Lesson Five. Calculus - Department of Mathematics
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1. 1 a 2 Find g1 x y dedy and 1 _evaluate Taylor Polynomials To find the Taylor series approximation of a function centered at x a to the nth degree use the command tayor function x a n Find the Taylor polynomial approximation of r x e centered at x to the 8th degree The output includes the order of an 8th degree polynomial This output is not a polynomial and if you wished to plot it you would have to delete that last expression To do so use convert expression polynom Then plot e and the Taylor polynomial on the same set of axes Use a range of 1 102. function of more than one variable Use the form limit function sequence of points whatever Find imit fI x y x 1 y 1 and limit gl x y x 0 y 0 What does the output of the second limit mean Y Derivatives The standard command for finding the derivative of a function is diff function variable or variables Find the derivatives of the functions f g and A You can find higher derivatives in different ways The most straight forward is by typing in the variable as a sequence as many times as necessary That is if you wish to find the 4th derivative of a function you type diff function x x x x Try it below on the function g However a more efficient way may be to use the repetition operator Try typing x 4 and executing it Now enter diff function x 4 One needs to be a little careful with the independent variable in the function under consideration Execute the following two statements and be sure you understand the different outputs May bad ae kf x dx dx Tt is often the case that we wish to find the derivative of a function defined implicitly in a relation The command to use is implicitdiffand it takes the form implicitdiff equation dependent variable independent variable by O 2 3 5 Find and of the relation xy x y 3x x dy How does Maple handle piecewise functions Check by executing the function below and then by finding the derivative First plot the fun
3. Lesson Five Calculus L Valdes Department of Mathematics San Jose State University Objectives To be able to find limits of functions find derivatives of functions find partial derivatives and derivatives of higher order integrate functions find the Taylor polynomial approximations of functions ARWNS Read Section 4 4 in User Manual To begin this lesson let us define some functions so they will be at the ready Say f x 3 rae ace 14 g x sin x cos 3 x and A x a 2 We will also need some x multivariable functions Type in fl x y xy z and gl x y sinf 5 x y There is also a package called student that we will be working with so let us open it now Remember how with student f x gt 3 x 19x 14 g x sin x cos 3 x h ea Ta X 3xy fl yi gt 4 4 a ty gl x y sin 1 Y Limits To find the limit of a function we write limit function x a direction where direction is right or left If direction is omitted then the limit from both directions is returned Find lim f x lim g x x gt x gt and lim h x lim h x x0 x07 The command Limit is used when you wish the function to remain inert that is to not return a value Try changing one of the expressions above so that it begins with L If you wish to use Limit you can follow that command line with value to obtain the value of the limit Tt may be the case that you wish to find the limit of a
4. ction with 1 lt x lt 6 to get a better idea of how the function behaves Finally use eval and the label of the output of the derivative command to find the values of the derivative atx 5 x 0 x 2 andx 4 gt gg x gt piecewise x lt 0 x 2 1 x lt 2 1 3 There is another command that works in much the same way as diff It is the differential operator D Use it instead of diff when you want to output a funcion In other words D is used for creating functions and diff is used to create expressions To find the derivative of the function f you enter D f Try to find the derivatives of f g and h using the differential operator D Then evaluate each of the derivatives atx 4 by using the statement D f 4 The correspondence between the two commands is D f x diff x x To use the operator D to yield higher order derivatives write D n function In other words D n function x diff function x n Find the following Do f D g and D A Partial Differentiation Maple handles partial differentiation in the suspected manner The independent variable is used as the second argument in the command diff Find et er a ene an Bp Gy Sy To find higher partial derivatives continue adding arguments For example difff x y an Find all the second partials of f7 and g1 Some people prefer using the differential operator D when finding partials You can be very explicit in which variable yo
5. u have chosen to differentiate by When given a function defined in terms ofn variables and you wish to find the partial with respect to the ith variable you type D i d Try finding D f7 x y below and compare it with diff f1 x y y Now if you wish to find function x y you can use the diff command or D 1 2 f x y dy Ox And if you wish to find the partial derivative repeatedly with respect to x say n times you can enter D x n x y Use D to find all second partials of f7 and g Let us look at one last fun item dealing with differentiation There is a command showtangent found in the student package Execute the following As you see the graph of f x was drawn along with the tangent line atx 1 Plot g x and the T tangent line atx 3 y y Integtration Integration is straight forward The format for a definite integral is intfnc x a b where fnc is a fucntion in x and x ranges from a to b Try integrating f x from 3 to 5 and then find the integral without evaluation Once again using nt instead of int will return the inert function Try it below You can use evalfto evaluate Int but this is not always a wise thing to do in that you may very well get erroneous results Integrating piecewise functions is no problem Integrate gg from 5 4 2 3 To find multiple integrals we nest the integrals To find f1 x y dx dy enter 1 0 22 ant int f1 x y x 0 3 y
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