Chapter 1.05

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#33 MTH/MTHT
#34

QA: Can you recite powers of 2 up to 2^16: 1, 2, 4, 8, 16, 32, . . . ? Practice!

QB: with an x window of [0,10] 
i)   graph x   * exp(-x) ; 
ii)  graph x^2 * exp(-x) ; 
iii) graph x^3 * exp(-x) ; 
iv)  graph x^4 * exp(-x) ; 
v) now, what is an appropriate x window for x^20 * exp(-x) ? 
vi) Describe how the shape changes as the power grows.

QC: WEP
i) Repeat problem 34 using 1/(1+exp(b*x+d)) ; 
ii)  if f(x)=1/(1+a*exp(b*x)) and g(x)=1/(1+exp(b*x+3)), what value must "a" have
if we want f(x) to equal g(x) everywhere?
iii) if f(x)=1/(1+4*exp(b*x)) and g(x)=1/(1+exp(b*x+d)), what value must "a" have
if we want f(x) to equal g(x) everywhere?
BTW, these types of functions are called "logistic" functions, not to be confused with "log"=logarithm functions.
The _shape_ of the graph is "sigmoid", but there are sigmoid-shaped functions that aren't logistic functions.
For example, arctangent has a sigmoid shape but isn't the logistic function.

QD: WEP
i) repeat problem 34 using 1/(1+exp(b*(x-c))); 
ii) if g(x)=1/(1+exp(2*x+3)) and h(x)=1/(1+exp(2*(x-c)), what value must "c" have
if we want g(x) to equal h(x) everywhere?
iii) if g(x)=1/(1+exp(2*x+d)) and h(x)=1/(1+exp(2*(x-7)), what value must "d" have
if we want g(x) to equal h(x) everywhere?
iv) What short phrase describes c? what short phrase describes b?