Calculus Theory
Calculus Theory: Limits. The calculus often starts with a discussion of one-sided limits and then introduces two-sided limits. Next, informally computing limits using graphs and tables is illustrated; and after this intuitive process, we discuss a technology dilemma. Indeed, computing limits with tables and graphs can produce inaccurate results. Next, infinite limits are illustrated and then different types of limits that do not exist are detailed. Finally, the formal definition of a limit is given and examples are shown how to make an epsilon-delta proof rigorously showing that a limit does exist.
Calculus Theory: Computing Limits. The basic rules that are used to compute limits are stated. Also included are examples on how to use them and limit rules for the trigonometric functions. One of the most important limit rules is that the limit of a continuous function, such as a polynomial or rational function, is the value of the function at the fixed given value. Finally, we state and use the squeeze rule for finding limits.
Calculus Theory: Continuity. Continuity of a function at a point, an important topic of calculus, is defined and extended to intervals. Several examples are given to illustrate what can go wrong with continuity; for example, "poles", "jumps", "holes", or some type of oscillating behavior. Then properties of continuous functions and one-sided continuity are discussed. This topic of calculus also illustrates how to define a function so that continuity on an interval or at a point can be assured.
Calculus Theory: Tangent Lines. The tangent line problem is widely considered to be the instigating idea behind the derivative. Computing the slope of a tangent line was a problem that the French mathematician Pierre de Fermat developed. Picking up on these ideas were Isaac Newton and Gottfried Leibniz, who then developed differential calculus.
Calculus Theory: Differentiation. Computing the limit of the difference quotient can be tedious and require ingenuity; fortunately for a large number of common function there is a better way to compute the derivative. In this topic of calculus, we detail the power rule, product rule and the quotient rule for differentiation. These rules greatly simplify the task of differentiation. We also give examples on how to find the tangent line give some geometric information; and to find the horizontal tangent lines to the graph of a given function.
Calculus Theory: Computing Derivatives. Formulas for finding the derivative of the trigonometric functions, inverse trigonometric functions, exponential functions, and logarithmic functions are given. These are examples of a special class of functions called transcendental functions. Examples are given for using algebraic techniques (quotient rule, product rule, etc.) with these transcendental functions.
Calculus Theory: Rates of Change. A rate of change is an important topic of calculus and we explain the difference between the average rate of change and instantaneous rate of change. We also illustrate the importance of the relative rate of change. Then rectilinear motion; and in particular the falling body problem and discussed. Basically, rectilinear motion refers to the motion of an object that can be modeled along a straight line; and the so-called falling body problems are a special type of rectilinear motion where the motion of an object is falling (or propelled) in a vertical direction.
Calculus Theory: Chain Rule. With a lot of work, we can find derivatives without using the chain rule either by expanding a polynomial, by using another differentiation rule, or maybe by using a trigonometric identity. The derivative would be the same in either approach; however, the chain rule allows us to find derivatives that would otherwise be very difficult to handle. The chain rule is an important topic of calculus, and we give plenty of examples, as well an easily understandable proof of the chain rule.
Calculus Theory: Implicit Differentiation. In this topic of calculus, the procedure of implicit differentiation is outlined and many examples are given. Proofs of the derivative formulas for the inverse trigonometric functions are provided and several examples of using them are given. Also detailed is the logarithmic differentiation procedure which simplifies the process of taking derivatives of equations involving products and quotients.
Calculus Theory: Related Rates. In this topic of calculus, we show how implicit differentiation and the chain rule can be used to calculate the rate of change of one variable in terms of the rate of change of another variable (which may be more easily measured). The procedure of solving a related rates problem is to find an equation that relates two quantities and then use the chain rule to differentiate both sides with respect to time. Then, from knowing the rate of change of one value at a point in time, we can calculate the rate of change of another quantity at that moment in time. The equation which describes the application is derived from the verbal description and is called a mathematical model of the problem. For these equations we can relate different rates of change by using the chain rule and implicit differentiation.
Calculus Theory: Approximations. Differentials are motivated, defined, and then used to approximate real numbers. The linearization of a function around a point (via the tangent line) is illustrated. Newton's method for approximating the zeros of a function is also detailed along with a sufficient condition on the convergence of the Newton method.
Calculus Theory: Optimization. The absolute maximum and absolute minimum of a function is defined. Examples are given of each type for some familiar functions. The extreme value theorem is stated and it is shown (through examples) how the assumptions in the extreme value theorem are necessary. Relative extremum of a function are also defined. The critical number theorem will state that a relative extrema of a continuous function can occur only at a critical number, but it does not say that a relative extrema must occur at each critical number. The classical example is the cubic function; the derivative is 0 at 0 but there is no relative extrema at 0. The cubic function merely has a horizontal tangent at 0. In summary, given a continuous function on a closed bounded interval, to find the absolute extrema we first find all critical numbers and then compute the values of the given function at these numbers and on the boundary.
Calculus Theory: Mean Value Theorem. In this topic of calculus, we detail Rolle's Theorem and the Mean Value Theorem. We provide examples and illustrate why the hypotheses of these two theorems are necessary. We also give applications and detail two other theorems which are consequences of the Mean Value Theorem. We also emphasize that the Mean Value Theorem tells us that between two fixed points of time, the instantaneous velocity is equal to the average velocity.
Calculus Theory: Numerical Integration. Numerical integration is important because of the fact that finding the antiderivative of a given function is not always simple. For example the natural exponential function to a (non-trivial) quadratic polynomial power is not simple. Numerical integration was first done by Archimedes; his, "method of exhaustion" has been around thousands of years. A calculus student is often expected to master four common techniques. The left and right rules are the most straight-forward to learn. They can be applied with either uniform width subintervals or varying width subintervals. They consist of using a Riemann sum where the subinterval representatives are chosen as the left-endpoints or the right-endpoints, respectively. Other methods of numerical integration include the midpoint rule, trapezoidal rule, and Simpson's rule.
Cite this as:Calculus Theory
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/calculus-theory.html
