Calculus Review

GLY-4822

Slope• Slope = rise/run • = y/x • = (y2 – y1)/(x2 – x1)

• Order of points 1 and 2 not critical, but keeping them together is

• Points may lie in any quadrant: slope will work out

• Leibniz notation for derivative based on y/x; the derivative is written dy/dx

Exponents

• x0 = 1

Derivative of a line• y = mx + b• slope m and y axis intercept b• derivative of y = axn + b with respect to x:• dy/dx = a n x(n-1) • Because b is a constant -- think of it as bx0 -- its

derivative is 0bx-1 = 0 • For a straight line, a = m and n = 1 so• dy/dx = m 1 x(0), or because x0 = 1, • dy/dx = m

Derivative of a polynomial

• In differential Calculus, we consider the slopes of curves rather than straight lines

• For polynomial y = axn + bxp + cxq + …• derivative with respect to x is • dy/dx = a n x(n-1) + b p x(p-1) + c q x(q-1) + …

Example

a 3 n 3 b 5 p 2 c 5 q 0

0

2

4

6

8

10

12

14

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

x

y

y = axn + bxp + cxq + …

-5

0

5

10

15

20

25

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

x

y

dy/dx = a n x(n-1) + b p x(p-1) + c q x(q-1) + …

Numerical Derivatives

• slope between points

Derivative of Sine and Cosine

• sin(0) = 0 • period of both sine and cosine is 2• d(sin(x))/dx = cos(x) • d(cos(x))/dx = -sin(x)

-1.5

-1

-0.5

0

0.5

1

1.5

0 1 2 3 4 5 6 7

Sin(x)Cos(x)

Partial Derivatives

• Functions of more than one variable• Example: C(x,y) = x4 + y3 + xy

1 4 7

10 13 16 19S1

S7

S13

S19

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

X

Y

2.5-32-2.51.5-21-1.50.5-10-0.5-0.5-0-1--0.5-1.5--1

Partial Derivatives

• Partial derivative of h with respect to x at a y location y0

• Notation h/x|y=y0

• Treat ys as constants• If these constants stand alone, they drop out

of the result• If they are in multiplicative terms involving x,

they are retained as constants

Partial Derivatives

• Example: • C(x,y) = x4 + y3 + xy • C/x|y=y0

= 4x3 + y0

1 4 7

10 13 16 19S1

S7

S13

S19

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

X

Y

2.5-32-2.51.5-21-1.50.5-10-0.5-0.5-0-1--0.5-1.5--1

WHY?

Gradients

• del h (or grad h)

• Flow (Darcy’s Law):

yh

xhh

ji

hKq

Gradients

• del C (or grad C)

• Diffusion (Fick’s 1st Law):

yC

xCC

ji

CDJ

Basic MATLAB

Matlab

• Programming environment• Post-processer• Graphics• Analytical solution comparisons

• Use File/Preferences/Font to adjust interface font size

Vectors>> a=[1 2 3 4]

a =

1 2 3 4

>> a'

ans =

1 2 3 4

Autofilling and addressing Vectors> a=[1:0.2:3]'

a =

1.0000 1.2000 1.4000 1.6000 1.8000 2.0000 2.2000 2.4000 2.6000 2.8000 3.0000

>> a(2:3)ans =

1.2000 1.4000

xy Plots

>> x=[1 3 6 8 10];>> y=[0 2 1 3 1];>> plot(x,y)

Matrices>> b=[1 2 3 4;5 6 7 8]

b =

1 2 3 4 5 6 7 8

>> b'

ans =

1 5 2 6 3 7 4 8

Matrices

>> b=2.2*ones(4,4)

b =

2.2000 2.2000 2.2000 2.2000 2.2000 2.2000 2.2000 2.2000 2.2000 2.2000 2.2000 2.2000 2.2000 2.2000 2.2000 2.2000

Reshape>> a=[1:9]

a =

1 2 3 4 5 6 7 8 9

>> bsquare=reshape(a,3,3)

bsquare =

1 4 7 2 5 8 3 6 9

>>

Load

• a = load(‘filename’); (semicolon suppresses echo)

If

• if(1)…else…end

For

• for i = 1:10• …• end

BMP Outputbsq=rand(100,100);

%bmp1 output

e(:,:,1)=1-bsq; %r e(:,:,2)=1-bsq; %g e(:,:,3)=ones(100,100); %b imwrite(e, 'junk.bmp','bmp');

image(imread('junk.bmp')) axis('equal')

Quiver (vector plots)

>> scale=10;>> d=rand(100,4);>> quiver(d(:,1),d(:,2),d(:,3),d(:,4),scale)

Contours

• h=[…];• Contour(h)• Or Contour(x,y,h)

Contours w/labels

• C=[…];• [c,d]=contour(C);• clabel(c,d), colorbar

Numerical Partial Derivatives

• slope between points • MATLAB

– h=[]; (order assumed to be low y on top to high y on bottom!)

– [dhdx,dhdy]=gradient(h,spacing)

– contour(x,y,h)– hold– quiver(x,y,-dhdx,-dhdy)

Gradient Function and Streamlines• [dhdx,dhdy]=gradient(h);• [Stream]= stream2(X,Y,U,V,STARTX,STARTY);• [Stream]= stream2(-dhdx,-dhdy,

[51:100],50*ones(50,1));• streamline(Stream)• (This is for streamlines starting at y = 50 from x

= 51 to 100 along the x axis. Different geometries will require different starting points.)

Stagnation Points

Integral Calculus

Cnxaxaxn

n

1

)1(

Cxaxax 2

2

Cxaxax 3

32

Integral Calculus: Special Case

Cnxaxaxn

n

1

)1(

????? 1 xax

Integral Calculus: Special Case

Cnxaxaxn

n

1

)1(

Cxaxax ln 1