Chapter 11: Symbolic Computing for Calculus MATLAB for Scientist and Engineers Using Symbolic Toolbox

You are going to See that MuPAD does calculus as we do Analyze functions by their plots, limits and derivatives Be glad that MuPAD does all complex integrations and differentiation for you.

Differentiation: Definition Differentiation by Definition

Functions and Expressions On Functions On Expressions

Hold actual evaluations Multiple Derivatives Hold actual evaluations Derivative of Symbolic Functions Multiple Derivatives $: Sequence Operator

Value of Derivative at a Point Functions Expressions

Multivariate Functions

Multivariate Functions (cont.) Partial Derivatives on x and y Partial Derivatives on 1st variable Partial Derivatives on 1st and 2nd variables

Jacobian Partial derivatives

Exercise Consider the function f : x → sin(x) /x. Compute first the value of f at the point x = 1.23, and then the derivative f′(x). Why does the following input not yield the desired result? f := sin(x)/x: x := 1.23: diff(f, x)

Exercise De l’Hospital’s rule states that Compute by applying this rule interactively. Use the function limit to check your result.

Exercise Determine the first and second order partial derivatives of f1(x1, x2) = sin(x1 x2) . Let x = x(t) = sin(t), y = y(t) = cos(t), and f2(x, y) = x2 y2. Compute the derivative of f2(x(t), y(t)) with respect to t.

Limit Limit

Left and Right Limit

Other Limits Conditional Limits Intervals

Exercise Use MuPAD to verify the following limits:

Integration Definite and Indefinite Integrations

Numeric Integration No Symbolic Solution

Integration with Real Parameters Use assume to set attributes of parameters.

Exercise Compute the following integrals: Use MuPAD to verify the following equality:

Exercise Use MuPAD to determine the following indefinite integrals:

Exercise The function intlib::changevar performs a change of variable in a symbolic integral. Read the corresponding help page. MuPAD cannot compute the integral Assist the system by using the substitution t = sin(x). Compare the value that you get to the numerical result returned by the function numeric::int.

Sum of Series

Exercise Use MuPAD to verify the following identity: Determine the values of the following series:

Calculus Example Asymptotes, Max, Min, Inflection Point Look at the overall characteristics of the function.

Asymptotes Horizontal Vertical

Min and Max Roots of the Derivative

Inflection Point Roots of the Second Derivative

Putting All Together Display the findings about the function.

Key Takeaways Now, you are able to find limit with optional left, and right approaches, get derivatives of functions and expressions, analyze functions by finding their asymptotes, maxima and minima, and to get definite and indefinite integrals of arbitrary functions.

Notes limit(f(x),x=infinity) diff(sin(x^2)^2,x) hold(expr) f := x -> x^2*sin(x) reset() f'(x) D([1,2],f) limit(1/x, x=0, Right) PI int(sin(x),x=0..PI) assume(a>0) int(x^n,x) assuming n <> -1 sum(k^2,k=1..n) sum(x^n/n!,n=0..infinity simplify(expr) numer(expr) denom(expr) solve(expr) op(sol,[2,1,1]) plot::Line2d([x1,y1],[x2,y2]) plot::PointList2d( [[x1,x2],..])

Notes