what is the relation between Limit and Derivative? - Mathematics Stack Exchange
After looking at limits, this chapter moves on to the idea of a derivative. An important result that is often used without statement is the relationship between. Definition: Let f(x) be a function of x, the derivative function of f at x is given by: . The relationship between differentiability, continuity, and having a limit is this. Limits. Continuous Functions. Applications of the Derivative . v from f will have f divided by time. The central question of calculus is the relation between v and f.
We work quite a few problems in this section so hopefully by the end of this section you will get a decent understanding on how these problems work. Higher Order Derivatives — In this section we define the concept of higher order derivatives and give a quick application of the second order derivative and show how implicit differentiation works for higher order derivatives.
Logarithmic Differentiation — In this section we will discuss logarithmic differentiation. Logarithmic differentiation gives an alternative method for differentiating products and quotients sometimes easier than using product and quotient rule. More importantly, however, is the fact that logarithm differentiation allows us to differentiate functions that are in the form of one function raised to another function, i.
Applications of Derivatives - In this chapter we will cover many of the major applications of derivatives. Critical Points — In this section we give the definition of critical points. Critical points will show up in most of the sections in this chapter, so it will be important to understand them and how to find them. We will work a number of examples illustrating how to find them for a wide variety of functions.
Minimum and Maximum Values — In this section we define absolute or global minimum and maximum values of a function and relative or local minimum and maximum values of a function.
- Differential calculus
We also give the Extreme Value Theorem and Fermat's Theorem, both of which are very important in the many of the applications we'll see in this chapter. Finding Absolute Extrema — In this section we discuss how to find the absolute or global minimum and maximum values of a function.
In other words, we will be finding the largest and smallest values that a function will have. The Shape of a Graph, Part I — In this section we will discuss what the first derivative of a function can tell us about the graph of a function. The first derivative will allow us to identify the relative or local minimum and maximum values of a function and where a function will be increasing and decreasing.
We will also give the First Derivative test which will allow us to classify critical points as relative minimums, relative maximums or neither a minimum or a maximum. The Shape of a Graph, Part II — In this section we will discuss what the second derivative of a function can tell us about the graph of a function.
The second derivative will allow us to determine where the graph of a function is concave up and concave down. The second derivative will also allow us to identify any inflection points i. We will also give the Second Derivative Test that will give an alternative method for identifying some critical points but not all as relative minimums or relative maximums.
With the Mean Value Theorem we will prove a couple of very nice facts, one of which will be very useful in the next chapter. We will discuss several methods for determining the absolute minimum or maximum of the function. Examples in this section tend to center around geometric objects such as squares, boxes, cylinders, etc.
More Optimization Problems — In this section we will continue working optimization problems.
The examples in this section tend to be a little more involved and will often involve situations that will be more easily described with a sketch as opposed to the 'simple' geometric objects we looked at in the previous section. Linear Approximations — In this section we discuss using the derivative to compute a linear approximation to a function.
We can use the linear approximation to a function to approximate values of the function at certain points. While it might not seem like a useful thing to do with when we have the function there really are reasons that one might want to do this. We give two ways this can be useful in the examples.
1. Limits and Differentiation
Differentials — In this section we will compute the differential for a function. We will give an application of differentials in this section. However, one of the more important uses of differentials will come in the next chapter and unfortunately we will not be able to discuss it until then.
Newton's Method is an application of derivatives will allow us to approximate solutions to an equation. There are many equations that cannot be solved directly and with this method we can get approximations to the solutions to many of those equations. Business Applications — In this section we will give a cursory discussion of some basic applications of derivatives to the business field.
Note that this section is only intended to introduce these concepts and not teach you everything about them. Integrals - In this chapter we will give an introduction to definite and indefinite integrals. We will discuss the definition and properties of each type of integral as well as how to compute them including the Substitution Rule. We will give the Fundamental Theorem of Calculus showing the relationship between derivatives and integrals. We will also discuss the Area Problem, an important interpretation of the definite integral.
Indefinite Integrals — In this section we will start off the chapter with the definition and properties of indefinite integrals. We will not be computing many indefinite integrals in this section.
This section is devoted to simply defining what an indefinite integral is and to give many of the properties of the indefinite integral. Actually computing indefinite integrals will start in the next section. Computing Indefinite Integrals — In this section we will compute some indefinite integrals.
The integrals in this section will tend to be those that do not require a lot of manipulation of the function we are integrating in order to actually compute the integral. As we will see starting in the next section many integrals do require some manipulation of the function before we can actually do the integral. We will also take a quick look at an application of indefinite integrals.
Substitution Rule for Indefinite Integrals — In this section we will start using one of the more common and useful integration techniques — The Substitution Rule. With the substitution rule we will be able integrate a wider variety of functions. The integrals in this section will all require some manipulation of the function prior to integrating unlike most of the integrals from the previous section where all we really needed were the basic integration formulas.
More Substitution Rule — In this section we will continue to look at the substitution rule.
The problems in this section will tend to be a little more involved than those in the previous section. Area Problem — In this section we start off with the motivation for definite integrals and give one of the interpretations of definite integrals. As we will see in the next section this problem will lead us to the definition of the definite integral and will be one of the main interpretations of the definite integral that we'll be looking at in this material.
Definition of the Definite Integral — In this section we will formally define the definite integral, give many of its properties and discuss a couple of interpretations of the definite integral.
Calculus I - Functions
We will also look at the first part of the Fundamental Theorem of Calculus which shows the very close relationship between derivatives and integrals. Computing Definite Integrals — In this section we will take a look at the second part of the Fundamental Theorem of Calculus. This will show us how we compute definite integrals without using the often very unpleasant definition. The examples in this section can all be done with a basic knowledge of indefinite integrals and will not require the use of the substitution rule.
Included in the examples in this section are computing definite integrals of piecewise and absolute value functions.
Substitution Rule for Definite Integrals — In this section we will revisit the substitution rule as it applies to definite integrals. The only real requirements to being able to do the examples in this section are being able to do the substitution rule for indefinite integrals and understanding how to compute definite integrals in general.
Applications of Integrals - In this chapter we will take a look at some applications of integrals. Average Function Value — In this section we will look at using definite integrals to determine the average value of a function on an interval.
We will also give the Mean Value Theorem for Integrals. We will determine the area of the region bounded by two curves. More Volume Problems — In the previous two sections we looked at solids that could be found by treating them as a solid of revolution.
Not all solids can be thought of as solids of revolution and, in fact, not all solids of revolution can be easily dealt with using the methods from the previous two sections.
Work — In this section we will look at is determining the amount of work required to move an object subject to a force over a given distance.
Also included are a brief review of summation notation, a discussion on the different 'types' of infinity and a discussion about a subtlety involved with the constant of integration from indefinite integrals. Proof of Various Limit Properties — In this section we prove several of the limit properties and facts that were given in various sections of the Limits chapter.
Proof of Trig Limits — In this section we give proofs for the two limits that are needed to find the derivative of the sine and cosine functions using the definition of the derivative. Area and Volume Formulas — In this section we derive the formulas for finding area between two curves and finding the volume of a solid of revolution.
Types of Infinity — In this section we have a discussion on the types of infinity and how these affect certain limits. Note that there is a lot of theory going on 'behind the scenes' so to speak that we are not going to cover in this section. This section is intended only to give you a feel for what is going on here. To get a fuller understanding of some of the ideas in this section you will need to take some upper level mathematics courses. Summation Notation — In this section we give a quick review of summation notation.
To complete the problem, here is a complete list of all the roots of this function. Also note that, for the sake of the practice, we broke up the compact form for the two roots of the quadratic. You will need to be able to do this so make sure that you can. This example had a couple of points other than finding roots of functions.
The first was to remind you of the quadratic formula. In fact, the answers in the above example are not really all that messy. So, here is fair warning. One of the more important ideas about functions is that of the domain and range of a function. In simplest terms the domain of a function is the set of all values that can be plugged into a function and have the function exist and have a real number for a value.
The range of a function is simply the set of all possible values that a function can take. Example 4 Find the domain and range of each of the following functions. This means that this function can take on any value and so the range is all real numbers. If we know the vertex we can then get the range. Example 5 Find the domain of each of the following functions. Recall that these points will be the only place where the function may change sign.
This means that all we need to do is break up a number line into the three regions that avoid these two points and test the sign of the function at a single point in each of the regions.