Chapter 3.04 Chain Rule

Here's an interesting link about visualizing the chain rule:
https://www.khanacademy.org/cs/challenge-visualizing-the-chain-rule/1180463364
In Calculus II, we use the chain rule in reverse to do anti-derivatives, and call it u-substitution.

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15 (hint: when you graph it, set k=1 and use an x window of [-1,5]; we
usually care mostly about just x>=0 with this function, though. )
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QA: Find the derivative of:
i) exp(a*x+b)
ii) sin(a*x+b)
iii) cos(a*x+b)
(we would have included Ln(a*x+b) but we won't see that
until Chapter 3.06)

QB: Find the derivative of:
i) exp( a*(x-c)+d )
ii) sin( a*(x-c)+d )
iii) cos( a*(x-c)+d )

QC: Find the derivative of:
i) exp(-x)
ii) exp(-x^2)
iii) exp(-x^2 / 2)
iv) exp( -( (x-mu)/sigma )^2 / 2 )
v) exp( -(x-mu)^2 / (2*sigma^2) )
These functions are important in statistics--they are the formula for the
usual bell-shaped curve.

QD: Find the derivative of these functions, and of course graph the function
and the derivative as you go along:
i) 1- 1/(1+exp(x) )
ii) exp(x)/(1+exp(x))^2
iii) optional: -exp(x)*(exp(x) - 1)/(1+exp(x))^3
Remember that (i) is the Logistic function, which has many uses in applied
math.

QE:
i) graph sin(0.5*x)+0.3*sin(20x) from x=-10 to +10;
which wave is more prominent, the low-frequency or high-frequency one?
ii) find the derivative of the function and graph it as well.
Now which wave is more prominent?
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Part II starts here
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75 hint: find y' and y'', then show that y''-4y'+3y all cancels out to give 0.
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91 MTH/MTHT
92 MTH/MTHT


QF: Recall that f(x)=sqrt(r^2-x^2) is the equation of the upper half of a
circle of radius r centered on the origin.
i) Find the equation of a normal line to the curve at x=a.
ii) What is the y-intercept of that line? How does it change if "a" changes?
or if "r" changes?
iii) Explain your answer to part (ii) geometrically.

QG: Suppose you start with a sine wave of amplitude 1 and frequency "omega":
f(t) = 1*sin(omega*t)
This could represent the sound of a flute, or the voltage of an incoming radio
signal, among other things. Then you take the derivative. 
i) How does the amplitude of the derivative change when you change omega?
Increasing omega ___(increases? decreases?)__ the amplitude
of the derivative.
ii) Are high frequencies more or less prominent than low frequencies after
taking the derivative? Explain.
iii) Applying an operation like a derivative to a signal is called
"filtering" it. Would you say the derivative is a "high-pass filter" (lets
high frequencies pass through, while filtering out low frequencies) or a 
"low-pass filter" ? Explain.


The "Applied Project" about "Where should a Pilot start Descent" might be a
good project for some people in our class. Skim through it and see if you
are interested.