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
x input output y output: y = f (1)=e -1 ~ a point on graph of f (x) input: x o = 1
x domain = set of inputs input output y
x domain = set of inputs input output y plot of f (x) range = set of outputs
functions of two variables - how do you evaluate this function? - how do you find the domain/range? - how do you plot the function?
x y z input: (x o,y o ) = (1,1) output: a point on graph of f (x,y)
x y z points on graph of f (x,y)
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
z=f (x,y) x y Topographic maps plot curves of equal altitude. These curves are also called contour lines or level curves.
x y z level z=z*=0.5
level curves/contour lines for z=0.5 :. y x
(x o,y o ) = (1,1) level curves/contour lines for z=z* :. contour plot of f (x,y) x y
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
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 )
surface plot DOSE RESPONSE CURVE
contour plot contour lines for r=r* :
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)
RECEPTIVE FIELDS x y
(.. or hand) x y The position of a stimulus is “encoded” in the activity level of a neuron.
surface plotcontour plot (level sets) x y v x y
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