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LIMITS INTRODUCTION WITH
- Introduction to Limits in Calculus Eight worked limit problems. Illustrates the concepts of limits using graphs. The limits of periodic functions, such as sine functions, are found using graphs and using the calculator. Rating A
- Limits of Basic Functions Just a few introductory examples that illustrate properties of limits, such as the limit of a constant function, the limit of a first degree equation.
- Introduction and Basic Definitions 5 worked limit problems. Includes Absolute Value of x over x and sin(1/x). Some more theoretical or thinking problems.
- Definition of a Limit A few examples related to negative infinity and positive infinity.
- Basic Limit Problems. 5 introductory limit problems are given for the beginner. Shows graphically that as x approaches 0, the sin(x)/x appoaches 1. However there is a hole at x=0.
- Some Basic Properties 5 limit problems solved. One problem requires that you know that limit of sin(x)/x = 1 and the limit cos(x) =1 both as x approaches 0. A fairly difficult limit problem is also given that requires rationalization of the denominator and numerator. One solution requires cross-multiplication and simplification.
- Find Limits of Functions in Calculus Fifteen limit problems with complete solutions with explanations. This section focuses on limits that involve absolute value, rational expressions, the ratio of two polynomials, as well as complex problems that involve the ratio of trigonometric expressions, natural logarithms, exponents and complex radicals. Various methods of solving for limits is illustrated, such as the use of L'Hopitals rule. Rating A
- Properties of Limits in Calculus Five examples illustrating the basic properties of limits: Setup well for illustrating the use of f(x) and g(x) involving a limit problem when its divided up into two or more functions: the limit of the sum of two functions, the limit of the product of two functions, the limit of the quotient of two functions, the limit of the nth root of a function. Basic formatting of the examples could use improvement. Rating C
- The Limit Includes five examples with solutions. Provides solutions to some more difficult limit problems for piecewise functions. The limit of the Heaviside function is shown to not exist. Describes the limit function from a graphical standpoint, but more in detail than section 1. Touches on the concept of epsilon.
- Limits an Introduction An elementary introduction to limits
- Understanding Epsilon-Delta proofs
- Finding Delta Given a Specific Epsilon. Four problems with worked solutions
- Finding Delta Given a Specific Epsilon. Ten problems with worked solutions These problems are relatively easy and serve as a good introduction to the basic solution to delta epsilon problems.
- Verification of Limits Through the Relationship between Epsilon and Delta. Eight problems with explained solutions
- Determination of Vertical Asymptote 10 problems with worked solutions
- Determination of Horizontal Asymptotes using Limits 10 problems with worked solutions.
- Definition of a Limit and Calculation of Delta for a Given Epsilon and a General Epsilon. Provides example problems with solutions
- The Definition of the Limit Describes analytically the definition of a limit. Lists nine definitions related to limits. Has six problems with explained solutions for limit proofs.
- One-Sided Limits Offers an explanation of one sided limits. Includes four multi-part problems. Provides a method for quick graphical analysis of limits.
- Limit Properties A concise list of nine basic limit properties. Three high level limit problems with complete solutions.
- Basic Limit Laws
- Zero over Zero
- Limits - Formal (e,d) Definitions
- Limits - Techniques 1 - Finite fixed point (x0), finite limit (y0)
- Computing Limits
- Introduction to Limits
- Infinite Limits
- Rules for the Determination of Horizontal and Vertical Asymptotes for Rational Functions. Includes ten worked examples that illustrates how to determine if the the function defines a hole or a vertical asymptote. Use of rules that pertain the degree of of the polynomial in the numerator and denominator are listed. Limits are not used for these examples.
- Definition and Determination of Asymptotes Through the Use of Calculus from Wikipedia. Includes section on oblique asymptotes, Specials cases for determination of horizontal asymptote for different polynomial degrees. Determination of the horizontal and vertical asymptote through transformation techniques. Arctanget asymptote is discussed. Taking the limit of a function as x approaches infinity and negative infinity to determine the horizontal asymptotes.
- Limits at Infinity 5 problems with complete solutions. Emphasizes the use of factoring to find limits. In this problem set, limits of functions as x approaches positive infinity and negative infinity to find the horizontal asymptotes of a function.
- Limits at Infinity Provides a few limit word problems with solutions. using exponential decay functions. Also uses limits evaluated at positive and negative infinity to find horizontal asymptotes. Vertical asymptotes, which are determined initially through examination of the denominator, are verified, by verifying that the limit approaches negative infinity and positive infinity as x approaches the value of the asymptote.
- Limits at Infinity Provides several problems and solutions with detailed explanations through graphs. Emphasizes dominating terms to logically evaluate limits. For example, the growth rate of the numerator function f(x) is compared against the denominator function growth rate, g(x). Provides basic rules for evaluating the limits of polynomial based rational expressions. Discusses the limits of periodic and oscillating functions as they approach infinity. Illustrates the squeeze theorem to find the limit of a function that is periodic as it approaches specific numbers, that is not negative or positive infinity.
- Limits At Infinity, Part I
- Limits At Infinity, Part II
- Limits of Functions as x approaches a Constant 15 problems with complete solutions for limits that often are mistakenly miscalculated at infinity. The use of L'Hopitals rule is used in the solution for some of the problems.
- Limits to Infinity
- Limits and Infinity
- Limits of Functions as X Approaches Infinity 20 problems with complete solutions as for limits approaching infinity
- Limits of Functions as X Approaches Infinity
- Precise Limits of Functions as x Approaches a Constant
- Squeeze Theorem- Provides solution to the limit of x^2*sin(1/x) through use of squeeze theorem. Also provides proof of the squeeze theorem, which is also known as the pinching theorem and the sandwich theorem.
- Use of Squeezing Theorem to Find Limits Intermediate problem illustrates the setup of a math function for evaluating the limit using the squeeze theorem also known as the sandwich thereon. Requires that you determine the range of one part of the function in order to bound the limit. Proof http://www.analyzemath.com/calculus/limits/squeezing.html
- The Pinching or Sandwich Theorem
- The Squeeze Theorem - More Advanced Problems. Features 3 problems with solutions that illustrate the use of the squeeze theorem. Provides an introduction that explains basic procedure for using the squeeze theorem.
- Limits of Functions using the Squeeze Principle
- Calculate Limits of Trigonometric Functions Seven examples complete with solutions for finding limits of complex trigonometric functions.
- Limit Problems with Solutions from Spark Notes
- Limits and L'Hopitals Rule
- Limits Test Review Summaries and Cheat Sheets A two page review sheet of the basic limit definitions, properties and theorems you will need to know to excel in limits. Summarizes the common types of continuous functions. Also includes several different procedures used for finding limits, factoring, rationalizing, rational expression combinations, limits of functions to remember, limits of piecewise functions.
- List of Unsolved Problems about Limits
CONTINUITY WITH PROBLEMS AND SOLUTIONS
- Continuity 1 Definitions of Continuity with Respect to Limits. Includes 5 worked example problems. Examples include Jump Discontinuities and problems related to the Intermediate Value Theorem. Recites that the continuity of function at a point is only guaranteed if the limit of the function as x approaches that point is equal to the same value of the function at that point. An example illustrates the use of the Intermediate Value Theorem to find the roots of a polynomial equation.
- Continuity 2 Illustrates graphs of different types of discontinuities for different functions. Includes jump discontinuities, removable discontinuities, for functions such as 1/x, sine functions, and piecewise functions. A few example problems with complete and graphed explanations and solutions. Expresses the point that the function, f(x), is continuous at c if and only if the right- and left-hand limits exist and are both equal to f(c)
- The Intermediate Value Theorem
- Continuity contains solved problems that require that the continuity of a specific function be discussed. Coefficients of a functions must be determined such that the piecewise functions is continuous at given point that they are approaching at the same time. Solutions to these problems can be done in two different ways. The solutions presented here takes the right and left hand limits of the functions and equates them.
- Continuity of Functions of One Variable Lists seven properties of continuous functions which are products, quotients, sums and differences of continuous functions. Also lists functions that are always continuous such as exponentials, sine and cosine trigonometric functions and polynomial functions. Provides 15 problems with detailed solutions.
- Continuous Functions in Calculus Introduction and definition of continuous functions. Provides examples of continuous and discontinuous functions. Defines three required conditions of continuity. Provides three example continuity problems with complete solutions. Problems emphasize continuity over intervals.
- Continuity Theorems and Their use in Calculus Provides five theorems used for the evaluation of continuous functions: polynomial, sine, cosine, arctan, e^x, continuity of a rational functions, continuity of composite functions. Also includes a few worked examples to clarify the main points of the five theorems.
- Continuity Problems and Solutions from Spark Notes: Provides 7 problems with complete explanations. Problems require that you determine if the function is continuous at a specific point or over a specific interval
- Continuity of Piecewise Defined Functions from the University of Tennessee at Knoxville - Provides 8 piecewise problems with complete solutions. Problems illustrate how to create a piecewise function that will be continuous over an interval or at a point.
- Differentiation and Continuity
- The Bisection Method
LIMITS AND DERIVITIVES WITH PROBLEMS AND SOLUTIONS
- The Definition of the Derivative Includes worked examples on use of the definition of the derivative to calculate the derivative. Also provides information on alternative notation used instead of the more common prime notation.
- Definition of the Derivative
- The Definition of the Derivative
- Using the Definition to Compute the Derivative
- Difference Quotient The difference quotient is defined as the slope of the secant line. Four example problems with solutions are given that make use of the definition as it applies to the limit of the difference quotient.
- Limit Definition of the Derivative - HMC Calculus Tutorial Emphasizes the geometric definition of the derivative through the secant and tangent. Includes some examples.
- Interpretation of the Derivative Relates the concept of rate of change, such as velocity, water flow and others to the derivative. Provides a method on how to solve for the derivative of a function from its graph.
- Use Definition to Find Derivative Provides 3 worked problems with solutions using the definition of a derivative to find the derivative of a functions.
- Derivatives Using the Limit Definition 12 problems with complete solutions focused on using the limit definition to compute the derivative. Also has example problems that illustrate the process of showing the a function is differentiable.
- Proof of Various Limit Properties Provides a list of 9 limit properties and provides complete proofs for each limit property. Proofs utilize triangle inequality, induction and the other properties of limits for the proofs.
- Proof of Two Commonly Used Trigonometric Limits A graphical illustration is included to improve understanding. Limits are for sin(x)/x and (cos( h) - 1)/ h
- Proof of Trigonometric Limits Provides another proof for the trigonometric limits of sin (x) / x and (cos( x) - 1)x
- Instantaneous Velocity as Defined in the Limit. Desribes instaneous velocity in the simpler of terms with a table and numerical calculations. Also defines instantaneous velocity as a limit of distance over time.
- Solving for Instaneous Velocity Using the Definition of a Limit. Provides a completely worked example that requires you solve for the instaneous velocity at a given time given a distance function.
- Applications of Limits to Find Instaneous Velocity. Average velocity is discussed in relation to the slope of secant line. Determining instaneous velocity at a given point in time from a function with a dependent variable of distance and an independent variable of time using the definition of a limit. Requires that you evaluate the limit as the final time approaches the beginning time or as delta time approaches zero. Solves the problem numerically. (s(t2) - s(t1)) /( t2 -t1) as t1 approaches t2 or delta t approaches zero.
- Derivative - Wikipedia, the free encyclopedia Provides a complete overview of continuity, limits and derivatives. Illustrates the geometric concept of a derivative through an animation of the tangent and secant functions.
REVIEW OF LIMITS AND CONTINIUITY
- Review for limits and continuity Includes worked problems, graphing facilities and calculator instructions. Visual explanations of epsilon-delta definition, numerical determination of limits, bisection method, horizontal asymptotes, vertical asymptotes
DERIVITIVE RULES AND PROCEDURES
- Differentiation and Continuity
- Calculus: Derivatives
- Calculus: Properties of Derivatives
- Solving for Derivitives of Polynnominals Using Basic Derivitive Rules.
- Rules of Differentiation of Functions in Calculus Basic rules of differentiation with example and solution for each rule. Includes power rule, sum rule, difference rule, product rule, quotient rule and constant rules of differentiation.
- Product and Quotient Rule Includes four product and quotient differentiation problems with solutions. Emphasizes a common mistake that students make when using the product and quotient rule. Also gives a related rates word problem that requires the use of the quotient rule to solve. Indicates that the product and quotient rule can be applied to products and quotients that consist of three or more functions.
- Product Rule for Derivatives Three product rule differentiation problems with solutions. Illustrates the product rule for finding derivatives that are the products of trigonometric functions and the exponential function.
- The Product Rule
- Proof of the Derivative Product Rule http://www.math.hmc.edu/calculus/tutorials/prodrule/proof.pdf
- Differentiation Using the Product Rule 19 product rule differentiation problems with solutions. Illustrates by example use of the triple product rule. As well, lines that are tangent and parallel to a function are determined with the product rule. Product rule derivative problems are determined with products that include exponentials, the exponential, natural logarithms, and trigonometric functions such as sine, cosine and tangent.
- Quotient Rule for Derivatives Two quotient rule derivative problems with solutions.
- Proof of the Derivative Quotient Rule http://www.math.hmc.edu/calculus/tutorials/quotient_rule/proof.pdf
- The Quotient Rule
- Differentiation Using the Quotient Rule Twenty-one quotient rule differentiation problems with solutions. Problems require the use of the quotient rule for derivatives to solve. Some problems also require that the chain rule also be applied.
- Differentiation Formulas
Ten differentiation problems using the sum and difference rules for taking derivatives. Also includes an in-depth problem related to velocity and direction of change of velocity.
- Differentiation Using the Chain Rule Twenty chain rule differentiation problems with solutions. Complete explanations are given. Chain rule differentiation problems include functions that include sine, cosine, natural log, arctangent, and exponential functions. Illustrates the chain rule to find a differentiable function and to find the value at a specific point of the function.
- Use the Chain Rule of Differentiation in Calculus Four chain rule differentiation problems with solutions. Shows solutions that use implement du substitution through the evaluation of du/dx and dy/du.
- Chain Rule Features two chain rule differentiation problems with solutions with du/dx and dy/du format. Also shows alternate notation often used to represent chain rule differentiation
- The Chain Rule
- The Chain Rule
- Partial Derivatives Examples
Finding Derivatives of Specific Types of Functions
- Find Derivatives of Functions in Calculus 11 derivative problems with solutions that are solved with the chain rule, product rule and quotient rule
- Differentiation of Trigonometry Functions 18 trigonometric derivative problems with solutions that make use of the derivatives for cosine, sine, tangent, cosecant, secant and cotangent. One problem requires that you find the line that is perpendicular to the tangent line of the trigonometric function.
- Differentiation of Trigonometric Functions 3 trigonometric derivative problems with explained solution.
- Derivatives of Trigonometric Functions 10 trigonometric derisive problems with solutions. Uses the theorems of limits for limit (sin x)/x = 1 and (cos (x)-1)/x = 0 to determine the derivatives. Also illustrates how to find the savings rate as a specific point in time when the financial savings rate is a trigonometric function.
- Derivatives of Trigonometric Functions Shows that limit (cos (x)-1)/x = 0 through the use of the fact that limit (sin x)/x = 1. Also has three derivative trigonometric problems with solutions.
- Differentiation of Inverse Trigonometric Functions 15 inverse trigonometric derivative problems with solutions. Inverse trig functions include arcsin, arccos, arctan, arccotan, arccsc and arcsec. One word problem is also solved. A problem that requires that you take the limit of a function that involves inverse trigonometric problems is also solved.
- Derivative of Inverse Trigonometric Functions 3 inverse trigonometric derivative problems with solutions.
- Derivatives of Inverse Trig Functions Provides a proof of the derivative for the arcsin using composite inverse definitions. Also provides a few examples for finding the derivatives of inverse trig functions.
- Derivative of Inverse Function Example showing how to find the derivative of the inverse of a function, given an inverse. That function is a linear function. Uses another example to show how the properties of inverse functions are used to find the derivatives of the arcsin, also known as the inverse sine function.
- Derivatives of Exponential and Logarithm Functions Basic formulas for finding depravities of exponentials and logarithms are given. Proofs are presented. Two sample problems are given as well as word problem about exponentials and velocity.
- Logarithmic Differentiation 8 examples for solving complex differentiations through the use of logarithmic differentiation.
- Logarithmic Differentiation 3 worked examples for solving complex differentiation through the use of logarithmic differentiation, or what might be referred to also as implicit differentiation
- Logarithmic Differentiation 7 problems which required logarithmic differentiation are explained and solved. These types of problems are identified by the fact that the variable in the equation is also used as an exponent for variable. In these cases the ordinary rules of differentiation do "not" apply and techniques of logarithmic differentiation and / or implicit differentiation are used. These types of problems are common in solutions for differential equations.
- Differentiation of Exponential Functions 4 example exponential differentiation problems with solutions. Functions of the form b^x are solved.
- Find Derivative of y = x^x. Classic example of using logarithmic differentiation to find the derivative of a function that raises the power of a variable to a multiple of the variable itself.
- Differentiation of Logarithmic Functions 4 examples that show how to take the derivative of a function of the form logb(x) where b is the base of the log. Formula for the derivative of a log of a given base is given. Examples provide solutions.
- Differentiation of Hyperbolic Functions Formulas with examples are given on the derivatives of hyperbolic functions. Graphs are given of the hyperbolic functions. sinh, cosh, tanh, sech, coth, csch
- Derivatives of Hyperbolic Functions The derivative of hyperbolic functions are derived with the hyperbolic exponential function definitions. List of derivatives of the the six hyperbolic functions. Graphs are included. Identities of the hyperbolic are given.
- Differentiating Inverse Functions
- Derivatives Involving Absolute Value 2 examples of absolute value derivative problems with complete solutions.
- Differentiating Special Functions A basic review of differentiation of trigonometric, logarithmic, and exponential functions. The derivatives of sine and cosine are derived through the definition of a limit.
- Techniques of Differentiation A few review problems with complete solutions. Problems include quotient and product rule. Also examples illustrate the use of the derivative to find the slope of a tangent line and the slope of a line perpendicular to the tangent lines.
- Derivite Calculator. This calcualtor will give the derivitve of a function entered. It will also plot the original function entered as well as the derivitive of that function.
- Fact, Formula, Test Review Sheets, Cheat Sheets
- Proofs of Derivative Applications Facts/Formulas Fermat’s Theorem, Rolles Theorem, Mean Value Theorem
LIMITS DETERMINATION WITH L'HOPITAL'S RULE, LIMITS BY DERIVITIVES
- L'Hopital's Rule 8 example problems with solutions that use L'Hopitals rules to find the limit of a function.
- L'Hopital's Rule And The Indeterminate forms 0 / 0 5 limit problems with explanations that illustrate finding the limits of functions with indeterminate form 0/0 with derivitives as defined by L'Hopitals rule.
- Indeterminate forms of Limits Five limit problems with explained solution that involve complex functions such as exponential, logs, and trigonometric functions. The seven forms of indeterminate limits are listed. L'Hopital's rule requires that the function has one of the indeterminate forms. The problems require the identification of the indeterminate form type and then use of L'Hopitals limit rule, when and only when, the rules indeterminate form requirement are met.
- L’Hospital’s Rule and Indeterminate Forms 7 limit problems with solutions. These problems are more difficult than most. An introduction to L'Hopitals rule and a list of the seven forms of the indeterminate limits are given. A procedure for converting a function that is a product of functions to a quotient of functions. One problem's solution requires that the natural log of the functions be taken first in order to evaluate form of the indeterminate limit.
TABLES OF DERIVITIVES
- TABLE OF DERIVITIVES A short table of about twenty functions and their derivatives
- Table of Derivatives
- Newton’s Method Two Newton Methods problems with explained solution. Includes introduction to Newton's Method and applications where it is used. Newton's method is a successive approximation method that lends itself as a computer software algorithm. A initial approximation to the numerical answer to the equation must be supplied. '
- Newton's Method to Find Zeros of a Function Three newton methods problems with explained solutions are given. The use of making a rough graph of the function to make an approximation to the numerical solution of the equation, also known as the zeros of the function.
- The Newton-Raphson Method
- The Tangent Line Approximation Explanation of how to find a tangent line at a point on a function
- Linear Approximation of Functions 3 linear approximation problems with solutions. The problems use derivatives to determine a linear approximation to a function at a point on the function
- Linear Approximations Two linear approximation problems with solutions. Includes graphs to illustrate the solution.
- Linear Approximations
- Solve Tangent Lines Problems in Calculus Two tangent line problems that require that a line parallel to the tangent be found.
- Slope of the tangent line and the normal line to a function is determined through the use of a derivitive http://www.cliffsnotes.com/study_guide/Tangent-and-Normal-Lines.topicArticleId-39909,articleId-39887.html
RATE OF CHANGE PROBLEMS
- Basic Related Rate Problems Using Derivitive to Solve. Shows an example and provides several problems with answers only.
- Solve Rate of Change Problems in Calculus 3 rate of change problems with graphical illustrated solutions. One fluid flow rate of change problem, one velocity and distance rate of change problem, and one electrical rate of change problem that involves resistances that change with time.
- Solve Rate of Change Problems in Calculus One rate of change problem with solution involving the rate of air flow and rate of volume change of a balloon. An in-depth and animated graphical illustration for determining the tangent line of a function at a given point.
- Rates of Change of Tangent Lines
- Rates of Change Two rates of change problems. One problem involves cars traveling in different directions.
- Tangent Lines and Rates of Change
RELATED RATE PROBLEMS
- Related Rates Related rate problem with solution. rate at which the radius of the balloon is increasing when the diameter of the balloon is at a given length
- Related Rates 6 multipart related rate problems with graphically illustrated solutions. Related rate classical ladder problem. Related rates of a distances as a function of angular direction Tank leading related rate problem, rate at which water depth is changing at a specific depth. Related rate of length of shadows changing.
MINIMUMS, MAXIMUMS, FIRST AND SECOND DERIVITIVE TEST
- Monotonicity and the Sign of the Derivative
- The First Derivative Maxima and Minima 3 example problems that involve finding the minimum and maximum of a function through the use of the first derivative.
- Critical Points 7 critical ;point problems with solutions. The critical points are found for a number of complex functions. Section also provides a definition for a critical point.
- Critical Points
- Plotting Functions By Using the First Derivitive of a Function. The Shape of the Graph. This web page has 5 examples that illustrate how you plot a graph of a given function by finding the critical points, the points where the maximum and minimum occur. The first deritivive of the function is taken and then set to zero and solved to determine the critical points. Then the direction of the slopes of the intervals between the critical points is evaluated using the the first derivitive and test points between each crtical point.
- Find Critical Numbers of Functions 4 critical number problems with solutions. Definition of critical number (critical point) is given.
- Derivative, Maximum, Minimum of Quadratic Functions Two maximum minimum problems with solutions for quadratic functions (second degree equations or more commonly known as the parabola).
- Minimum and Maximum Values 6 maximum and minimum problems with graphical explained solutions. Includes the mathematical definition for maximum and minimum. Text based and graphical definition of global and local minimum and maximums also called absolute and relative minimum and maximums. Extreme Value theorem and Fermet's theorem are discussed.
- Finding Absolute Extrema 5 absolute extrema problems with solutions.
- Global Extrema
- Concavity and Points of Inflection
- Determine the Concavity of Quadratic Functions 2 concavity examples that illustrate how to determine the concavity of a parabola.
- Concavity and the Second Derivative Test 2 example problems. Explanation and definition of concavity and inflection points. Use of the second derivative test to determine where a function is concave down and concave up and where the inflection points are.
- The Shape of a Graph, Part I 5 examples with solutions that make use of the first derivitive to determine if a function is increasing or decreasing over an interval and if a function ever increases or decrease. Critical points are also used in the solutions to the problem.
- The Shape of a Graph, Part II 3 concavity examples with solutions. Gives a practical working definition of concavity. Also gives the mathematical definition of concavity. Discusses misconceptions related to concave up and concave down.
- First, Second Derivatives and Graphs Of Functions 2 examples of using the first and second derivatives to graph a function. The examples are organized in steps and include graphs to illustrate the somewhat lengthy process.
- Graphing Functions using First and Second Derivatives 11 examples for graphing functions using the first and second derivatives. Horizontal and vertical asymptotes are found in the procedure. Problems, which come compete with worked solutions, include the most basic to the most difficult. Problem number 11 is one of the more interesting problems.
MAXIMUM / MINIMUM OPTIMIZATION PROBLEMS(Using first and second derivitve tests)
- Maximum and Minimum Problems 21 solved maximum and minimum problems. The problem set focuses on solving a variety of word problems that require that you find the minimum or maximum. Section also includes a six step procedure for solving minimum and maximum optimization problems. .
- Minimum Distance Problem Provides two methods to solve a classical minimum distance problem. This minimum distance problem is often seen on many tests, Knowing how to work this problem is often essential is passing a Calculus course.
- Maximum Area of Rectangle - Problem with Solution The example illustrates how to find the largest rectangle that can be inscribed in a given triangle. A graphically illustrated solution is presented.
- Maximum Radius of Circle - Problem with Solution This example illustrates how to find the largest circle that can be inscribed in a right triangle.
- Maximum Area of Triangle - Problem with Solution This example illustrates how to find the largest triangle that can be inscribed in a circle. As with all minimum and maximum problems the first derivative test is used.
- Maximum Area of Triangle - Problem with Solution
- Maximize Volume of a Box This example illustrates how to find the maximum area of a box that can be made from a piece of metal of a given size. This classical problem includes a graphic of the metal sheet as well as a plot of the volume of the box versus the length of the corner cutout on the box.
- Maximize Power Delivered to Circuits This example is a must for electrical engineers studying calculus. The example illustrates the process of finding the resistor value needed to maximize the power delivered to a load. An electrical schematic is given along with the graph of power versus resistance for the circuit.
- Use First Derivative to Minimize Area of Pyramid This example illustrates how to find the minimum surface area of a pyramid that has a square base and a fixed volume of 1000 cubic centimeters.
- Use Derivatives to solve problems: Distance-time Optimization For those wanting to minimize their travel time, this illustrated and animated example shows how to use derivatives to find the fastest route to travel. The solution provides an animated applet that lets the student visualize the results. A plot of the time it takes for traversing the different route options is presented in the applet. The point should be dragged slowly in the Applet for best results. Also be sure to turn on the radio button on the graph. A complete step -by-step analytical solution is also presented.
- Use Derivatives to solve problems: Area Optimization This example illustrates how to find the dimensions of a rectangle that has a fixed perimeter that will maximize the area of the rectangle. An interactive applet illustrates the function that describes the area as a function of the width and length of the rectangle. A complete-step-by-step analytical solution is presented. Be sure to turn the graph radio button on to use the applet. Also move the point on the applet slowly to improve the response of the applet.
- Use Derivatives to solve problems: Area Optimization
- Optimization Problems 6 area optimization and volume optimization problems are presented. They are solved using four different methods, the first derivative method, a variant of the first derivative method, the absolute extream method, and the second derivative method. Problems focus on finding the minimum amount of materials needed to construct geometric objects with given areas or volumes.
- More Optimization Problems 8 optimization problems are worked through with complete solutions. Most of the problems relate to wire length optimization. These problems can be used for minimizing costs when wires from utility poles must be staked to the ground. Problems have a relatively high level of difficulty. The problem set features a classical wire minimization problem. This problem requires that you determine where a wire must be cut to form a triangle and a square that has the maximum area. Other optimization problems presented relate to the fields of architecture and design. A window must be built that lets the maximum amount of light in, given that only a fixed amount of framing materials is available.
- Business Applications
For those studying business and business calculus, this section features 8 optimization problems with solutions that provide the methods to maximize revenue and profit and minimize costs based on given business models. Profit, cost and profit general math equations are used in these solutions along with the derivative. The profit function, the demand function, the price function, the marginal revenue function and the marginal profit function are all discussed and used to solve these business problems.
MEAN VALUE THEOREM
- Mean Value Thereon An explanation of the Mean Value Theorem as an extension of Rolle's theorem.
- Mean Value Definition and Concept This section explains the mean value theorem and has a interactive graphical tutorial to further explain the concept. The mean value theorem is considered one of the most important concepts in calculus. It states that a tangent to a function will have a secant line, which has its end points on the function, that is parallel
- The Mean Value Theorem 4 problems which require the use of the Mean Value Theorem are solved. The problems presented give insight into the type of problems that the mean value theorem can be used to solve.
- Mean Value Theorem Problems 2 problems are solved that use the Mean Value Theorem
- The Mean-Value Theorem
- Mean Value Theorems for Integrals
- Implicit Differentiation One example that solves for dy/dx using implicit differentiation technique.
- Implicit Differentiation Six implicit differentiation problems of varying levels of difficulty. Complete with solutions. Two methods to solve the implicit differentiation problems are discussed. Implicit differentiation is applied to complex functions that involve exponentials, natural logs and trigonometric functions.
- Implicit Differentiation Problems 16 solved implicit differentiation problems of varying degrees of difficulty. An introduction to implicit differentiation is given with an involved example. It is noted that because many equations can not be solved explicitly for y, the derivative can not be found explicitly. This requires the use of implicit differentiation techniques, which makes use of the Chain rule.
- Implicit Differentiation 4 examples of implicit differentiation problems, one that includes using implicit differentiation to find the tangent line to an ellipse are solved.
- Arc Length This page, which takes a little while to load up, is worth the wait. A proof is given for the derivation of the arc length formula for a given function over a specific interval. The Mean Value Theorem and Riemnes sums are used in the proof. An example is also given and solved. The example is illustrated with an animated mathematical java applet. The other problems are illustrated with well defined graphics and the mathematical symbols used are clear.
- Differentials Three solved differential problems are solved. Computation of delta x and delta y for given variations in an independent variable. Differentials are used to estimate change in dimensions, such as in the volume problem that is solved and also estimate error.
- Higher Order Derivatives Solving for the second and third derivative of a function. Three higher order derivative problems are solved. Implicit differentiation is applied to find second devitrifies and third derivatives of complex functions.
DERIVITITIVE TEST REVIEW SUMMARY AND CHEAT SHEETS
- Derivative Test Review Summary and Cheat Sheet A four page review of derivatives with the most common definitions, procedures, applications and concepts. Also illustrates the basic terms with small graphs and helpful reminders. Definitely a must before your first midterm in Calculus I.
- More Problems on the Derivative
- Area Problem The basics concepts of using Riemann Sums to estimate the area underneath a function. Different ways to define Riemann Sums for higher levels of accuracy. Illustrated with numerous graphs to instruct on finding the area. Summation notation is also defined and the use of the limit of the summation to find the actual area under a curve.
- The Algebra of Summation Notation 14 summation problems with solutions. Basic summation formulas are given for common functions. Telescomping series, limits and partial fractions are also used to solve the problems. http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/summationdirectory/Summation.html#PROBLEM 1
- The Fundamental Theorem of Calculus
- Antiderivatives A proof related to antideritives is given as well as an example.
- Riemanns Sums A graphical illustrated introduction to Riemanns Sums and the notation used to represent the area of function in the notation for Riemann sums. Rules related to the calculation of common Riemann sum sections.
- Calculus: Construction of the Riemann Integral
- Calculus: Antiderivatives and the Riemann Integral
- The Limit Definition of a Definite Integral 15 solved summation problems using the limits definition of a definite integral
- Properties of Integrals - Tutorial 3 solved integration problems are given that are solved with basic integration techniques: substitution, integration by parts, the integration of the sum of functions and the integration of the difference of the sum of functions.
- Evaluate integrals 4 solved integral problems that involve trigonometric functions and exponential functions.
- Table of Integrals http://www.analyzemath.com/calculus/Integrals/table_integrals.html
- Indefinite Integrals 4 solved indefinite integral problems for polynomial functions. Basic properties of indefinite integrals are discussed. Links to proof these properties. http://tutorial.math.lamar.edu/Classes/CalcI/ProofIntProp.aspx#Extras_IntPf_Indef1 http://tutorial.math.lamar.edu/Classes/CalcI/ProofIntProp.aspx#Extras_IntPf_Indef2
- Computing Indefinite Integrals 4 multipart indefinite integral problems are solved
- Substitution Rule for Indefinite Integrals
- More Substitution Rule 5 multi-part indefinite integrals are solved with the substitution rule. Several trigonometric integrals are found using identities and substitutions.
- Definition of the Definite Integral 7 solved problems that involve definite integrals. The definition of the definite integral and Riemanns sums are used. The Fundamental Theorem of Calculus is discussed. Several properties of integrals are listed and explained.
- Properties of the Definite Integral
- Computing Definite Integrals 4 multipart solved definite integral problems. Includes integrals of piece wise functions and absolute value functions.
METHODS OF INTEGRATION
- Substitution Rule for Definite Integrals 5 multipart definite integral problems are solved through single and multivariable substitution. Odd and even function integration solutions are discussed.
- The Method of U-Substitution 18 indefinite integrals are solved through substitution techniques. Discussion of anti-derivitive to determine the terms to substitute for U.
- Computing Integrals by Substitution 4 definite integral problems are solved through substitution techniques. Two methods are give for some of the problems.
- Integration Substitution Methods
- Calculus: U-Substitution
- Numerical Integration
- Problems on Techniques of Integration
INTEGRATION BY COMPLETING THE SQUARE
- Computing Integrals by Completing the Square The square of a function is completed to obtain the form needed for integration that equates to the arccos and arctan. The proof the quadratic formula is given by completing the square.
INTEGRATION OF EXPONENTIAL FUNCTIONS
- The integration of Exponential Functions12 indefinite integral problems are solved. Two formulas are given for the integral of natural (base e) logarithm and other exponential functions.
INTEGRATION OF LOGARITHM FUNCTIONS
- Evaluate Integrals Involving Logarithms - Tutorial 3 solved indefinite logarithm integral problems. Substitution is used to solve
- Calculus: The Natural Logarithm Function
INTEGRATION OF NON-ELEMENTARY FUNCTIONS
- Integration of Non elementary Functions
INTEGRATION OF RATIONAL FUNCTIONS
- Integration of Rational Functions
- Integrating Rational Functions
INTEGRATION OF TRIGONOMETRIC FUNCTIONS
- Integration of Trigonometric Integrals 27 trigonometric integration problems are solved. Basic integrals for trigonometric functions are listed along with the trigonometric identities often used to solve trigonometric integral problems
- Trigonometric Substitution Two common integration problems are solved through the use of trigonometric substitution Guidelines for substitutions are given using right triangle sketches.
- Trigonometric Substitutions A non-pdf solution of the previous problems
- Special Trigonometric Integrals Formulas used for the evaluation of trigonometric functions that are of the form sin(mx), cos(nx) and use various sum and difference trigonometric identities.
- Trigonometric Substitution
- Rational Expressions of Trigonometric Functions
- Integrating Powers and Product of Trigonometric Functions
- More on Product of Sines and Cosines
- Other Trigonometric Powers
INTEGRATION BY PARTS
- The Method of Integration by Parts 23 indefinite integral problems are solved with the integration by parts method. The formula for integration by parts is also given.
- Integration by Parts 2 integration by parts are solved. One involves the arctan function and the other is an exponential sinusoid.
- The integration of Rational Functions Resulting in Logarithmic or Arctangent functions 23 integration problems with solutions that involve rational functions and result in an answer that involves the logarithmic or arctangent functions.
- Integration by Parts
- Integration By Parts
- Calculus: Integration by parts
PARTIAL FRACTION INTEGRATION
- Partial Fractions Comprehensive examples are given for finding the integrals through partial fraction decomposition
- Partial Fractions Decompositions. 3 problems that use the method of partial fraction decomposition to find the integral of different types of the form P(x)/Q(x).
- The Method of Integration by Partial Fractions 20 problems are solved through partial decomposition techniques.
- Method of Partial Fractions
POWER SUBSTITUTION INTEGRATION
- The Method of Integration by Power Substitution 14 integral problems are solved that use the power substitution process, a process used when the integral to be solved contains many variables that are to the nth root, such as x^1/4.
ACCELERATION, VELOCITY AND OTHER PHYSICS PROBLEMS
- Average Function Value Two solved problems that use the average function value or Mean Value Theorem for Integrals to find the average value of a function over an interval.
- Work 2 problems are solved applying the integral to solve for work given the force function and the distance an object moves.
AREA INTEGRATION PROBLEMS
- More on the Area Problem Two solved area integration problems are presented. Both involve the finding the area of a region, one which is bounded by two curves and the other region, a triangle, bounded by three lines. Graphics are included.
- Find Area Under Curve 2 problems are solved using integrals to find the area between two curves and within a polygonal region.
- The Area Problem and the Definite Integral Three example area problems are solved using relations related to the Riemann sum. Advanced notation is used here to describe the relationship between sums, limits and integration. The problems here and presentation level are rated at a high difficulty level.
- Area Between Two Curves 7 solved area integral problems. The section explains how to setup different types of area problems and when to integrate over the x axis, dx, and when to integrated across the y axis, dy.
- Find The Area of Circle Using Integrals in Calculus Formula for the area of a circle is derived using integrals. Using integrals to prove the formula for an area of a circle is involved and requires splitting the circle into quadrants and the use of trigonometric identities.
- Find The Area of an Ellipse Using Calculus The formula for the area of the ellipse is derived through integration. The problem is solved through symmetry and the use of trigonometric identities The area of one-quadrant of the ellipse is found first.
- Volumes of Solids of Revolution / Method of Rings 4 volumes defined by rotating a function about an axis are found with the method of rings integration, which uses the inner and outer radius. The lengthy solutions include descriptive 2D and 3D graphics to illustrate the shape of the revolved function and the original function
- Volumes of Solids of Revolution / Method of Cylinders 4 solved problems are presented here that use the method of cylinders, also known as the method of shells, to find that volume of complex functions that are rotated about the axis. 3D graphics are provided that enable the visualization of the shape after it is rotated. Emphasis is on the concept of using different types of cross-sections to simplify the mathematics involved in the integration. s
- Find The Volume of a Frustum Using Calculus The volume of a frustum is derived by revolving the edge of the frustum about the x-axis and integrating the result.
- Find The Volume of a Square Pyramid Using Integrals The formula of the volume for a pyramid with a square base is derived through integration. The cross-sections uses are squares that are parallel to the pyramid's base.
- Find The Volume of a Solid of Revolution 4 problems that require revolving a function about an axis to find the volume. The problem set, which comes with illustrated solutions, shows how the volume can be found by either rotating about the x-axis or the y-axis.
- Volume, Volume of a Sphere The volume of a sphere is found by three different methods, the method of discs, the methods of washers and the method of cylindrical shells.
- More Volume Problems 6 advanced volume by cross-sectional area problems are solved with complete explanations and 3D graphics. The volumes solved for include the cap of a sphere, a square pyramid, a torus, a solid that has a disk base and whose cross-sections are equilateral triangles, a cylinder as well as wedge cut from a cylinder. These volume problems cannot be solved through standard volumes of solid of revolutions techniques.
REVIEW SHEETS AND TABLES
- Table of Integrals
- Derivatives and Integration Review and Summery, Test Review Sheets, Cheat Sheets Basic properties of Derivatives, Common Derivatives, Properties, Formulas, and Rules of Integrals, Standard Integration Techniques, Integration of Products and Quotients of Trigonometric Functions, Standard Right Triangle Trigonometric Substitutions for Integrals
- Integral Review and Summery, Test Review Sheets, Cheat Sheets Integral definition, Fundamental Theorem of Calculus, Properties of Integrals, Common Integrals, Standard Integration Techniques, Integration by Parts, Products and Quotients and Trigonometric Functions, Trigonometric Substitutions, Partial Fractions, Application of Integrals, Area Between Curves, Volumes of Revolution, Arc Length Surface Area, Improper Integrals, Discontinuous Integrand, Approximating Definite Integrals, Midpoint Rule, Trapezoid Rule, Simpson's Rule
- Table of Common Integrals Lists 18 commonly used integrals. Also gives an overview of the different techniques used for integrating specific types of functions and expressions, such as rational expressions.
- Notes for First Year Calculus A 78 page review of the basic calculus areas covered in the first year of calculus.
INTEGRATION AND INTEGRAL PROOFS, FORMULAS
- Fundamental Theorem of Calculus
CALCULUS AND RELATIONSHIP TO INFINITY
Plots of Common Functions, and about common functions such as arctan
About the Arctan Function
SERIES AND CONVERGENCE
Constant of Integration
OTHER MATHEMATICAL TABLES
Table of Laplace Transforms
Table of Fourier Transforms
Fourier Transform of Rectangular Functions
Tables of Mathematical Formulas
Series Convergence and Tests
Definition of a Series
Limit Comparison Test
Alternating Series Test
Absolute Convergence Test
Using Series Tests
Finding the Form of a Series
Finding the Form a Series, No N Powers
Finding the Form a Series, N Only in the Power
Finding the Form a Series, N in the Power and in the Base
Finding the Form a Series, N Factorial in the Series
Power Series Introduction
Taylor Polynomials and Error Bounds
Power Series as Functions
To solve for the integral of 1/(1+x^2)
Examples of Sequences http://maths.abdn.ac.uk/~igc/tch/ma2001/notes/node18.html
Substitute x=tan(z), where z is the angle Zeta
then dx = sec2(z) dz
Substitute back into the original integral
so you now find the integral of (sec2(z)*dz)/(1+tan2(z))
since 1 + sec2(z) = tan2(z)
we have the integral of (tan2(z)*dz/(tan2(z)
the integral simplifes to the integral of dz, since the tan2(z) terms cancel
The integral of dz is z a, since
Since z is equal to the the arctan (x), the integral of 1/(1+x^2) is simply the arctan(x)
Trigonometric Identities http://www.clarku.edu/~djoyce/trig/identities.html