2

( ) xy f x e

functions of single variables

inputoutput

- how do you evaluate this function?- how do you find the domain/range?- how do you plot the function?

MAT17C – LEWIS – 04/02/12

x

2

( ) xy f x e

inputoutput

y

1, 1,o ox y e

output: y = f (1)=e-1~0.3679a point on graph of f (x)

input: xo = 1

xdomain = set of inputs

2

( ) xy f x e

inputoutput

x

y

xdomain = set of inputs

2

( ) xy f x e

inputoutput

x

y

plot of f (x)

range = set of outputs

10 y

2 2

( , )x y

z f x y e

functions of two variables

- how do you evaluate this function?- how do you find the domain/range?- how do you plot the function?

2 2

( , )x y

z f x y e

x

y

z

input: (xo,yo) = (1,1)

output:

0135.0e)1,1( 2 fz

2, , 1,1,o o ox y z ea point on graph of f (x,y)

2 2

( , )x y

z f x y e

x

y

z

points on graph of f (x,y)

2 2

( , )x y

z f x y e

x

y

z

2, , 1,1,o o ox y z ea point on graph of f (x,y)

surface plot of f (x,y)

range = set of outputs

domain = set of inputs

yx,

10 z

http://nrs.ucdavis.edu/stebbins/maps/map.htm

z=f (x,y)

x

yTopographic maps plot curves of equal altitude. These curvesare also called contour lines or level curves.

2 2

( , )x y

z f x y e

x

y

z

level z=z*=0.5

2 2

( , )x y

z f x y e

level curves/contour lines for z=0.5 : 2 2ln 0.5 x y

.

y

x

222 8325.0 yx

2 2

( , )x y

z f x y e

(xo,yo) = (1,1)

level curves/contour lines for z=z* : * 2 2lnc z x y

.

contour plot of f (x,y)

x

y

0135.0e)1,1( 2 fz

2

22( )

50

dr f d

d

DOSE RESPONSE CURVE

d = dose of toxin (mg)

r = fraction of individuals registering the standard response (e.g., death)

d

r

LD50

median lethal dose

2

22 23

( , )d

r f d wd w

DOSE RESPONSE CURVE

family of curves

d

r

note: LD50= 2/3 w

w=45

w=50

w=120

w=150

d = dose of toxin (mg)

w = weight of individual (kg)

r = fraction of individuals registering the standard response (e.g., death)

2

22 23

( , )d

r f d wd w

surface plot

DOSE RESPONSE CURVE

2

22 23

( , )d

r f d wd w

DOSE RESPONSE CURVE

contour plot

contour lines for r=r* : dr

rw

*

*1

2

3

0,4

9),(2

2

v

yxyxfv

• What are the level sets (contours) of f(x,y)? Draw the contour plot.

• What is the change in v along contours?

• What are the “shape” of the contours?

• What is the “shape” of f(x,y)?

• What is the shape of the function restricted to y=1?

• What is the shape of the function restricted to ?

• What is the rate of change of v with changes in x at (x=-1,y=1)? What is the rate of change of v with changes in x at (x=-1,y=1)? … what about in other directions?

2xy

v = activity of a neuron (Hz)(x,y) = position of stimulus on hand (or retina)

RECEPTIVE FIELDS

x

y

RECEPTIVE FIELDS

(.. or hand)

(.. o

r ha

nd)

x

y

The position of a stimulus is “encoded” in the activity level of a neuron.

49),(

22 y

xyxfv

surface plot contour plot(level sets)

xy

v

x

y

?