another homework dump?
I'll get you started.
A: Sketch a function that behaves as described (separately for i,ii); also, on the same axes, sketch the derivative of that function.
i) increasing on [0,5], f '(5)=0, then decreasing on [5,∞)
ii) increasing on [0,5], f(5) = ∞, and increasing on (5,∞)
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i) so, basically bell-shaped, peaking at x=5. Now consider 1/(x^2+1) which is bell-shaped, but peaks at x=0. Shifting it 5 to the right, that would be
f(x) = 1/((x-5)^2 + 1)
ii) clearly there is an asymptote at x=5. Consider -1/x, which has its asymptote at x=0. Shift that right by 5, and you have
f(x) = -1/(x-5) = 1/(5-x)
Now go to your favorite online online graphing site and check the graphs.
See
https://www.wolframalpha.com/input/?i=plot+y%3D1%2F%28%28x-5%29%5E2%2B1%29%2C+y%3D1%2F%285-x%29
What do you have to say on the other parts?
QA: Sketch a function that behaves as described (separately for i,ii); also, on the same axes, sketch the derivative of that function.
i) increasing on [0,5], f ‘(5)=0, then decreasing on [5,infinity)
ii) increasing on [0,5], derivative = “infinity” at x=5, and increasing on (5,infinity)
QB: Consider f(x)=sin(x)+x/2 (note that this is different than (sin(x)+x)/2 ). Would you say that f is increasing on [0,20] (try plotting it)? Would most people who haven’t taken a calc class say that f is increasing on [0,20] ? Explain.
QC: Sketch a continuous function on [0,10] that has a global minimum at x=3, a local minimum at x=8, and a global max (on the interval) at x=10. Can you do it without having a local max in the interval?
QD: (i) Find all critical numbers of exp(-x)*x^3 on [0,infinity]. Note that the x^3 is not in the exponent of the “e”: it’s not exp(-x*x^3). While I usually avoid problems where you have to factor things, factoring will help a lot on this problem. Also, you might want the logic that if a*b*c=0, then at least one of a, b, or c must be 0. This shape is related to a “Gamma distribution” in probability/statistics.
(ii) Now find all the critical numbers of exp(-x^10)*x^9 on [0,infinity]. Again, note that the x^9 is not part of the exponent. This shape is related to a “Weibull distribution” in probability/statistics.
(iii) What do you notice about the shape of the two curves above, compared to each other (ignoring their size and spread)?
QE: Find the local minimum of f(x)=x - k*Ln(x), for constant k>0. This is related to “interior point methods” in optimization algorithms (which I teach about in Math 560). It’s a way of saying “minimize f(x)=x for x>=0” without having a separate constraint like “x>=0”; it turns a minimum-is-at-an-endpoint kind of problem into a minimum-is-in-the-interior-of-the-interval problem. This is handy if you have more than one dimension and finding the cornerpoints would take too long.On the other hand, it turns a linear problem (with constraints) into a nonlinear problem, which for decades seemed like a bad trade-off to algorithm designers, but turns out it can be a good idea.
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