Interpreting the Derivative
Rules for Derivative Discovery
Optimization
Exponents and Logarithms
Nico
100
This value represents the rate of change of x at a given point. 100
What is f'(x)? or What is the derivative of x? Found via simple definition of the derivative.
100
The algebraic expression of the power rule, for any constant real number n.
What is d/dx (x^n) = nx^n-1? Solution: Definition of power rule. A bit easy, but it's the 100 question, and it's a fantastically important tool to remember. You could properly prove it in the mathematical sense with a convoluted application of the binomial theorem, but that wasn't even covered in the book except in passing and would take too long, so it's not strictly necessary.
100
A condition where global extrema are guaranteed to exist.
What is a closed interval a =< x =< b, where f is continuous? Solution: Logically speaking, the Extreme Value Theorem is rather intuitive. Cut off any finite chunk of a continuous function line, and you'll logically get points higher/lower than or equal to the rest. Even in a perfectly flat line, every point counts as both global minimum and maximum.
100
This famous mathematician got himself immortalized in the base of the natural logarithm, even though he didn't discover it.
Who is Leonhard Euler? Solution: The explanation that e is named as such because it's called Euler's number can be found in the book, but what the book won't tell you was that it was first precisely identified as important by Jacob Bernoulli as the limit of (1+1/n)^n years before Euler was even born. Euler thought writing that was a hassle, made up e, and used it in enough of his publications that the mathematics community shrugged and accepted it.
100
These slips of paper are undoubtedly Nico's greatest downfall in math class, but which he believes he has made a sincere effort to more consistently fill out.
What are the warm-up and exit tickets? Solution: Don't think I need to directly show this one, but it is something I hope I have improved on over the year.
200
This is the unit of measurement for (f^-1)'(x), when f(x)=the number of cookies I have eaten x days into December.
What is cookies/day when days=x to days=x+1? f(x)= units are days (x) and cookies (f(x)). Ergo, f'(x)= days per cookie, after x cookies consumed (ergo, next cookie will take f'(x) days of time to consume). The inverse of that function can be expressed simply by swapping the units around.
200
The value of d/dx e^(x+3) at x=2.
What is 148.413.... or e^5? Solution: Seems convoluted, but really isn't. e^x is it's own derivative, ergo just solve for e^5.
200
This value is the global maxima for the sine function (y= sin x) in a single 2pi period.
What is 1? Solution: Even not knowing the precise location of the interval, our solution is guaranteed. An interval of 2pi insures that the sine function passes through every y value it will ever have, as that is the period. Ergo, it much reach the top, which as we remember is 1=sin pi/2 + or - any number of 2pi.
200
The number of cell divisions a cell would need to perform before the total cell count exceeded that of the human body (37.2 trillion)?
What is 37.2 trillion and one divisions? Solution: Another trick question. Each cell division is an independent event, and even if they went of in the staggered bursts 2^x would imply, that would count bursts, not total divisions. Because this was cheeky, the 'proper' answer of 45 division instances would also suffice.
200
Nico's stunning success in this important test made him appreciate all he had learned at Zoo School and feel much more confident about college.
What is the ACT Mathematics Section? Solution: I did spectacularly on all sections of the ACT, but the math portion was the only one I got a perfect on. Thanks for everything, Matt!
300
This value is Santa's instantaneous velocity as he passes over Greenwich, England, precisely halfway (in time) into his 24 hour world tour, where his overall miles traveled can be given by y=(43x^2)+10x, where y=miles after takeoff and x=time in hours, his speed naturally increasing as he sheds the weight of all of the presents.
What is 1,042 miles per hour? Solution: If Santa's speed can be expressed via a polynomial, then the derivative of that polynomial (in this case applying power rule gets us y=86x+10) will give us the instantaneous velocity. Thus, we simply plug in 12 hours (halfway into our flight), giving us 86*12=1032+10=1042 MPH.
300
The derivative of z/(5z+2).
What is 2/(5z+2)^2? Solution: Quotient rule gives us d/dz z*5z+2 - z*d/dz 5z+2/ (5z+2)^2. Product rule gives us 1*5z+2 - z(5)/ (5z+2)^2. Simplified into 2/(5z+2)^2
300
In total, I have 50 hats and jackets. This value is the maximum possible number of unique outfits I could create, given an optimal hat/jacket distribution.
What is 625? Solution: The difficulty of this question lies in the lack of information, meaning we have to treat hats and jackets like variables. Ergo, x+y=50 and we're looking to maximize A=xy. Solve constraint for x. y=50-x A=x(50-x). so A=50x-x^2. ID critical points by finding derivative A'(x)=50-2x 50-2x=0. x=25. Intuitively speaking, we know A''(x)=-2, so concave down, meaning this value is as high as we're going ot get. Thus, 25 hats * (50-25=) 25 jackets = 625
300
This must be the half life of my cocoa, given that I left it lying out 4.68 minutes ago and the fluid mysteriously dropped from 9 ounces to 6 ounces.
What is 8 minutes? Solution: Solved by backtracking exponential decay. coco=9 oz*2^-time/half-life. 6 coco=9 coco*2^-4.68/half-life. 2^-4.68/half-life=.6666666 simple algebra gets us 8. Somehow I doubt the cause was radioactive coco, however....
300
Of all the portfolios Nico had to put together, this one was the first he got done, redeeming last year's catastrophe.
What is the AP Calculus portfolio? Solution: I included this one as a homage to my appreciation for the simplicity of this assignment and how well I felt the Jeapardy theme went with the way we did math this year. It also goes to show that I really think I genuinely enjoyed math this time around, and have hopefully matured in my work habits.
400
The function one would use to chart the steady decrease in the seasonal growth of the snowfall rate every winter, as a product of global warming, expressed as a simple change from the original function of the amount of snow s(t) at time t.
What is the third derivative of s(t) or s'''(t)? This is a bit complicated, but basically, s(t) gives us snow at time t. s'(t) would thus give us snowfall rate at time t (instantaneous velocity, so to say), s''(t) shows us how that changes as write approaches (acceleration in snowfall rate), and our final, third derivative shows us how THAT ALSO changes over time.
400
This surprisingly long function is the derivative of y=cos^2 (x^3).
What is -6x^2 *cos(x^3) * sin(x^3)? Solution: This requires multiple applications of the chain rule as well as the trigonometric rules. cos^2 (x^3)=[cos (x^3)]^2 by definition. chain rule gives us 2(cos(x^3)) * d/dx (cos(x^3)). chain again and d/dx cos rules 2 cos(x^3) * -sin(x^3) * d/dx (x^3) power rule turns final x^3 into 3x^2. Simplify and multiply for final answer.
400
By pushing a rectangle into a circle with a radius of one meter, you can cover at most this much surface area.
What is 2 square meters? Solution: This is easier to show via graphing, but that doesn't work in this format, so this'll get complicated. Imagining our circle as one centered on the point of origin in a standard x,y coordinate grid, we can define the rectangle we're trying to construct as an extrapolation of a single point, (a vertex of the rectangel), in coordinate 1 (on the edge of the circle), as the perfect symmetry of our circle would naturally leave the other 3 corners obvious. This gives us two nonnegative x,y values to shoot for. The area we're looking to maximize can be defined as A=2x*2y or 4xy Optimizing for x gives us x= sqrt(1-y^2), so we can give area as 4y*sqrt(1-y^2). Derivative=-4y^2/sqrt(1-y^2), thus 4-8y^2/sqrt(1-y^2), and at this point it's fairly obvious our critical points are 1 and sqrt(1/2) (we can dispose of the extraneous negative answers, as we're in the first quadrant). y being at 1 would naturally give us a perfect line of a rectangle, ergo we plug in sqrt(1/2) into a(y), giving us our final value of 2.
400
Sqrt [(xy^2)/z^8] in terms of log x, y, and z.
What is 1/2 log x + log y - 4 log z?
400
This helpful mathematical rule was the most difficult concept for Nico to grapple with this year.
What is the chain rule? Solution: The Chain Rule is admittedly a fantastically helpful tool in finding derivatives, but personally I found the entire thing a tremendous hassle, particularly in what counted as inside and outside the function. Using it has a habit of extending equations into convoluted blobs of numbers and variables, and in particular it lacked the neatness the power rule had. Wasn't so much difficult, as tedious.
500
A philosophical quandary raised by Zeno of Elea, arguing that motion is impossible because nothing can move in a precise instant, which is intuitively (if not physically) addressed by the concept of derivatives.
What is the Arrow Paradox or Fletcher's Paradox? Solution: The paradox is named as such because the analogy was one of an arrow. At an absolute instant, what speed does it have? Admittedly, this one is a bit unfair, as it requires you to know something (the name of a paradox) not exactly covered in the book (which is why it's the 500 question). But it's important in that it gets to the heart of what a derivative is and why it matters. A derivative is an expression of the rate of change at an infintesimal. It is what breathes life into the arrow even as it hovers in the frame of an instant and gives the curve of an arc its' perfectly fluid slope.
500
DOUBLE JEOPARDY This equation is of the tangent line at x=2 for y= 4sqrt(2x) - 6e^(2-x).
What is y=8x-14. Solution: When in doubt, chain rule it out. We need the derivative before we can find the tangent line at a given point, so firstly 4*(1/2)*(2x)^(-1/2) *(2) - 6e^(2-x) *-1 = 4(2x)^-1/2 + 6e^2-x = 4/sqrt(2x) + 6e^2-x which has mercifully remained untouched, thanks to e. Evaluating at x=2, 4/(sqrt2*2=2) + 6e^(2-2=0) = 8 giving us our slope. Go back 2 steps 2 -(8*2)=-14 and we have all the parts for our final equation.
500
The dimensions for the zoo's new Red Panda habitat, built to maximize area given only 60 meters of wall to box in the critters.
What is a 60 meter large circle? Solution: This is a bit of a trick question, as it leads you to think you should solve for the optimum rectangle ('box in"), like similar optimization problems. However, it never specifies the shape, and the true key to optimization is to think outside the box. In this case, recognizing the age-old engineering principle of a circle being optimal for area.
500
By this fateful year, the popularity of Kwanzaa (6.4 adherents as of now, exponential decay of -5% yearly) will be exceeded by Pastafarianism (1.01 million 'adherents', 12% annual growth).
When is 2029? Solution: The key to this problem is turning our holiday trends into exponential decay and growth functions. For Kwanzaa, we can represent it as K=6.40*(1-.05=).95^t, where t=time after 2017 and K= population in millions. The growth of His Noodly Revelation can be represented by P=1.01*1.12^t. Then we just graph and find when the two trends converge, which is at x=11.216, or a little over 11 years into the future. Given that the year is almost over, Pastafarianism should eclipse Kwanzaa by the year 2029
500
Between witty put-downs and inexplicable Disney music numbers, Nico has felt he has really gotten to know this superb teacher.
Who is Matt Johnson? Solution: Growth as a person can be measured in many ways, and I think one of the most important ways I've grown over the last semester is in connecting to other people, including you. For so long I've been so disengaged from classes, adrift in a sea of my own thoughts. Some days, I still am, but I hope that in general that's changed for the better.
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