chapter 11: symbolic computing for calculus matlab for scientist and engineers using symbolic...
TRANSCRIPT
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 inte-
grations and differentiation for you.
3
Differentiation: Definition
Definition
Differentiation by Definition
4
Functions and Expressions
On Functions On Expressions
5
Multiple Derivatives
Derivative of Symbolic Functions
Multiple Derivatives
Hold actual evaluations
$: 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
( , , )
( , , )
x y zJ
r u v
10
Exercise
Consider the function f : x → sin(x) /x. Com-pute 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 de-sired 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 re-spect 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 indef-inite integrals:
22
Exercise The function intlib::changevar performs a change
of variable in a symbolic integral. Read the correspond-ing 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 characteristicsof 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 arbi-
trary functions.
31
Notes
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) PIlimit(1/x, x=0, Right)
int(sin(x),x=0..PI) int(x^n,x) assuming n <> -1assume(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)
32
Notes