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

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<ul><li>Slide 1</li></ul> <p>Chapter 11: Symbolic Computing for Calculus MATLAB for Scientist and Engineers Using Symbolic Toolbox Slide 2 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. 2 Slide 3 Differentiation: Definition Definition Differentiation by Definition 3 Slide 4 Functions and Expressions On Functions 4 On Expressions Slide 5 Multiple Derivatives Derivative of Symbolic Functions Multiple Derivatives 5 Hold actual evaluations \$: Sequence Operator Slide 6 Value of Derivative at a Point Functions Expressions 6 Slide 7 Multivariate Functions 7 Slide 8 Multivariate Functions (cont.) 8 Partial Derivatives on x and y Partial Derivatives on 1 st variable Partial Derivatives on 1 st and 2 nd variables Slide 9 Jacobian Partial derivatives 9 Slide 10 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) 10 Slide 11 Exercise De lHospitals rule states that Compute by applying this rule interactively. Use the function limit to check your result. 11 Slide 12 Exercise Determine the first and second order partial derivatives of f 1 (x 1, x 2 ) = sin(x 1 x 2 ). Let x = x(t) = sin(t), y = y(t) = cos(t), and f 2 (x, y) = x 2 y 2. Compute the derivative of f 2 (x(t), y(t)) with respect to t. 12 Slide 13 Limit 13 Slide 14 Left and Right Limit 14 Slide 15 Other Limits Conditional Limits Intervals 15 Slide 16 Exercise Use MuPAD to verify the following limits: 16 Slide 17 Integration Definite and Indefinite Integrations 17 Slide 18 Numeric Integration No Symbolic Solution 18 Slide 19 Integration with Real Parameters Use assume to set attributes of parameters. 19 Slide 20 Exercise Compute the following integrals: Use MuPAD to verify the following equality: 20 Slide 21 Exercise Use MuPAD to determine the following indefinite integrals: 21 Slide 22 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. 22 Slide 23 Sum of Series 23 Slide 24 Exercise Use MuPAD to verify the following identity: Determine the values of the following series: 24 Slide 25 Calculus Example Asymptotes, Max, Min, Inflection Point 25 Look at the overall characteristics of the function. Look at the overall characteristics of the function. Slide 26 Asymptotes Horizontal Vertical 26 Slide 27 Min and Max Roots of the Derivative 27 Slide 28 Inflection Point Roots of the Second Derivative 28 Slide 29 Putting All Together Display the findings about the function. 29 Slide 30 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. 30 Slide 31 Notes 31 limit(f(x),x=infinity) diff(sin(x^2)^2,x) diff(sin(x^2)^2,x \$ 3) hold(expr) reset() f := x -&gt; x^2*sin(x) f'(x) PI limit(1/x, x=0, Right) int(sin(x),x=0..PI) int(x^n,x) assuming n-1 assume(a&gt;0) sum(k^2,k=1..n) simplify(expr) sum(x^n/n!,n=0..infinity numer(expr) op(sol,[2,1,1]) solve(expr) plot::Line2d([x1,y1],[x2,y2]) plot::PointList2d( [[x1,x2],..]) D([1,2],f) denom(expr) Slide 32 Notes 32 </p>