CALC 3 CONCEPT PACKET Complete

Transcription

1 CALC 3 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: 4

5 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

6 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 Exponential Functions Examples 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. 6

7 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 7

8 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. 8

9 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. 9

10 Vectors Vectors are different from Scalars in that they have both a magnitude and a direction. Position, Velocity, Acceleration, and Force are all Vectors. Stating a vector requires at least two or more values, which depend on the form. Two Dimensional Vector Component Form or Vector Magnitude Direction Form The Vector Component Form states a vector by listing its amount in each of the coordinate axis directions. is the amount of the vector along the axis and is the unit vector that points in the direction is the amount of the vector along the axis and is the unit vector that points in the direction The Vector Magnitude Direction Form states a vector by listing its amount and angle to the coordinate axis directions. Conversion from Two Dimensional Component Form to Magnitude Direction Form The Components of a Vector are perpendicular and form a right triangle, leaving the Magnitude and Direction relations for a Vector with Orientation for a Vector with Orientation for a Vector with Orientation for a Vector with Orientation Orientation for a Vector that points towards the first quadrant, it will have a component and a component Orientation for a Vector that points towards the second quadrant, it will have a component and a component Orientation for a Vector that points towards the third quadrant, it will have a component and a component Orientation for a Vector that points towards the fourth quadrant, it will have a component and a component Conversion from Two Dimensional Magnitude Direction Form to Component Form If the angle is measured to the If the angle is measured to the nearest axis nearest axis If the angle axis is measured to the Sign determined by Orientation Sign determined by Orientation Sign is automatically determined Three Dimensional Vector Component Form and Magnitude The Vector Component Form states a vector by listing its amount in each of the coordinate axis directions. is the amount of the vector along the axis and is the unit vector that points in the direction is the amount of the vector along the axis and is the unit vector that points in the direction is the amount of the vector along the axis and is the unit vector that points in the direction The Components of a Vector are perpendicular and form a right triangle, leaving the Magnitude relation Vector and Distance Between Two Points A Vector between an initial point and a final point can be formed by the following The distance between an initial point and a final point can be calculated by the following 10

11 Unit Vector and Vector Scaling A Unit Vector has a magnitude of exactly one while still being in the same direction as some other vector. A Vector of magnitude but in the direction of some other vector can be calculated through a Unit Vector as Vector Products There are two different vector products: The Dot Product or Scalar Product and the Cross Product or Vector Product. Dot Product or Scalar Product The Dot Product or Scalar Product produces a scalar quantity that is an unscaled projection of each vector in the product onto the other vector in the product. The Dot Product or Scalar Product is used for vector components and projections. Cross Product or Vector Product The Cross Product or Vector Product produces a vector quantity that is simultaneously perpendicular to each one of the original vectors in the product. The Cross Product or Vector Product is used to calculate perpendicular direction vectors. Vector Component and Vector Projection Vector Component and Vector Projection are scalar and vector projections of one vector onto a second vector. Vector Component or Parallel Scalar Projection Vector Component or Parallel Scalar Projection is a scalar amount of one vector parallel to a second vector. Parallel Vector Projection Parallel Vector Projection is a vector amount of one vector parallel to a second vector. Perpendicular Scalar Projection Perpendicular Vector Projection is a projection vector amount of one vector perpendicular to a second vector. Perpendicular Vector Projection Perpendicular Vector Projection is a projection vector amount of one vector perpendicular to a second vector. Vector Area and Volume Computations Vector Products can be used to calculate the area of a triangle or parallelogram with two of its sides formed by two nonparallel vectors or the volume of a parallelepiped with three of its sides formed by three nonparallel vectors. 11

12 Points, Lines, and Planes in Space Points, Lines, and Planes in Space are a set of related points that are defined by a specific set of necessary conditions. Point in Space A Point in Space is a single location defined by an ordered combination of components in one of the forms: Ordered Coordinate Form Ordered Coordinate Form locates the Point in Space by stating in order its value along each of the axes Parametric Vector Form Parametric Vector Form relates Space Vector from the Origin to point Line in Space A Line in Space is the set of all points such that the vector formed between any one arbitrary coordinates point and a different fixed coordinates point are parallel to a fixed components vector. The fixed coordinates point and the parallel fixed components vector are either given or must be determined from the statements given in the problem. The definition of a Line in Space requires one of the forms: Parametric Vector Form Parametric Vector Form relates Space Vector from the Origin to point and parallel vector Parametric Components Form Parametric Components Form relates components to point and parallel vector Symmetric Components Form Symmetric Components Form relates components to point and parallel vector through Plane in Space A Plane in Space is the set of all points such that the vector formed between any one arbitrary coordinates point and a different fixed coordinates point are perpendicular to a fixed components vector. The fixed coordinates point and the parallel fixed components vector are either given or must be determined from the statements given in the problem. The definition of a Plane in Space requires one of the forms: Compact Scalar Product Form Compact Scalar Product Form relates normal vector and vector in the plane with point Expanded Scalar Product Form Expanded Scalar Product Form relates normal vector and Vector in the Plane with point Standard Form Standard Form relates normal vector and variable point in the plane with point 12

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14 Calculation of Line Equations The Calculation of Line Equations finds the equation of a line in parametric or symmetric form under the conditions: Equation of a Line with a point and parallel to a vector Line with a point and parallel to a vector 1. Use the point on the line and the parallel vector to form the equations for the line. Equation of a Line with two points Line with two points, a point and a point 1. Calculate the parallel vector using the two point vectors 2. Use the point on the line and the parallel vector to form the equations for the line. Equation of a Line with a point and parallel to another line Line with a point and parallel to another line either with a vector or with a set of parametric equations or with the symmetric equations 1. Use the point on the line and the parallel vector to form the equations for the line. Equation of a Line with a point and perpendicular to two lines Equation of a line with a point and perpendicular to one line either with a point and a parallel vector or with a set of parametric equations or with the symmetric equations and perpendicular to another line either with a point and parallel vector or with a set of parametric equations or with the symmetric equations 1. Calculate the parallel vector by the cross product of the line vectors 2. Use the point on the line and the parallel vector to form the equations for the line. Equation of a Line with a point and perpendicular to a plane Equation of a line with a point and perpendicular to a plane either with a point and a normal vector or with standard equation where 1. Calculate the parallel vector by setting it equal to the normal vector of the plane 2. Use the point on the line and the parallel vector to form the equations for the line. Equation of a Line at intersection of two planes Equation of a line at the intersection of a plane either with a point and a normal vector or with standard equation where and another plane either with a point and a normal vector or with standard equation where 1. Calculate the parallel vector by the cross product of the normal vectors If cross product is zero the planes are parallel. 2. Calculate a point on the line of intersection by choosing choosing a value for any one of that is in both plane equations and solving the two standard equations and for the other values to form 3. Use the point on the line and the parallel vector to form the equations for the line. 14

15 Calculation of Plane Equations The Calculation of Plane Equations finds the equation of a plane in standard form under the conditions: Equation of a Plane with a point and normal to a vector Plane with a point and perpendicular or normal to a vector 1. Use the point on the plane and normal vector to form the equation for the plane. Equation of a Plane with three points Plane with three points, a point, a point, and a point 1. Calculate two vectors that are in the plane and using any two of the three point vectors such as and 2. Calculate the normal vector by the cross product of the two vectors and as 3. Use the point on the plane and the normal vector to form the equatios for the plane. Equation of a Plane with a point normal to a line Plane with a point and perpendicular or normal to a line either with a point and a parallel vector or with a set of parametric equations or with the symmetric equations 1. Use the point on the plane and normal vector to form the equation for the plane. Equation of a Plane with a point and a line Plane with a point and containing a line either with a point and a parallel vector or with a set of parametric equations or with the symmetric equations 1. Calculate a vector in the plane using the two point vectors 2. Calculate the normal vector by the cross product of the two vectors and in the relation 3. Use the point on the plane and the normal vector to form the equation for the plane. Equation of a Plane with a point and parallel to a plane Plane with a point and parallel to another plane with a point and a normal vector or with standard equation where 1. Use the point on the plane and the normal vector to form the equation for the plane. Equation of a Plane with a point and a line of intersection of two planes Plane with a point and containing the line at the intersection of a plane either with a point and a normal vector or with standard equation where and another plane either with a point and a normal vector or with standard equation where 1. Calculate the parallel vector by the cross product of the normal vectors If cross product is zero the planes are parallel. 2. Calculate a point on the line of intersection and therefore also on the plane by choosing choosing a value for any one of that is in both plane equations and solving the two standard equations and for the other values to form 3. Calculate a vector in the plane using the two point vectors 4. Calculate the normal vector by the cross product of the two vectors and in the relation 5. Use the point on the plane and the normal vector to form the equation for the plane. 15

16 Intersection of Objects in Space or Parallel and Skew Test Intersection of Objects in Space is the intersection points between any two lines, a line and a plane, or two planes. Line with Line Intersection Point or Parallel and Skew Line Test Intersection between a line either with a point and a parallel vector or with a set of parametric equations or with the symmetric equations and another line either with a point and parallel vector or with a set of parametric equations or with the symmetric equations 1. If one or both of the lines are not in parametric equation form, convert them into parametric equation form. 2. Choose any two of the three relation equations and solve for the parameters and from two of 3. Plug results for and into the third unused equation to check if the third equation and the solution holds true. If the third equation also holds true for the same and, the lines do intersect at a point so continue to step 4. If the equations have infinity solutions for the and, the lines do intersect but the lines are the same line. If third equation does not hold true for the same and, the lines do not intersect at all but are one of either: The lines are parallel if the lines do not intersect and The lines are skew if the lines do not intersect and 4. Plug the results for into or plug the results for into for the point of intersection coordinates. Line with Plane Intersection Point or Parallel Line and Plane Test Intersection between a line either with a point and a parallel vector or with a set of parametric equations and a plane either with a point and a normal vector or with standard equation where 1. If the line is not in parametric equation form, convert it into parametric equation form. If the plane is not in standard equation form, convert it into standard equation form. 2. Plug the parametric equations of the line into the plane standard equation for to form 3. Solve the equation for the parameter. If the equation has one solution for the, the line and plane do intersect at a point so continue to step 4. If the equation has infinity solutions for the, the line and plane do intersect but the line is in the plane. If the equation has no solution for the, the line and plane do not intersect at all but are parallel. 4. Plug the into for the point of intersection coordinates. Plane with Plane Intersection Line or Parallel Planes Test Intersection between a plane either with a point and a normal vector or with standard equation where and another plane either with a point and a normal vector or with standard equation where 1. Calculate the parallel vector for the line of intersection by cross product If cross product is nonzero the planes intersect, continue to step 2. If cross product is zero the planes are parallel. 2. Calculate a point on the line of intersection by choosing a value for any one of and solving the two standard equations and for the other values to form 3. Use the point on the line and the parallel vector to form the equations for the line of intersection. 16

17 Angle between Objects in Space or Parallel and Skew Test Angle between Objects in Space is the intersection angle between any two lines, a line and a plane, or two planes. Line with Line Intersection Angle or Parallel and Perpendicular Test The angle between a line either with a point and a parallel vector or with a set of parametric equations or with the symmetric equations and another line either with a point and parallel vector or with a set of parametric equations or with the symmetric equations Be sure that the two lines do in fact intersect and are neither parallel nor skew before using either of these formulas! If the two lines are identical if they intersect and the two lines are parallel if they do not intersect. If the two lines are at an angle if they intersect and the two lines are not related if they are skew. If the two lines are perpendicular if they intersect and the two lines are not related if they are skew. Line with Plane Intersection Angle or Parallel and Perpendicular Test The angle between a line either with a point and a parallel vector or with a set of parametric equations and a plane either with a point and a normal vector or with standard equation Be sure that the line and plane do in fact intersect and are not parallel before using either of these formulas! If the line is in the plane if they intersect and the line and the plane are parallel if they do not intersect. If the line and the plane are at an angle as the line and the plane will definitely intersect. If the line and the plane are perpendicular as the line and the plane will definitely intersect. Plane with Plane Intersection Angle or Parallel and Perpendicular Test The angle between a plane either with a point and a normal vector or with standard equation where and another plane either with a point and a normal vector or with standard equation where Be sure that the two planes do in fact intersect and are not parallel before using either of these formulas! If the two planes are identical if they intersect and the two planes are parallel if they do not intersect. If the two planes are at an angle as the two planes will definitely intersect. If the two planes are perpendicular as the two planes will definitely intersect. 17

18 Distance between Objects in Space The Distance between Objects in Space is the shortest and therefore perpendicular distance between any two of points, lines, or planes. The perpendicular distance is the magnitude length of the line segment that intersects one object at each of its two ends and intersects each object in a perpendicular orientation. Point to Point Distance Distance between a point with coordinates and another point with coordinates Point to Line Distance Distance between a point with coordinates and a line either with with a point and a parallel vector or with a set of parametric equations or with a set of symmetric equations Point to Plane Distance Distance between a point with coordinates and a plane either with a point and a normal vector or with standard equation by choosing values or being given values for any two of and solving the standard equation for the third value to form Line to Skew or Parallel Line Distance Distance between a line either with a point and a parallel vector or with a set of parametric equations or with the symmetric equations and another line either with a point and parallel vector or with a set of parametric equations or with the symmetric equations Line to Parallel Plane Distance Distance between a line either with a point and a parallel vector or with a set of parametric equations or with the symmetric equations and a plane either with a point and a normal vector or with standard equation by choosing values or being given values for any two of and solving the standard equation for the third value to form Plane to Parallel Plane Distance Distance between a plane either with a point and a normal vector or with standard equation where and another plane either with a point and a normal vector or with standard equation where 18

19 Cylinders, Plane Curves, and Quadric Surfaces Cylinders and Quadric Surfaces are two types of very common surfaces of either one, two, or three variables. Completing the Square converts a general second order polynomial of each variable x variable into a perfect square factor. 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. Cylinders A Cylinder is an equation of exactly two variables and a surface graph of exactly two variables in three dimensions or in the space of three variables. Whichever one of the three variables is missing from the Cylinder equation is the orientation axis. A Cylinder retains its same cross section shape along this axis of orientation. The Cylinder Form is x axis circular cylinder The x axis circular cylinder will keep its circular cross section graph shape found in the plane along the entire axis. y axis circular cylinder The y axis circular cylinder will keep its circular cross section graph shape found in the plane along the entire axis. z axis circular cylinder The z axis circular cylinder will keep its circular cross section graph shape found in the plane along the entire axis. x axis elliptical cylinder The x axis circular cylinder will keep its elliptical cross section graph shape found in the plane along the entire axis. y axis elliptical cylinder The y axis circular cylinder will keep its elliptical cross section graph shape found in the plane along the entire axis. z axis elliptical cylinder The z axis circular cylinder will keep its elliptical cross section graph shape found in the plane along the entire axis. x axis function cylinder The x axis function cylinder will keep its cross section graph shape that is found in the plane along the entire axis. y axis function cylinder The y axis function cylinder will keep its cross section graph shape that is found in the plane along the entire axis. z axis function cylinder The z axis function cylinder will keep its cross section graph shape that is found in the plane along the entire axis. 19

20 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 Quadric Surfaces A Quadratic Surface is a second degree polynomial equation of exactly three variables and a surface graph of exactly three variables in three dimensions or in the space of three variables. The General Quadric Surface form is With a condition that certain coefficients of the General Quadric Surface Form are zero leads to some common surfaces. 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. Elliptical or Circular Ellipsoid Cross Sections: Circle if, Ellipse if, Sphere if. Elliptical or Circular Hyperboloid of One Sheet Cross Sections: Circle, Hyperbola if, Ellipse, Hyperbola if. Elliptical or Circular Hyperboloid of Two Sheets Cross Sections: Circle, Hyperbola if, Ellipse, Hyperbola if. Elliptical or Circular Cone Cross Sections: Circle, Line if, Ellipse, Line if. Elliptical or Circular Paraboloid Cross Sections: Circle, Parabola if, Ellipse, Parabola if. Hyperbolic Paraboloid Cross Sections: Parabola, Hyperbola if, Parabola, Hyperbola if. 20

21 Cylindrical Coordinates Cylindrical Coordinates are a special type of parametric equations used for describing the relationship function for a graph with pole symmetry and curvilinear repetition about an axis. Cylindrical Coordinates allow for more possible relations. For pole symmetry and curvilinear repetition about an axis, Cylindrical Coordinates require a factorable term within one variable expression and a factorable term within another variable with the pole variable left equal. Cylindrical Coordinates differ from Rectangular Coordinates. Rectangular Coordinates axes,, and are static and do not move. Cylindrical Coordinates axes and are not static but rotate to an angle counterclockwise for positive and clockwise for negative such that a point is always located above or below the axis, forward along it for positive and backward along it for negative. The coordinates state the distance along the axis, the angle that the axis has been rotated in the plane from one axis, and the distance along axis. Conversion of Rectangular Coordinates and Graphs to Cylindrical Coordinates and Graphs Rectangular Coordinates can be converted to equivalent Cylindrical Coordinates by the relations: Conversion of Cylindrical Coordinates and Graphs to Rectangular Coordinates and Graphs Cylindrical Coordinates can be converted to equivalent Rectangular Coordinates by the relations: Conversion of z axis Cylindrical Coordinates and Graphs to Spherical Coordinates and Graphs z axis Cylindrical Coordinates can be converted to equivalent Spherical Coordinates by the relations: Conversion of Spherical Coordinates and Graphs to z axis Cylindrical Coordinates and Graphs Spherical Coordinates can be converted to equivalent z axis Cylindrical Coordinates by the relations: 21

22 Common Cylindrical Coordinate Forms The following are the most common Cylindrical Coordinate Forms and should be memorized: Circular Cylinder with x axis pole Circular Cylinder with x axis pole and radius has general form and cylindrical coordinate form: Circular Cylinder with y axis pole Circular Cylinder with y axis pole and radius has general form and cylindrical coordinate form: Circular Cylinder with z axis pole Circular Cylinder with z axis pole and radius has general form and cylindrical coordinate form: Circular Cone with x axis pole Circular Cone with x axis pole and constant has general form and cylindrical coordinate form: Circular Cone with y axis pole Circular Cone with y axis pole and constant has general form and cylindrical coordinate form: Circular Cone with z axis pole Circular Cone with z axis pole and constant has general form and cylindrical coordinate form: Circular Paraboloid with x axis pole Circular Paraboloid with x axis pole and constant has general form and cylindrical coordinate form: Circular Paraboloid with y axis pole Circular Paraboloid with y axis pole and constant has general form and cylindrical coordinate form: Circular Paraboloid with z axis pole Circular Paraboloid with z axis pole and constant has general form and cylindrical coordinate form: 22

23 Spherical Coordinates Spherical Coordinates are a special type of parametric equations used for describing the relationship function for a graph with radial point symmetry about the origin. Spherical Coordinates allow for more possible relations. For radial point symmetry about the origin, Spherical Coordinates require a factorable term within one variable expression, a factorable term within another variable, and a factorable term within another variable. Spherical Coordinates differ from Rectangular Coordinates. Rectangular Coordinates axes,, and are static and do not move. Spherical Coordinates axes,, and are not static but will rotate in the plane to an angle counterclockwise for positive and clockwise for negative and will rotate down from the axis to an angle down for positive or for negative in such a way that a point is always located on the axis, forward along it for positive and backward along it for negative. The coordinates state the distance along the axis, the angle that the axis shadow in the plane has been rotated from the axis, and the angle that the axis has been rotated down from the axis. The value is the same on both triangles and are used to set the triangle relations equal Conversion of Rectangular Coordinates and Graphs to Spherical Coordinates and Graphs Rectangular Coordinates can be converted to equivalent Spherical Coordinates by the relations: Conversion of Spherical Coordinates and Graphs to Rectangular Coordinates and Graphs Spherical Coordinates can be converted to equivalent Rectangular Coordinates by the relations: Conversion of z axis Cylindrical Coordinates and Graphs to Spherical Coordinates and Graphs z axis Cylindrical Coordinates can be converted to equivalent Spherical Coordinates by the relations: Conversion of Spherical Coordinates and Graphs to z axis Cylindrical Coordinates and Graphs Spherical Coordinates can be converted to equivalent z axis Cylindrical Coordinates by the relations: 23

24 Common Spherical Coordinate Forms The following are the most common Spherical Coordinate Forms and should be memorized: Sphere with origin center Sphere with origin center and radius has general form and spherical coordinate form: Circular Cone with x axis pole Circular Cone with x axis pole and constant has general form and cylindrical coordinate form: Circular Cone with y axis pole Circular Cone with y axis pole and constant has general form and cylindrical coordinate form: Circular Cone with z axis pole Circular Cone with z axis pole and constant has general form and cylindrical coordinate form: 24

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26 Line Parametric Equations Line Parametric Equations describe the relationship function between two or three variables,, and through a single parameter whose graph is a straight or curvilinear line in three dimensions known as a Space Curve. Each value of the parameter in the domain of,, produces exactly one ordered triplet on the Space Curve graph where is the coordinate function, is the coordinate function, and is the coordinate function. Line Parametric Equations can be expressed in parametric function form or vector parametric form: Line Parametric Function Form Line Parametric Function Form expresses each of the variables as a separate function of parameter. Line Parametric Vector Form Line Parametric Vector Form expresses variables as a vector from the origin to the point. Common Line Parametric Equation Forms and Conversion to Parametric Equations The following are the most common Line Parametric Equation Forms and should be memorized: Straight Line Straight Line with point and vector has general form and parametric form: Explicit Functions of x Explicit Function of x has general form and and parametric form: Explicit Functions of y Explicit Function of y has general form and and parametric form: Explicit Functions of z Explicit Function of z has general form and and parametric form: Circle parallel to yz plane with x axis center Circle with center and radius has general form and parametric form: Circle parallel to xz plane with y axis center Circle with center and radius has general form and parametric form: Circle parallel to xy plane with z axis center Circle with center and radius has general form and parametric form: 26

27 Circle parallel to yz plane with general center Circle with center and radius has general form and parametric form: Circle parallel to xz plane with general center Circle with center and radius has general form and parametric form: Circle parallel to xy plane with general center Circle with center and radius has general form and parametric form: Ellipse parallel to yz plane with x axis center Ellipse with center and axes and has general form and parametric form: Ellipse parallel to xz plane with y axis center Ellipse with center and axes and has general form and parametric form: Ellipse parallel to xy plane with z axis center Ellipse with center and axes and has general form and parametric form: Ellipse parallel to yz plane with general center Ellipse with center and axes and has general form and parametric form: Ellipse parallel to xz plane with general center Ellipse with center and axes and has general form and parametric form: Ellipse parallel to xy plane with general center Ellipse with center and axes and has general form and parametric form: Cycloid in the xy plane 27

28 Hyperbola parallel to yz plane with x axis center Hyperbola with center and axes and has general form or and parametric form: Hyperbola parallel to xz plane with y axis center Hyperbola with center and axes and has general form or and parametric form: Hyperbola parallel to xy plane with z axis center Hyperbola with center and axes and has general form or and parametric form: Function Helix with x axis center Function Helix with y axis center Function Helix with z axis center Conversion of Parametric Equations to Explicit Equations or Implicit Equations The conversion of a Parametric Equation to an Explicit Equation or an Implicit Equation can sometimes be accomplished: If at most one of the parametric equations is a trigonometric function, then this method may be possible 1. Choose the simpler function of either or 2. Algebraically solve for the parameter in terms of the dependent variable if is the simpler function, or algebraically solve for the parameter in terms of the dependent variable if is the simpler function 3. Plug the parameter in terms of the dependent variable into the other function if is the simpler function, or plug the parameter in terms of the dependent variable into the other function if is the simpler function. The parameter has now been eliminated. 4. Simplify the result into an Explicit Equation by solving for either variable or in terms of the other variable. If both of the parametric equations are trigonometric functions, then this method may be possible 1. If the trig functions have phases that differ by a scalar multiple, apply the Double Angle Identities or Half Angle Identities as needed until the trig functions have identical phases. 2. Algebraically solve for whatever trig function of parameter exists from the relation in terms of the dependent variable and algebraically solve for whatever trig function of parameter exists from the relation in terms the dependent variable. Do not solve for the parameter itself, just the trig functions of. 3. If the trig functions from and are a and pair, plug the results into the Pythagorean Identity: If the trig functions from and are a and pair, plug the results into the Pythagorean Identity: The parameter has now been eliminated. 4. Simplify the result into an Implicit Equation. 28