Download presentation

Presentation is loading. Please wait.

1
**Chapter 11: Symbolic Computing for Calculus**

MATLAB for Scientist and Engineers Using Symbolic Toolbox

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.

3
**Differentiation: Definition**

Differentiation by Definition

4
**Functions and Expressions**

On Functions On Expressions

5
**Hold actual evaluations**

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

6
**Value of Derivative at a Point**

Functions Expressions

7
**Multivariate Functions**

8
**Multivariate Functions (cont.)**

Partial Derivatives on x and y Partial Derivatives on 1st variable Partial Derivatives on 1st and 2nd variables

9
Jacobian Partial derivatives

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)

11
**Exercise De l’Hospital’s rule states that**

Compute by applying this rule interactively. Use the function limit to check your result.

12
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.

13
Limit Limit

14
Left and Right Limit

15
Other Limits Conditional Limits Intervals

16
Exercise Use MuPAD to verify the following limits:

17
Integration Definite and Indefinite Integrations

18
Numeric Integration No Symbolic Solution

19
**Integration with Real Parameters**

Use assume to set attributes of parameters.

20
**Exercise Compute the following integrals:**

Use MuPAD to verify the following equality:

21
Exercise Use MuPAD to determine the following indefinite integrals:

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.

23
Sum of Series

24
**Exercise Use MuPAD to verify the following identity:**

Determine the values of the following series:

25
**Calculus Example Asymptotes, Max, Min, Inflection Point**

Look at the overall characteristics of the function.

26
Asymptotes Horizontal Vertical

27
Min and Max Roots of the Derivative

28
Inflection Point Roots of the Second Derivative

29
Putting All Together Display the findings about the function.

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.

31
**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],..])

32
Notes

Similar presentations

OK

Advanced Engineering Mathematics, 7 th Edition Peter V. O’Neil © 2012 Cengage Learning Engineering. All Rights Reserved. CHAPTER 4 Series Solutions.

Advanced Engineering Mathematics, 7 th Edition Peter V. O’Neil © 2012 Cengage Learning Engineering. All Rights Reserved. CHAPTER 4 Series Solutions.

© 2018 SlidePlayer.com Inc.

All rights reserved.

To ensure the functioning of the site, we use **cookies**. We share information about your activities on the site with our partners and Google partners: social networks and companies engaged in advertising and web analytics. For more information, see the Privacy Policy and Google Privacy & Terms.
Your consent to our cookies if you continue to use this website.

Ads by Google

Ppt on employee provident fund act 1952 republican Download best ppt on cloud computing Ppt on new zealand culture dance Ppt on model view controller python Ppt on hlookup in excel Ppt on bluetooth communication standard Ppt on ministry of corporate affairs goi Ppt on argand plane Ppt on differential used in automobiles Ppt on bluetooth broadcasting