1. functions of single variables input output - how do you evaluate this function? - how do you find the domain/range? - how do you plot the function? MAT17C – LEWIS – 04/02/12

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

3. x domain = set of inputs input output y

4. x domain = set of inputs input output y plot of f (x) range = set of outputs

5. functions of two variables - how do you evaluate this function? - how do you find the domain/range? - how do you plot the function?

6. x y z input: (x o,y o ) = (1,1) output: a point on graph of f (x,y)

7. x y z points on graph of f (x,y)

8. x y z a point on graph of f (x,y) surface plot of f (x,y) range = set of outputs domain = set of inputs

10. http://nrs.ucdavis.edu/stebbins/maps/map.htm z=f (x,y) x y Topographic maps plot curves of equal altitude. These curves are also called contour lines or level curves.

12. x y z level z=z*=0.5

13. level curves/contour lines for z=0.5 :. y x

14. (x o,y o ) = (1,1) level curves/contour lines for z=z* :. contour plot of f (x,y) x y

15. DOSE RESPONSE CURVE d = dose of toxin (mg) r = fraction of individuals registering the standard response (e.g., death ) d r LD 50 median lethal dose

16. DOSE RESPONSE CURVE family of curves d r note: LD 50 = 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 )

17. surface plot DOSE RESPONSE CURVE

18. contour plot contour lines for r=r* :

19. 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 about in other directions? v = activity of a neuron (Hz) (x,y) = position of stimulus on hand (or retina)

20. RECEPTIVE FIELDS x y

21. (.. or hand) x y The position of a stimulus is “encoded” in the activity level of a neuron.

23. surface plotcontour plot (level sets) x y v x y

24. ?