Calculus I

1. Course information:

 

a) Course Description

This course is designed to develop the topics of differential and integral calculus. Emphasis is placed on limits, continuity, derivatives and integrals of algebraic and transcendental functions of one variable.

b) Course ObjectivesThe Objectives of this course to introduce limits and continuity, and develop skills for their determination. And provide students with skills related to applications of the derivative after introduce the derivative. Introduce the definite and indefinite integrals, and develop skills for their evaluation. Finally, students Provide skills related to some applications of the integral.

c) Course Learning Outcomes

At the end of the course, the students should be able to Apply the definition of limit to evaluate limits by multiple methods and use it to derive the definition and rules for differentiation and integration. Solve basic differential equations. Use the fundamental theorem of calculus to solve some problems. Deal with transcendental functions and Find the derivative of trigonometric, exponential, and logarithmic functions.

2. Course content:

Definition of Function and inverse Function

A function is a special relationship where each input has a single output. It is often written as “f(x)” where x is the input value.

An inverse function is a function

that undoes the action of another function. A function is the inverse of a function if whenever y = f ( x ) then x = g ( y ) . In other words, applying and then is the same thing as doing nothing.

Limits and Continuity:

The concept of the limit is one of the most crucial things to understand in order to prepare for calculus. A limit is a number that a function approaches as the independent variable of the function approaches a given value. For example, given the function f (x) = 3x, you could say, “The limit of f (x) as x approaches 2 is 6.” Symbolically, this is written http://img.sparknotes.com/figures/7/79e791d6cecbfa0779fcde1c44c51e99/latex_img12.giff (x) = 6.

So:

if we can make the values of arbitrarily close to L (as close to L as we like) by taking x to be sufficiently close to a (on either side of a) but not equal to a.

Limits at Infinity

The symbols and are not regarded as real numbers. They are symbols to indicate that a number increases or decreases indefinitely. When the limit of a function is or no limit exists; the symbol is used for convenience only.

Limits properties

  • the limit of a sum is the sum of the limits.
  • The limit of a difference is the difference of the limits.
  • The limit of a constant times a function is the constant times the limit of the function.
  • The limit of a product is the product of the limits.
  • The limit of a quotient is the quotient of the limits (provided that the limit of the denominator is not 0).

Continuity is another far-reaching concept in calculus. A function can either be continuous or discontinuous. One easy way to test for the continuity of a function is to see whether the graph of a function can be traced with a pen without lifting the pen from the paper. For the math that we are doing in precalculus and calculus, a conceptual definition of continuity like this one is probably sufficient, but for higher math, a more technical definition is needed. Using limits, we’ll learn a better and far more precise way of defining continuity as well. With an understanding of the concepts of limits and continuity, you are ready for calculus.

A function f (x) is continuous if and only if

Some Properties of continuity

  • Polynomials are continuous functions.
  • Rational functions are continuous everywhere except at the points, where the denominator is zero
  • If the function g is continuous at the point c and the function f is continuous at the point g(c), then the composition is continuous at c.
  • If a function f is continuous and has an inverse, then is also continuous.

Differentiation

Differentiation, in mathematics, process of finding the derivative, or rate of change, of a function. In contrast to the abstract nature of the theory behind it, the practical technique of differentiation can be carried out by purely algebraic manipulations, using three basic derivatives, four rules of operation, and a knowledge of how to manipulate functions.

The function Is called the derivative with respect to x of the function f, the domain of consists of all the points for which the limit exists.

A function f(x) is called differentiable at x=a if f′(a) exists and f(x) is called differentiable on an interval if the derivative exists for each point in that interval.

If f(x) is differentiable at x=a then f(x) is continuous at x=a.

Some Differentiation equation:

Applications of Derivatives

Applications of the Derivative identifies was that this concept is used in everyday life such as determining concavity, curve sketching and optimization. Topics include:

Rates of Change: can be positive or negative. This corresponds to an increase or decrease in the -value between the two data points. When a quantity does not change over time, it is called zero rate of change.

Critical Points: Critical points are places where the derivative of a function is either zero or undefined. These critical points are places on the graph where the slope of the function is zero. All relative maxima and relative minima are critical points, but the reverse is not true.

The three points are critical points of the green function. The red point is a local maximum and the blue point is a local minimum. The yellow point is neither a maximum or a minimum

Minimum and Maximum Values: Finding relative maxima and minima of a function can be done by looking at a graph of the function. A relative maximum is a point that is higher than the points directly beside it on both sides, and a relative minimum is a point that is lower than the points directly beside it on both sides. Relative maxima and minima are important points in curve sketching, and they can be found by either the first or the second derivative test.

intervals od increase and Decrease: Finding intervals of increase and decrease of a function can be done using either a graph of the function or its derivative. These intervals of increase and decrease are important in finding critical points, and are also a key part of defining relative maxima and minima and inflection points.

L’Hospital’s Rule:

Newton’s Method: Newton’s Method, also known as the Newton Raphson Method, is a way of approximating numerical solutions (i.e., x-intercepts or zeros or roots) to equations that are too hard for us to solve by hand.

The idea behind Newton’s Method is to start with an initial guess which is reasonably close to the true root, and then to use the tangent line to obtain another x-intercept that is even better than our initial guess.

Integral

An integral is a mathematical object that can be interpreted as an area or a generalization of area. Integrals, together with derivatives, are the fundamental objects of calculus. Other words for integral include antiderivative and primitive.

The integral expression is:

The Applications of Integration is:

Here we have some properties of integral function:

Transcendental function: In mathematics, a function not expressible as a finite combination of the algebraic operations of addition, subtraction, multiplication, division, raising to a power, and extracting a root. Examples include the functions log x, sin x, cos x, ex and any functions containing them. Such functions are expressible in algebraic terms only as infinite series. In general, the term transcendental means non-algebraic.

3. Summary

In this course, students have a good idea about the basic of Function, Limits, Integral, derivative.

And can analysis and solve some problems using that concepts.

4. References

[1] James Stewart, Calculus Early Transcendentals, 6e, Thomson Brooks/Cole,2008.

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