CALC 2 CONCEPT PACKET Complete

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1 CALC 2 CONCEPT PACKET Complete Written by Jeremy Robinson, Head Instructor Find Out More +Private Instruction +Review Sessions

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3 Important Derivative and Integral Formulas Derivative Formulas where u is a function of x Integral Formulas where u is a function of x Common Integral Formulas to avoid u substitution The following formulas can be used in place of the above formulas for the common case where is linear, Properties of Exponential Functions Properties of Logarithmic Functions

4 Unit Circle The Unit Circle is a graphical device to determine trig functions and inverse trig functions for certain common angles The angles are in red for positive or counterclockwise angles and are in green for negative or clockwise angles: The ordered pair coordinates on the circle for an angle correspond to the cosine and the sine for that particular angle: The four other fraction trig functions can also be determined from the values of and in the relations: For angles outside the range of the unit circle it is possible to determine an equivalent angle from the relations: The angles and trig function values can be determined in the other quadrants from the first quadrant reference angle as: for Orientation with a component and a component for Orientation with a component and a component for Orientation with a component and a component for Orientation with a component and a component for Orientation with a component and a component for Orientation with a component and a component for Orientation with a component and a component for Orientation with a component and a component 4

5 Completing the Square and Plane Curves Completing the Square on general equations of Plane Curves allows for the graphing and integration of the equation. Completing the Square Completing the Square converts a general second order polynomial of each variable into a perfect square factor. x variable y variable z variable Where may be positive or negative, may be positive, negative, or zero and each sign is retained throughout. For each of the variable expressions the result is a second order polynomial shifted in that variable direction. Plane Curves Plane Curves are common geometric shapes in a plane and common cross sections of surfaces parallel to a plane. If the expression contains term and term, complete the square to form shifted units along the x axis. If the expression contains term and term, complete the square to form shifted units along the y axis. If the expression contains term and term, complete the square to form shifted units along the z axis. Parabola Circle Ellipse Hyperbola 5

6 Inverse Functions The Inverse Function exactly reverses the operation of its corresponding Function making the two functions Inverse Pairs with each other. The Inverse Function or reverse operation of a Function requires the reverse steps in a reverse order. One-to-One and Invertible Functions For a function to be One-to-One, each value in the range must have exactly one value in the domain directly mapped to it as an ordered coordinate pair. A function must be One-to-One to be Invertible with a corresponding Inverse Function. To check if a function is One-to-One and Invertible: Analytically A function is one-to-one and therefore invertible if it has exactly one unique value mapping to each value. 1. Set and solve for. 2. If has only one unique solution value of exactly then the function is one-to-one and is invertible over its entire domain. It will also be one-to-one and invertible over any smaller subintervals of the domain. 3. If has multiple solution values of such as then the function is not one-to-one and not invertible. However, it may be one-to-one and invertible over smaller subintervals of the domain. Break the domain into smaller subintervals that pass the unique mapping test, and verify each subinterval with the steps. Graphically A function is one-to-one and therefore invertible if and only if it passes the horizontal line test. 1. Graph the function. 2. If every horizontal line intersects its graph at most once, then the function is one-to-one and is invertible over its entire domain. It will also be one-to-one and invertible over any smaller subintervals of the domain. 3. If any horizontal line intersects its graph more than once, then the function is not one-to-one and is not invertible. However, it may be one-to-one and invertible over smaller subintervals of the domain. Break the domain into smaller subintervals that pass the horizontal line test, and verify each subinterval with the steps. Defining the Inverse Function Let be a one-to-one function with domain and range. The inverse function is one-to-one with domain and range. While maps an value to a corresponding value, maps a value to a corresponding value defined by Finding the Inverse Function of a One-to-One Function Analytically 1. Write the original function replacing with. 2. Interchange the variables and. 3. Solve for in terms of keeping the correct solution and replace with. This is the inverse function. Graphically The graph of can be obtained by reflecting or taking the mirror image of the graph of about the line For the mirror image about the line, interchange the coordinate and coordinate in all ordered pairs and graph. Derivative of an Inverse Function Directly 1. Find the inverse function of the function 2. Take the derivative of the inverse function with respect to the independent variable. Derivative Rule If the inverse function cannot be found directly, the derivative of the inverse function can still be found at a coordinate by calculating the derivative of the original function and evaluating at the corresponding coordinate for a point : The inverse function derivative is evaluated at a value while the original function derivative is evaluated at an value! 6

7 Exponential Functions and Derivatives An exponential function has a constant base raised to a power that is a function of the independent variable. Definition of the Natural Base Number The Natural Base is the exact number when used as a base of an exponential function results in the graph having a tangent line slope or derivative with the exact value of 1 at the coordinate as seen by the derivative definition: The value of the Natural Base Number is irrational but can be found by definition to be All other exponential functions with a different base are then simply a scalar multiple of the Natural Base exponential. The Natural Base has several other relationships that result from its definition. These include: Properties of Exponential Functions All exponential functions with any base have the following properties The Limit of an Exponential Function 1. Directly evaluate the function at the limit by plugging in the value and using the results If the result is a finite number, that finite number is the value of the limit. If the result is or the value of the limit is If the result is,, or the value of the limit is indeterminate so continue to step Apply the exponential properties to simplify the equation and retry to directly evaluate the limit. 3. Continue applying other exponential properties until the limit is determinate. The Derivative of an Exponential Function The derivative of an Exponential Function is given by the following formulas: Where is any function of and is the derivative of to complete the chain rule within the derivative. The first formula is the derivative of the Natural Base Exponential and the second is the derivative of any Exponential. The Integral of an Exponential Function The integral of an Exponential Function is given by the following formulas: Where is any function of and is the differential of to complete the u substitution within the integral. The first formula is the integral of the Natural Base Exponential and the second is the integral of any Exponential. 7

8 Logarithmic Functions and Derivatives Logarithmic Functions of base are the inverse of exponential functions of base and are related by the following: The Natural Logarithmic Function and Change of Base Formula The Logarithm with Natural Base has unique properties and is called the Natural Logarithm. It has the special notation: A Logarithm of any base can be expressed as a Logarithm of the Natural Base by the following Change of Base Formula: Properties of Logarithmic Functions All Logarithmic Functions with any base have the following properties The Limit of a Logarithmic Function 1. Directly evaluate the function at the limit by plugging in the value and using the results If the result is a finite number, that finite number is the value of the limit. If the result is or the value of the limit is If the result is,, or the value of the limit is indeterminate so continue to step Apply the logarithmic or exponential properties to simplify the equation and retry to directly evaluate the limit. 3. Continue applying other logarithm or exponential properties until the limit is determinate. The Derivative of a Logarithmic Function The derivative of a Logarithmic Function is given by the following formulas: Where is any function of and is the derivative of to complete the chain rule within the derivative. The first formula is the derivative of the Natural Base Logarithm and the second is the derivative of any Logarithm. The Integral resulting in a Logarithmic Function The integral resulting in a Logarithmic Function is given by the following formula: Where is any function of and is the differential of to complete the u substitution within the integral. Logarithmic Differentiation Logarithmic Differentiation is a process through which to calculate the derivative of one of following functions: The Power Exponential Function A Multiple Complex Power Function 1. Take the Natural Logarithm of both sides of the relation function. Simplify the side using the Properties of Logarithmic Functions, especially watching for the following 2. Differentiate both sides implicitly with respect to, resulting in on the side of the relation function. 3. Solve the resulting equation for by multiplying both sides with the variable. 4. Replace all terms with the original relation function for the explicit derivative in terms of. 8

9 Inverse Trigonometric Functions Inverse Trigonometric Functions are the inverse of trigonometric functions, but only over the subintervals of the domain on which the trigonometric functions are one-to-one. They are related by the following in the subintervals: Properties of Inverse Trigonometric Functions All Inverse Trigonometric Functions have the following properties Inverse Trigonometric and Trigonometric Cancellation Formulas All Inverse Trigonometric and Trigonometric Function Pairs have the following cancellation formulas If the expression is made up of a combination of two different Inverse Trigonometric and Trigonometric Functions it will be necessary to sketch a Relation Triangle to simplify the expression following the definitions of the trig functions: The Derivative of an Inverse Trigonometric Function The derivative of an Inverse Trigonometric Function is given by the following formulas: Where is any function of and is the derivative of to complete the chain rule within the derivative. The Integral resulting in an Inverse Trigonometric Function The integral resulting in an Inverse Trigonometric Function is given by the following formulas: Where is any function of and is the differential of to complete the u substitution within the integral. 9

10 Integration by u Substitution Integration by u Substitution is used for integrals that contain products of functions that are related by one being the derivative of the other, or at least one being a constant multiple of the derivative of the other. The method makes use of the integral equivalent of the Chain Rule for the derivative. This method is especially useful for products of any one function and its derivative from the following: Power Functions Examples Trig Functions, Examples Inverse Trig Functions,, Examples Exponential Functions Examples Logarithmic Functions Examples Be sure to watch for a function and at least the variable part of its derivative together in the integral for u substitution. How to Choose the Function Correctly While the u Substitution Method is trial and error, there are some tips on choosing the proper Function. The choice of is determined by two different functions, one being the itself and one being the variable part of the derivative. If the given choice of does not work, try other possibilities. If still unable to find, move on to a different method, most likely By Parts. With the following tips, it is possible to pick out the proper for every integral encountered. 1) Trig Functions, If there is only one trig function, choose to be the quantity inside of the trig function, but not the trig function itself. If there are two or more trig functions, choose to be one of the trig functions such that the other is its derivative. 2) Inverse Trig Functions,, If there is only one inverse trig function and no denominator, choose to be the quantity inside of the inverse trig function, not the inverse trig function itself. If there exists a denominator together with the inverse trig function, choose to be the entire inverse trig function and the denominator may be exactly the variable form of its derivative. 3) Exponential Functions If there is only one exponential function, choose to be the exponent. If there are two or more exponential functions, choose to be one of the exponential expressions especially if it is in the denominator, the other is its derivative. 4) Logarithmic Functions If there is only one natural log function and no denominator, choose to be the quantity inside of the natural log function, but not the natural log function itself. If there exists a denominator together with the natural log function, choose to be the natural log function and the denominator may be exactly its variable form of the derivative. 5) Power Functions where n is an integer or a fraction, and may be either positive or negative Choose to be the expression being raised to the power, especially if the expression is in the denominator. The u Substitution Method 1) Choose the proper following the rules such that the variable part of its derivative is also present in the integral. 2) Differentiate the to find and divide or multiply any constants over with the. The function that and its constants equal to is the variable part of the and will always contain the. This expression must be in the integral! 3) Replace expressions of with and the exact variable part of the derivative with. Simplify this expression with algebra, and the result will always be one or more of the integrals from the Important Integral Formulas Table. 4) Integrate the functions of, then replace every in the integral result with the chosen from step

11 Integration by Parts Integration by Parts is used for integrals that contain products of functions that are not related by one being the derivative of the other (for which case u Substitution is used). The method makes use of the integral equivalent of the Product Rule for the derivative. This method is especially useful for products of two functions from the following: Power Functions Examples Trig Functions, Examples Inverse Trig Functions,, Examples Exponential Functions Logarithmic Functions Examples Examples How to Choose the Function Correctly While the Parts Method is trial and error, there are some tips on choosing the proper Function. It is easiest to remember the following mnemonic for the choice of : L I P E T. Whichever function is first in the list on this mnemonic should be the choice for in the integral. Whatever is left which includes the will be part of the. Here is the list: Logarithmic Functions Examples Inverse Trig Functions,, Examples Power Functions Examples Exponential Functions Examples Trig Functions, Examples The Parts Method 1) Start with the integral in the form of a product of two functions from the above list 2) Choose one of these functions to be, differentiate the to find the which should always include the 3) Whatever function is left should always include the and is equal to the, integrate this to find the 4) Put the results of these four quantities into the Parts Integral Formula The method does not directly solve the integral, but rather replaces the integral with a term plus a different integral. The hope is that this new integral can be integrated as is, or is at least simpler after algebra simplification than the original integral. If it is more complicated than the original integral, than the choices for and are probably not correct. Choose a different and, and retry the Parts Integral Formula. It is possible that while the resultant integral is simpler than the original integral, it still contains a product of two functions from the above list that cannot be integrated by u substitution. In this case it is necessary to run through the Parts Method a second time on this resultant integral. If again it contains a product of two functions from the above list that cannot be done by u substitution, it is necessary to run through the Parts Method a third time, etc. until an answer is obtained from an integral that eventually does not require the Parts Method. It is recommended to first find the complete expression for the answer before plugging in any bounds if it is a definite integral. 11

12 Integration of Trig Integration of Trig is used for any integrals that contain either solo trig functions, or multiplication and division combinations of trig functions in the numerator and denominator, usually either combinations of and together, or combinations of and together. The methods make use of the following Trig Identities: Pythagorean Trig Identities One of these two identities may be used to convert one squared trig function into another squared trig function. Half Angle Identities One of these two identities may be used to convert an even power trig function into an odd power trig function. Double Angle Identity This identity may be used to convert any Double Angle created from either Half Angle Identity back into a Single Angle. The Trig Method The actual Trig Method used depends on whether the integral is combinations of and alone or together, or combinations of and alone or together. The integral will be one of two possibilities: 1) Combinations of and either alone or together in the form of a) If either is odd, is odd, or both are odd, it is possible to do this integral through u substitution. Choose whichever trig function has the odd power, remove a single power term and put it with aside with the to represent the for the integral. Convert the remaining even powers of this trig function over to the other trig function through the use of the Pythagorean Trig Identities. Choose as this other trig function, and will be that set aside trig function with. b) If both and are even, this integral can be done directly after the conversion of the even powers into odd powers, and eventually into the first power by the use of the Half Angle Identities. If there is yet another even power after the first conversion using the Half Angle Identity, it may be necessary to convert a second time, etc. before the integration. 2) Combinations of and either alone or together in the form of a) If the is odd, it is possible to do this integral through u substitution. Remove a single power of both and, and put both of them aside with the to represent the for the integral. Convert the remaining even powers of over to through the use of the Pythagorean Trig Identities. Choose as, and will be. b) If the is even, it is possible to do this integral through u substitution. Remove two powers of, and put them aside with the to represent the for the integral. Convert the remaining even powers of over to through the use of the Pythagorean Trig Identities. Choose as, and will be. c) If both is even and is odd, it will be necessary to convert the integral over into and by replacing with and with, simplifying the expression, and then going back to and combination method. d) If and, convert using the Pythagorean Trig Identities and integrate 12

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14 Integration by Trig Substitution Integration by Trig Substitution is used for integrals that contain any one of the following forms found in the numerator or in the denominator and may also be found within any radicals or powers Where the represents a constant and represents a function of (which sometimes may be just itself). This method simplifies the expression through the use of two of the three Pythagorean Trig Identities: One of these two identities will appear in every integral problem where trig substitution is used. The identity will help simplify the function within the integral to put it into a form that can done directly or through u substitution. The Trig Substitution Method 1) Start with an integral that contains one of the above three forms, and decide the function for and the value for 2) Decide which one of the three forms the integral contains and assign a relation between,, and according to: then the relationship should be then the relationship should be then the relationship should be 3) Replace with its equivalent function of and replace with its constant value in the chosen relation, and then solve for the producing a relation between just and. Plug this expression into the integral for each and every contained within both the numerator and denominator of it and within any radicals or powers. 4) Take the derivative of this relation between and, which will produce a relation between,, and. Plug this into the integral for the contained within the numerator of it. 5) The integral should now only contain the variable contained in a variety of trig functions. It is best at this point to do some algebraic simplifications, including factoring out of any power or radical expressions, and the cancellations of terms from both the numerator and denominator. Watch for the Pythagorean Trig Identity that will occur once in every problem. Replace this expression with its equivalent term from one of the two Pythagorean Trig Identities found above. 6) The integral procedure from here follows Integration of Trig (see), which itself may require u substitution. 7) Once the integral has been calculated, it is necessary to back substitute to get the expression in terms of the original variable, usually. If u substitution was used to perform the Integration of Trig, the first step of back substitution would be to replace with its equivalent expression in terms of and then use the sine Double Angle Identity for any terms Once the integral result is in terms of only, it is ready for the Trig back substitution, which makes use of two things: For any terms that are not contained in a trig function Find the original relation between just and, found in step 3. Solve for in terms of from this relation, which will always involve an inverse trig function. This relation can be used to back substitute in for any terms that are not contained within a trig expression. These terms will always become inverse trig expressions in the answer. For any terms that are contained in a trig function Find the original relation between just and, found in step 3. Solve for the trig function of (but not for itself) in terms of which will always be a fraction expression, and then proceed to draw a Relation Triangle with a right angle and the angle. Two sides of the triangle will be known from the relation between and, and the third side can be found from the Pythagorean Theorem. With all sides of the triangle known, it is possible to find all necessary Trig Functions of in terms of to be back substituted in the integral result. Replace all trig functions with terms. 14

15 Integration by Partial Fractions Integration by Partial Fractions is used for any integrals that contain only Rational Functions, which are functions that have integer power function expressions or power functions with integer exponents in the numerator and in the denominator. The power function in the denominator must factor into two or more expressions to use this method. Partial Fraction Functions Examples Where each Integer Power Function is defined by Integer Power Function Examples The Partial Fraction Function The Partial Fraction Method is an algebra method that is used to convert a Partial Fraction Function into an addition or subtraction of several smaller Rational Functions that can therefore each be integrated separately. The method is actually the exact opposite operation of the common denominator operation in algebra and involves breaking up a large fraction into an addition or subtraction of smaller fractions. Once converted into the Small Rational Functions, each added or subtracted term will take one of the following forms and can be integrated separately: The Partial Fraction Integrals The integrals here are the only ones used in the Partial Fraction Method, though u substitution is often needed. The Partial Fraction Method 1) If the numerator Integer Power Function has an equal or higher degree than the denominator Integer Power Function, long divide the denominator into the numerator. The dividend terms can be integrated using either The Constant Rule or The Power Rule. The remainder term is a Partial Fraction Function, and can be integrated with steps. 2) Completely Factor the denominator. It will factor completely into either first order or second order factors only. 3) Break the Large Fraction Function into an addition of Small Fraction Functions with arbitrary coefficient numerators. A repeated factor in the denominator of the Large Fraction Function will break up into a number of Small Fraction Functions equal to the number of repetitions or power of the factor in the Large Fraction Function, with each fraction having a successively higher power than the one before it. The degree of the numerators of all Small Fraction Functions will always be exactly one degree lower than the factor in its denominator, ignoring any power repetitions of that factor. 4) Put the Small Fraction Functions with arbitrary coefficient numerators over a common denominator (which will match the denominator of the Large Fraction Function), and combine them together into a single fraction. This single fraction will already have its denominator equal to the Large Fraction Function denominator, so now it is possible to set the numerators equal to each other and solve for the arbitrary coefficients necessary to make them match. 5) Replace these coefficients back into the addition of Small Fraction Functions; each will have one of the integral forms. 6) Integrate each term separately following The Partial Fraction Integrals above, using u substitution whenever needed. 15

16 Limit Results and Indeterminate Forms indicates a boundless large positive number and indicates a boundless large negative number. indicates a very small positive number near 0 and indicates a very small negative number near 0. represents a finite nonzero positive constant. Results involving a small positive number or a small negative number =small positive number/positive constant=small positive number= = =small negative number/positive constant=small negative number= = =positive constant/small positive number=large positive number= =positive constant/small negative number=large negative number= =square root of a small positive number=small positive number= = =square root of a small negative number= or Does Not Exist: negative number not in the domain = natural log of a small positive number=large negative number= = natural log of a small negative number= or Does Not Exist: negative number not in the domain =sine of a small positive number=small positive number= = =sine of a small negative number=small negative number= = =cosine of a small positive number=positive number slightly less than 1= = =cosine of a small negative number=positive number slightly less than 1= = =arctangent of a small positive number=small positive number= = =arctangent of a small negative number=small negative number= = Results involving a large positive number or a large negative number =large positive number/positive constant=large positive number= =large negative number/positive constant=large negative number= =positive constant/large positive number=small positive number= = =positive constant/large negative number=small negative number = =positive constant raised to a large positive number= if k>1 and = if 0<k<1 =positive constant raised to a large negative number=small positive number= = =square root of a large positive number=large positive number= =natural log of a large positive number=large positive number= =sine of a large positive or negative number= or Does Not Exist because of oscillation =cosine of a large positive or negative number= or Does Not Exist because of oscillation =arctangent of a large positive number= =arctangent of a large negative number= Results involving combinations of,,, and =small positive or negative number/large positive or negative number=small positive number= = =small positive number raised to a large positive number=small positive number= = =small positive number raised to a large negative number=large positive number= =small positive or negative number/small positive or negative number= =large positive or negative number/large positive or negative number= =small positive or negative number*large negative or positive number= =large positive number-large positive number= =small positive or negative number raised to small positive or negative number= =large positive or negative number raised to small positive or negative number= =number close to 1 but greater than 1 raised to a large positive or negative number= =number close to 1 but lesser than 1 raised to a large positive or negative number= 16

17 L'Hospitals Rule and Leading Term Analysis L Hospitals Rule and Leading Term Analysis are methods to evaluate limits that result in an form. L Hospitals Rule L Hospitals Rule can be used only when the limit results in either one of the two If a limit results in either of the two, the limit can be calculated following the steps: 1. Verify the limit results in one of the two. 2. Separately calculate the derivative of the numerator and denominator of the expression and retry the limit: 3. Repeat the derivative of the numerator and denominator of the expression and retry the limit until the result is no longer a and instead will be the final result of the limit. It is possible to convert most other forms into a L Hospitals Form including: This can be achieved by the following processes depending on the type of For results of this type take either one of the two function terms that are being multiplied, reciprocate it and place it into the denominator. This will not change the value of the expression, only its form. The form should now be one of the depending on which one of the two terms is reciprocated. For results of this type, several possibilities exist to change its form depending on the two subtracting terms: 1) If either of the two subtracting expressions has a denominator, find a common denominator. The form should now be one of the. Remember,,, and all have a denominator. 2)If one or both of the two subtracting terms are square roots, the form can be changed by multiplying the expression by their root conjugate over itself. The root conjugate is the same as the original expression with a plus sign between the two terms instead of a minus sign. The form should now be one of the. 3)If the terms have a common factor, factor it. The form should now be one of the. For results of this type, raise the elementary base to the natural log of the expression in the limit and bring the exponent out to the front of the natural log. The form will become which has a method above. The answer is always of the form so that,,, and are all possible answers. Leading Term Analysis Leading Term Analysis is used for infinite limits where L'Hospital's Rule fails. Leading Term Analysis involves calculating the limit after dividing both the numerator and denominator of the fraction function by the leading term in the denominator or simply after taking the ratio of the individual leading terms from the numerator and denominator. Leading Terms from greatest to least for 1) Power Exponential, which can be subordered by the value of the multiplier. For example 2) Quadratic Exponential which can be subordered by the value of the base. For example 3) Factorial, which can be subordered by the value of the multiplier. For example 4) Linear Exponential which can be subordered by the value of the base. For example 5) Power which can be subordered by the value of the power. For example 6) Logarithm, which are all of the same order regardless of the value of the base. 7) Bounded Functions 17

18 Improper Integrals An improper integral is any integral that has an infinite value at either or both of its two bounds, or any value equal to or between the bounds that causes an infinite value in the integrand, the function that is to be integrated. To solve these integrals, it is necessary to replace each value that either is infinity or creates an infinity with an arbitrary variable and evaluate the limit of the integral result as the arbitrary variable approaches the value. The steps of solution are: 1. Determine which of the three types below is the improper integral to be calculated and rewrite it as a limit integral: a. The first type of improper integral has either a lower bound of, an upper bound of, or both. If it has a lower bound of, replace that bound with an arbitrary variable (often ) and rewrite the integral with a limit of the arbitrary variable as it approaches the infinite value of. If it has an upper bound of, replace that bound with an arbitrary variable (often ) and rewrite the integral with an infinite limit of the arbitrary variable as it approaches the infinite value of. If it has both a lower bound of and an upper bound of, split the integral at any value between the two bounds (often chosen to be 0), replace each infinite bound with a different arbitrary variable (often and ) and rewrite each integral with a limit of the arbitrary variable as it approaches the infinite values and. b. The second type of improper integral has either a lower bound or an upper bound creating infinity in the integrand. If it has a lower bound creating infinity in the integrand, replace that bound with an arbitrary variable (often ) and rewrite the integral with a one sided limit of the arbitrary variable as it approaches the value of. If it has an upper bound creating infinity in the integrand, replace that bound with an arbitrary variable (often ) and rewrite the integral with a one sided limit of the arbitrary variable as it approaches the value of. c. The third type of improper integral has a value between the bounds creating infinity in the integrand. Split the integral at the value between the two bounds, replace each bound with a different arbitrary variable (often and ) and rewrite each integral with a one sided limit of the arbitrary variable as it approaches the value of. 2. Evaluate the integral(s) by whatever method necessary, and then plug in the upper and lower bounds of which one is a number and one is the arbitrary variable involved in the limit statement. 3. Calculate the limit of each integral result for the final answer. Each integral is Convergent if the limit is finite and each integral is Divergent if the limit is infinite or does not exist ( ). If every integral is Convergent, the final answer is Convergent to the total value. If one or more integrals are Divergent, the final answer is Divergent. The p integral is an improper integral of a power function in the denominator with either zero or infinity as a bound: Comparison Test for Integrals The Comparison Test is used to determine if an integral that cannot be evaluated directly is Convergent or Divergent: If the functions and are related by over the interval, the integrals are related by If the integral is Convergent, then the integral is also Convergent If the integral is Divergent, then the integral is also Divergent 18

19 Infinite Sequences An Infinite Sequence is an infinitely long set or list of terms that are not added together but are separated by commas. An Infinite Sequence is of particular interest since it has an infinite number of terms and therefore has no end. Even though it has no end, it is still possible to investigate the value of the infinitely large index terms or the individual terms that exist an infinite distance along the list. In order to do this, the sequence must follow a pattern. Term Index The Term Index value is the place number of the term in the sequence list. The initial term has an index value of 1, the next term has an index value of 2, the next term has an index value of 3, etc. If the terms follow a pattern, the entire sequence can be expressed as a compact formula in two different ways: Direct Formula A Direct Formula of the index, where each index term of the sequence is generated by the corresponding index value being plugged into the direct function formula. The Direct Formula will take the following form: Where the Direct Formula can be a combination of the following: Power Functions Examples Trig Functions, Examples Inverse Trig Functions,, Examples Exponential Functions Logarithmic Functions Examples Examples Recursion Formula A Recursion Formula of, where each term of the sequence is generated by one or more of the terms preceding it in the sequence list. Each term depends on one or more terms with a lower index value so it is necessary to have lower index terms to calculate successive higher index terms. The Recursion Formula will take one of the following forms: Where the Recursion Formula will usually involve arithmetic combinations of the preceding terms. Sequence Convergence or Divergence The Convergence or Divergence of the infinitely large index terms requires calculating the infinite limit on either the Direct Formula or a Recursion Formula: If the sequence has finite value terms that approach the same value an infinite distance along the list, the sequence is Convergent and approaches the common value of the terms an infinite distance along the list. If the sequence has infinite value terms an infinite distance along the list, the sequence is Divergent. If the sequence has finite value terms that oscillate an infinite distance along the list, the sequence is Divergent. Direct Formula In the infinite limit of a Direct Formula, it will have a Limit Result. If the Limit Result is finite, the sequence converges to that value. If the Limit Result is Infinite or Does Not Exist, the sequence Diverges. It may be necessary to use L'Hospital's Rule or Leading Term Analysis if the limit produces an from the previous sections. Recursion Formula In the infinite limit of a Recursion Formula, name all terms the same value regardless of index, and then solve the resulting expression for. If a solution exists, the sequence Converges to that value. If no solution exists, it Diverges. 19

20 Infinite Series An Infinite Series is an infinitely long set or list of terms added together and beginning with an index value of as follows Even though the Infinite Series has no end, it is possible to find the summation trend of the infinite number of terms. Term Index The Term Index value is the place number of the term in the series list. The initial term has an index value of, the next term has an index value of, the next term has an index value of, etc. Partial Sum Index The Partial Sum Index value is the place number of the last term in the summation list. The initial sum has an index value of, the next sum has an index value of, the next sum has an index value of, and the entire Series sum has an index value of. Series Convergence or Divergence Quick Test Guide The appropriate test for Convergence or Divergence of the Infinite Series depends on the form of the as follows: Geometric Series Test Telescoping Series Test Divergence Test Integral Test Limit Comparison Test Direct Comparison Test Alternating Series Test Ratio Test Root Test 20

21 Geometric Series Test The Form Notes The sum of the series can be found for all geometric series. A linear exponential is an exponential function of any constant base but with a linear function as an exponent. If the series has two terms in the numerator, split into two separate geometric series and test the Convergence of each. If both converge, the total series Converges to the total sum. If either one or both diverge, the total series Diverges. The Test 1. Calculate the Ratio 2. Compare the Ratio to 1 If the Geometric Series Converges and with converges to the value If Telescoping Series Test The Form the Geometric Series Diverges Notes The sum of the series can be found for all telescoping series. If the series is in the fraction form, it will be necessary to use Partial Fractions to put it into the subtraction form. The Test 1. Be sure the series is in subtraction form. Use Partial Fractions on fraction form to put into subtraction form. 2. Write out the nth Partial Sum, or each of the terms of the series added together out to an arbitrary value. 3. Cancel out all middle terms within the nth Partial Sum. Each negative part of a term should cancel with a positive part of a term found terms after it. Each positive part of a term should cancel with a negative part of a term found terms before it. After cancelling all middle terms, there should remain exactly the first terms of the Partial Sum and exactly the last terms of the Partial Sum that are not able to cancel out. 4. Take the infinite limit of the nth Partial Sum. If the Limit Result is finite, the series converges to that value. If the Limit Result is Infinite or Does Not Exist, the series Diverges. It may be necessary to use L'Hospital's Rule for any. Divergence Test The Form Notes This test can only be used to show divergence, and will be inconclusive if the limit is equal to zero. The contrapositive of the statement in the test should be remembered: If the series diverges, but if it is already known that the series converges then The Test 1. Take the infinite limit of the nth term. 2. Compare the result to 0. If the series Diverges. If the test is inconclusive and another test should be used. 21

22 Integral Test The Form Notes The Integral Test is especially used for functions involving together with or together with The integral will always have the same bounds as the series, making it improper with an upper bound of Integral Series Test Minimum Error Term and Maximum Error Term for a partial sum with finite terms The Test 1. Calculate the definite integral with the same bounds as the series 2. If the integral converges, so does the series but not to the same value. If the integral diverges, so does the series Limit Comparison Test The Form Notes If the numerator function is the higher function or equal to the denominator function, use the Divergence Test instead This test may be used for many other functions that are not of the normal type. The known series that will be used to compare the unknown series to depends on the form of one of the first two types stated above: p Series The first type or the Power Fraction Function will be compared to a p series result as the known series Geometric Series The second type or the Exponential Fraction Function will be compared to a geometric series as the known series The Test 1. Find a known series to compare to the original unknown series by taking the fraction of the leading term in the numerator over the leading term in the denominator which always forms a p series for the first type, and a geometric series for the second type. The third type should be compared to a p series of one degree higher. 2. Decide the trend of the known series by the p series result or by the geometric series test. 3. Calculate the infinite limit of the ratio of the of the unknown series nth term over the known series nth term. It may be necessary to use L'Hospital's Rule for any. 4. If the result of the limit is a finite positive value both series have the same trend. If the known function is found by the method described, the result of the limit should always be a finite positive value. If the known series converges, the unknown series also converges. If the known series diverges, the unknown series also converges. 22

23 Direct Comparison Test The Form Notes This test is used for any terms that contain except for that of together with for which the integral test is used For this test the unknown series must be either lesser than a convergent series or greater than a divergent series. The most common bounded functions together with the bounds of each are the following: The Test 1. Find a known series to compare to the original unknown series by taking the bounded function together with both its lower and upper bounds and algebraically manipulating the center function and identically manipulating each of the bounds until the center function appears exactly like the unknown series at which point each of the two bounds will now also be series named and.in the form: 2. Choose one of the two bounds or to be the known series for a direct inequality comparison to the original unknown series. Only one of the two bounds will work since they will always have identical trends. If both of the bounds and converge choose the right side bound, ignore the left side bound, and the unknown series will then converge by the direct comparison test to the right side bound series. If both of the bounds and diverge choose the left side bound, ignore the right side bound, and the unknown series will then diverge by the direct comparison test to the right side bound series. Alternating Series Test The Form Notes This test only shows Conditional Convergence and the series must be alternating in order to use this test. To show Absolute Convergence, use any one of the other tests except the Alternating Series Test. Alternating Series Test Maximum Error Term for a partial sum with finite terms The Test 1. If necessary replace with and factor out of any of to produce one of 2. Find, the nth term of the Alternating Series after removing the alternating sign of Determine if the Alternating Series Converges or Diverges by testing the nth term under two conditions: a. The Infinite Limit Determine if b. Decreasing: Calculate the term and compare to the term. If the nth term decreases. Or calculate the derivative of the term and compare to zero. If the nth term decreases. If both the infinite limit equals zero and the nth term decreases the series converges. If either the infinite limit does not equal zero or the nth term does not decrease the series diverges. 23

24 Ratio Test The Form Notes For the ratio test to work, it must contain at least one exponential or factorial in either the numerator or denominator. If the series has power functions in both the numerator and the denominator, use the Limit Comparison Test instead. The Test 1. Calculate the term by replacing each in the term with an 2. Calculate and simplify the absolute value ratio of by reciprocal multiplication and division cancellation 3. Calculate the Infinite Limit of the ratio If If If the series converges the series diverges the test is inconclusive and another test should be used, likely Limit Comparison Test Root Test The Form Notes For the root test to work, it must contain all terms in either the numerator or denominator raised to the power of. If the series has power functions in both the numerator and the denominator, use the Limit Comparison Test instead. The Test 1. Calculate and simplify the absolute value root of by exponent multiplication and cancellation 2. Calculate the Infinite Limit of the absolute value root If If If the series converges the series diverges Absolute Convergence and Conditional Covergence Process The Form the test is inconclusive and another test should be used, likely Limit Comparison Test Notes It must be an Alternating Series to test for both Absolute Convergence and Conditional Convergence. The Test 1. Test Absolute Convergence using any appropriate test except the Alternating Series Test for the series If necessary replace with and factor out of any of to produce one of and then remove the alternating sign of before testing If Convergent, it is Absolutely Convergent. If Divergent, continue to step Test Conditional Convergence using the Alternating Series Test for the series If Convergent, it is Conditionally Convergent. If Divergent, it is Divergent. 24

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26 Series Analysis Series Analysis is used to determine absolute convergence, conditional convergence, or divergence of series without the use of any tests. Series Analysis involves calculating the convergence or divergence dependent only on the resulting form of the series after taking the ratio of the individual leading terms from the numerator and denominator. Leading Terms from greatest to least for 1) Power Exponential, which can be subordered by the value of the multiplier. For example 2) Quadratic Exponential which can be subordered by the value of the base. For example 3) Factorial, which can be subordered by the value of the multiplier. For example 4) Linear Exponential which can be subordered by the value of the base. For example 5) Power which can be subordered by the value of the power. For example 6) Logarithm, which are all of the same order regardless of the value of the base. 7) Bounded Functions Not Alternating and Leading Term in numerator and denominator are not both power functions or natural logs Alternating and Leading Term in numerator and denominator are not both power functions or natural logs Leading Term in numerator is a power function and Leading term in denominator is a power function Leading Term in numerator is a constant and Leading term in denominator is a power function Leading Term in numerator is a constant and Leading Term in denominator is a power x with a power natural log Leading Term in numerator is a power natural log and Leading term in denominator is a power function Leading Term in numerator is a constant and Leading Term in denominator is a power natural log with x Leading Term in numerator is a constant and Leading Term in denominator is a power natural log 26