line is in reference to differentiation.

There, we will prove the validity of such a statement. How do we prove this notion? Well, we simply prove the physical concept of what being tangential is: if we have a perfectly round surface, we can imagine balancing a ruler on it and that the ruler is only balancing on one point. This physical description implies that there is a gap everywhere else, and this concept of a "gap" will be expressed mathematically by the inequality between the value of the tangent line to the curved function.

To begin our proof, we must first simplify the problem and assign a direction to our proof. These three theorems all relate to how a tangent relates to intersections of the function and the graph. Rolle's Theorem is the introductory theorem for derivatives. This proof will open up access to many other proofs in this chapter, especially the next big differentiation theorem found in the next section, Mean Value Theorem.

Rolle's Theorem is described below. We will present two proofs for Rolle's Theorem. The first utilizes the Existence of Minimum and Maximum Points Theorem, while the second utilizes a contradiction. We will first remove an obvious case. In other words, we know that there exists a supremum that exists in the continuous function as a consequence of the Minimum-maximum theorem.

This supremum mimics the maximum point by definition. The minimum works the same way, except that we will find an infimum instead of a supremum and that the infimum definition will match up with the minimum point. In other words, because we have, though an application of a theorem, found a supremum that is greater than every value in the function and we ruled out the case that the function is constant, we can safely assert that this supremum is greater than at least the endpoints, by definition.

It is named the mean value theorem because graphically, this theorem states that if you have a function and its secant line, there will always be a tangent line somewhere on the function that is parallel to the secant line. Because it is so similar, the proof is surprisingly similar in structure as well. Pricing of derivatives may feature a strike price; This is the price at which it can be exercised. In reference to fixed income derivatives investing, there is also a call price at which the issuer can convert security.

Investors can take different positions- long or short. Long position means you are purchasing the derivative and short position denotes you are the seller. Derivatives investing have 3 primary uses in the market: hedging a position, increasing leverage or speculating movement of an asset.

Hedging positions are usually carried out to offer protection against the risk of an asset. If one owns shares of a stock and wants a safeguard against the possibility that stock prices will fall, the put option has to be purchased. In such a scenario, if stock prices rise, there is again as one owns shares, and if they fall, you gain because of the put option.

Potential security associated loss is hedged using an options position. Leverage can be really enhanced through the use of derivatives investing. These options are most valuable when the market is at its volatile best. The movement of the option is magnified if the price of an underlying asset moves strongly in a favorable direction. Through this technique, investors bet on the future price of the asset. As options offer investors the chance to leverage their position, large speculative plays can be carried out at a modest cost.

OTC derivatives investing are contracts that are made between parties on a private basis, for example, swap agreements. This is a market which is the larger of the two and relatively unregulated. Derivatives trading on the exchange, in contrast, are standardized contracts. The largest difference between the two markets is in the case of OTC contracts where there may risk associated with counterparts as contracts are created privately and therefore not subject to regulation.

This derivatives investing risk is missing from exchange derivatives investing, where the clearinghouse acts as the go-between. There are variations in each case. Just like television has different channels, tuning in to any of these contracts will yield differential results.

While options are contracts that permit but do not carry the obligation to purchase or sell an asset, thereby being used by those investors who do not want to risk position in asset outright and want exposure in case of large price movements in the underlying asset. Image source: pixabay. This is an OTC derivative aiming to transfer credit risk from one party to another.

Through synthetically creating or eliminating of credit exposures, institutions are able to manage credit risks more easily. Credit derivatives investing come in many avatars- 3 key ones are total return, credit linked swap and credit default. Whether the derivatives investing are plain vanilla or more complicated and exotic, the distinction is mainly on account of how differentiated or complex the derivative us. Derivatives investing show an agreement between parties which is hugely flexible and has a starting and ending date.

Investors can try derivatives investing for capturing the profit emanating from price variations within the investment. This is why derivatives investing are referred to as-as leveraged investments. If the leverage is employed, both positive and negative results matter in an even bigger way.

Investors are using price variations in underlying investment to capture profits through derivatives investing. Another reason why derivatives investing are a better investment is because you can control more asset for less cash wherein the fraction of the price of the underlying asset is used to derive profits. Derivatives investing are also extremely flexible because anything can be used as an underlying asset, even the weather or the rate of crop growth.

Most common derivatives investing are options, while warrants and swap contracts are some other. Derivatives investing are contracts that are formed between buyers and sellers. Also known as wasting assets because they have limited shelf life, with a set expiry date, their value decreases as expiry nears.

The payoff is based on the expiration date, though this is not so in all cases. Money is not always exchanged at the start of the contract through cash, or spot price is the price an investor has to pay up for immediate purchase. Much like the underlying cash instruments, certain derivatives investing are traded on established stock exchanges such as NYSE or Chicago Board of Trade. Exchange traded derivatives investing provide clearing and regulatory safeguards for investors.

Certain other derivative instruments such as forwards, swaps and exotic derivatives investing are traded outside formal exchanges. Many large organizations work as derivatives investing dealers, customizing derivatives investing to be tailor-made to suit the needs of the client.

A swap often occurs as one party has a comparative advantage in a certain area while another can borrow more freely at fixed interest rates as opposed to variable. Such contracts are used for hedging as well as speculating with respect to future crisis. Forwards and futures contracts differ in that futures refer to standardized contracts that are traded on the exchange while nonstandardized, OTC contracts are forwards.

Financial instruments such as stocks, currencies, bonds, and commodities are primary or cash based.

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Matching Function Graphs With Their Derivative Graphs

Technical analysis is an investing method that is used to evaluate investments and identify trading opportunities in price trends and patterns by using charts. Options Trading and Portfolio Investment Analysis and Design Tools by Peter Hoadley. Software tools for derivatives traders and portfolio investors. present study investigated engineering students' conceptions and misconceptions related to derivative, particularly interpreting the graph of a function.