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

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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

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Differentiation: Definition Definition Differentiation by Definition 3

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Functions and Expressions On Functions 4 On Expressions

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Multiple Derivatives Derivative of Symbolic Functions Multiple Derivatives 5 Hold actual evaluations $: Sequence Operator

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Value of Derivative at a Point Functions Expressions 6

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Multivariate Functions 7

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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

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Jacobian Partial derivatives 9

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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

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Exercise De l’Hospital’s rule states that Compute by applying this rule interactively. Use the function limit to check your result. 11

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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

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Limit 13

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Left and Right Limit 14

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Other Limits Conditional Limits Intervals 15

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Exercise Use MuPAD to verify the following limits: 16

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Integration Definite and Indefinite Integrations 17

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Numeric Integration No Symbolic Solution 18

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Integration with Real Parameters Use assume to set attributes of parameters. 19

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Exercise Compute the following integrals: Use MuPAD to verify the following equality: 20

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Exercise Use MuPAD to determine the following indefinite integrals: 21

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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

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Sum of Series 23

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Exercise Use MuPAD to verify the following identity: Determine the values of the following series: 24

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Calculus Example Asymptotes, Max, Min, Inflection Point 25 Look at the overall characteristics of the function. Look at the overall characteristics of the function.

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Asymptotes Horizontal Vertical 26

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Min and Max Roots of the Derivative 27

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Inflection Point Roots of the Second Derivative 28

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Putting All Together Display the findings about the function. 29

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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

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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 -> 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>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)

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Notes 32

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