Students should bear in mind that the main purpose of learning calculus is not just knowing how Accompanying the pdf file of this book is a set of Mathematica. computerescue.info Table of Contents. Preface. Here are my online notes for my Calculus I course that I teach here at Lamar University . Calculus Made Easy has long been the most populal' calculus pl'imcl~ In this cmatician, Ilcarned the elements of calculus f,'om S. P. ompson's engaging.
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book on the Calculus, basedon the method of limits, that should be within the capacity of In both the Differential and Integral Calculus, examples illustrat-. The right way to begin a calculus book is with calculus. This chapter will jump directly into the two problems that the subject was invented to solve. You will see. Exercises and Problems in Calculus. John M. Erdman. Portland State University. Version August 1, c John M. Erdman. E-mail address: [email protected]
By reading the book carefully, students should be able to understand the concepts introduced and know how to answer questions with justification. At the end of each section except the last few , there is an exercise.
Students are advised to do as many questions as possible. Most of the exercises are simple drills. Such exercises may not help students understand the concepts; however, without practices, students may find it difficult to continue reading the subsequent sections. Preview the PDF.
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You should come see our Mathematics documents. Infinite Limits — In this section we will look at limits that have a value of infinity or negative infinity.
We will concentrate on polynomials and rational expressions in this section. Continuity — In this section we will introduce the concept of continuity and how it relates to limits. We will also see the Intermediate Value Theorem in this section and how it can be used to determine if functions have solutions in a given interval. The Definition of the Limit — In this section we will give a precise definition of several of the limits covered in this section.
We will work several basic examples illustrating how to use this precise definition to compute a limit.
Derivatives - In this chapter we introduce Derivatives. We cover the standard derivatives formulas including the product rule, quotient rule and chain rule as well as derivatives of polynomials, roots, trig functions, inverse trig functions, hyperbolic functions, exponential functions and logarithm functions. We also cover implicit differentiation, related rates, higher order derivatives and logarithmic differentiation. The Definition of the Derivative — In this section we define the derivative, give various notations for the derivative and work a few problems illustrating how to use the definition of the derivative to actually compute the derivative of a function.
Interpretation of the Derivative — In this section we give several of the more important interpretations of the derivative. We discuss the rate of change of a function, the velocity of a moving object and the slope of the tangent line to a graph of a function.
Differentiation Formulas — In this section we give most of the general derivative formulas and properties used when taking the derivative of a function. Examples in this section concentrate mostly on polynomials, roots and more generally variables raised to powers. Product and Quotient Rule — In this section we will give two of the more important formulas for differentiating functions.
We will discuss the Product Rule and the Quotient Rule allowing us to differentiate functions that, up to this point, we were unable to differentiate.
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Derivatives of Trig Functions — In this section we will discuss differentiating trig functions. Derivatives of Exponential and Logarithm Functions — In this section we derive the formulas for the derivatives of the exponential and logarithm functions.
Derivatives of Inverse Trig Functions — In this section we give the derivatives of all six inverse trig functions. We show the derivation of the formulas for inverse sine, inverse cosine and inverse tangent.
Derivatives of Hyperbolic Functions — In this section we define the hyperbolic functions, give the relationships between them and some of the basic facts involving hyperbolic functions. We also give the derivatives of each of the six hyperbolic functions and show the derivation of the formula for hyperbolic sine. Chain Rule — In this section we discuss one of the more useful and important differentiation formulas, The Chain Rule.
With the chain rule in hand we will be able to differentiate a much wider variety of functions. As you will see throughout the rest of your Calculus courses a great many of derivatives you take will involve the chain rule!
Implicit Differentiation — In this section we will discuss implicit differentiation. Not every function can be explicitly written in terms of the independent variable, e.
Implicit differentiation will allow us to find the derivative in these cases. Knowing implicit differentiation will allow us to do one of the more important applications of derivatives, Related Rates the next section.
Related Rates — In this section we will discuss the only application of derivatives in this section, Related Rates. In related rates problems we are give the rate of change of one quantity in a problem and asked to determine the rate of one or more quantities in the problem. This is often one of the more difficult sections for students.
(x4 x2 6)x-1/3 dx (x4 x-1/3 x2 x-1/3 6 x-1/3 ) dx x11/3dx x5/3dx 6 x-1/3dx
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. 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.
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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.
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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.
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.It is never too late to start learning and it would be a shame to miss an opportunity to learn a tutorial or course that can be so useful as Understanding Basic Calculus especially when it is free! We will concentrate on polynomials and rational expressions in this section. Limits and continuity Types of discontinuities: Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.
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.
Students are advised to do as many questions as possible. Chung Downloads: We will cover the basic notation, relationship between the trig functions, the right triangle definition of the trig functions.
Linear Approximations — In this section we discuss using the derivative to compute a linear approximation to a function.