Calculus II

1. Course information:

Course description

This is the second course of a three-part Calculus sequence; this course is an introduction to integral calculus. It develops the concept of the integral and its applications. Other topics include techniques of integration, improper integrals, sequences and series of numbers, Taylor series, polar coordinates, parametric equations, and separable differential equations.

b) COURSE OBJECTIVES

The objective of this course for students is to introduce inverse Trigonometric and Hyperbolic functions and their properties. Having a good information about techniques of integration and using this technique to analyze and solve a lot of problems in calculus.

Introduce sequences and use it to develop the study of properties of infinite series. And introduce polar coordinate system and find the tangent lines and arc length for parametric and polar curves.

C) COURSE LEARNING OUTCOMES

By the end of the course, the students should be able to Identify the properties of inverse trigonometric functions, hyperbolic, and inverse hyperbolic functions, the derivatives and integrals of these functions. understanding different integration techniques and how can use it to solve simplify integral problems.

At the end of this course, students also will be able to Identify the properties of sequences and infinite series and their limits. Use Taylor and Maclaurin series to approximate functions and use parametric and polar equations to solve applied problems including area and arc length.

2. COURSE CONTENTS:

Trigonometric function, In mathematics, one of six functions (sine, cosine, tangent, cotangent, secant, and cosecant) that represent ratios of sides of right triangles. They are also known as the circular functions, since their values can be defined as ratios of the x and y coordinates of points on a circle of radius 1 that correspond to angles in standard positions.

The fundamental trigonometric identity is , in which θ is an angle. Certain intrinsic qualities of the trigonometric functions make them useful in mathematical analysis. In particular, their derivatives form patterns useful for solving differential equations.

In the next figure, we have the Trigonometric and invers function.

Her we have some properties of trigonometric function

Hyperbolic functions

The hyperbolic functions have similar names to the trigonometric functions, but they are defined in terms of the exponential function. we define the three main hyperbolic functions, their graphs and mention their inverse functions and reciprocal functions.

Hyperbolic graph

Invers hyperbolic function formula and his graph

Techniques of Integration:

  • Substitution:

to replace expressions involving square roots with expressions that involve standard trigonometric functions, but no square roots. Integrals involving trigonometric functions are often easier to solve than integrals involving square roots

  • Powers of sine and cosine

Functions consisting of products of the sine and cosine can be integrated by using substitution and trigonometric identities. These can sometimes be tedious, but the technique is straightforward.

Example:

Integration by Parts: there is a technique that will often help to uncover the product rule.

Start with the product rule:

This technique for turning one integral into another is called integration by parts, and is usually written in more compact form. If we let u = f (x) and v = g (x) then

  • Rational Functions: A rational function is a fraction with polynomials in the numerator and denominator. For example,

There is a general technique called “partial fractions” that, in principle, allows us to integrate any rational function. The algebraic steps in the technique is rather cumbersome if the polynomial in the denominator has degree more than 2, and the technique requires that we factor the denominator, something that is not always possible. However, in practice one does not often run across rational functions with high degree polynomials in the denominator for which one has to find the antiderivative.

So, all we need to do is find A and B so that 7x − 6 = (A + B) x + 3A − 2B, which is to

say, we need 7 = A + B and −6 = 3A − 2B.

Numerical Approximations

It is often the case, when evaluating definite integrals, that an antiderivative for the integrand cannot be found, or is extremely difficult to find. In some instances, a numerical approximation to the value of the definite value will suffice. The following techniques can be used.

  • Riemann Sum
  • Trapezoidal Rule
  • Simpson’s Rule

Parametric Equations and Polar Coordinates

A parametric equation is where the x and y coordinates are both written in terms of another letter. This is called a parameter and is usually given the letter t or θ. (θ is normally used when the parameter is an angle, and is measured from the positive x-axis.)

The parametric equations of a curve are the pair:

For example:

Chain rule is

Each value of t defines a point (x,y)=(f(t),g(t)) that we can plot. The collection of points that we get by letting t be all possible values is the graph of the parametric equations and is called the parametric curve.

Polar Coordinate and Curve: he polar coordinate system consists of an origin, or pole O and the polar axis, which is usually chosen to be the horizontal axis.

The Polar Coordinates of a point P are specified by two numbers, . Here r is the distance OP from the origin to the point, and µ is the angle measure, in radians, between the polar axis and the line OP.

To convert from use

To convert from use

You must choose the value of based on the appropriate quadrant in which the point (x; y) lies. This is because the inverse tangent function is not single-valued everywhere.

Area Formulae

We can use the idea that the area of a circular sector of radius r and central angle is to prove the following area formula:

The area of the region enclosed by the polar curve between the rays is given by:

Sequences and Infinite Series:

he lists of numbers you generate using a numerical method like Newton’s method to get better and better approximations to the root of an equation are examples of (mathematical) sequences.

Sequences are infinite lists of numbers

The feeling we have about numerical methods like Newton’s method and the bisection method is that if we continue the iteration process more and more times, we would get numbers that are closer and closer to the actual root of the equation. In other words:

Sequences for which exists and is finite are called convergent sequences, and other sequence are called divergent sequences.

A series is any “infinite sum” of numbers. Usually there is some pattern to the numbers, so we can communicate the pattern either by giving the first few numbers, or by giving an actual formula for the nth number in the list. For example, we could write

Series Limit: It is only natural to define the sum or limit of the series to be equal to the limit of the sequence of its partial sums, if the latter limit exists.

So, for the Meg Ryan series, we really do have

Geometric series: One kind of series for which we can find the partial sums is the geometric series. The Meg Ryan series is a specific example of a geometric series.

A geometric series has terms that are (possibly a constant times) the successive powers of a number. The Meg Ryan series has successive powers of .

Taylor Series: is a representation of a function as an infinite sum of terms that are calculated from the values of the function’s derivatives at a single point.

https://archive.is/SGDCU/b824d1cf803a02e1f5ad8ae083edd0acdecb0c95.gif

The Taylor Series for some function is:

.

 3. Summary:

In this course we have some good information about technique of integral, Trigonometric and Hyperbolic function, Sequence and infinite Series that can help student to solve problem that related to these topics.

4. References:

[1] James Stewart, Calculus: Early Transcendentals,1983

[2] Bruce H Edwards, Ron Larson, Calculus: Early Transcendental Functions,1987

[3] James Stewart, Essential Calculus: Early Transcendentals,2006.

[4] George B. Thomas, Thomas’ Calculus,1951.

 

اترك تعليقاً

لن يتم نشر عنوان بريدك الإلكتروني. الحقول الإلزامية مشار إليها بـ *

WhatsApp chat