Nice assignment, but how do you expect us to do this when it clearly involves using a graphing app ?
I will do #2, since it is a classic question that simply follows the definition of an ellipse.
(A good teacher will actually demonstrate this for the class on the blackboard )
Make a sketch, let the points given be A(-3,0) and B(3,0
let any point which satisfies the given condition as P(x,y)
then AP + BP = 8
√( (x+3)^2 + (y-0)^2 ) + √((x-3)^2 + (y-0)^2 ) = 8
√( (x+3)^2 + y^2 ) + √((x-3)^2 + y^2 ) = 8
simplify if needed
result: ellipse
If the difference between AP and BP is to be 2, wouldn't that just be
√( (x+3)^2 + y^2 ) - √((x-3)^2 + y^2 ) = 2
result: hyperbola
QA: “The Softmax/Hockey Stick question”:
Functions with these shapes are useful in applications where something had been roughly constant but then starts growing linearly, or vice versa, or has some other piecewise-linear behavior. Examples include the famous “hockey stick” global warming graph. The
For each of these functions, find the derivative. Graph f and f’ on the same graph, on the domain [-3,3]
i) Ln(1+e^x)
ii) (x+sqrt(1+x^2))/2
iii) x*exp(a*x)/(1+exp(a*x)) ; when graphing, perhaps start with a=2. This one is called “softmax” and is useful in AI/Machine Learning/Artificial Neural Networks.
iv) try graphing those 3 original functions (not their derivatives) all on the same graph, on the domain [-3,3].
QB:
i) Suppose I put two thumbtacks into a piece of paper at (-3cm,0) and (+3cm,0).
Then I take a string of length 8cm and tie each end of it to a thumbtack, leaving a length of 8 cm of string from one thumbtack to the other. Then I use a pencil point to pull the string taut and move it around the pencil around, keeping the string taut, so the distance from one tack to the pencil point to the other tack is always 8 cm. Write an implicit equation that describes this situation, which we can rephrase as:
the sum of the distances from the pencil point to the two thumbtacks is always 8 cm.
ii) What geometric figure do you get from that situation?
iii) Ignore the string analogy for now. What if the _difference_ of the distances is always 2 cm, rather than the sum being 8 cm? What equation do you get?
iv) What geometric figure do you get from that situation?
QD: what is the derivative of 6000-4000*cos(2*pi*(t-3)/24) ? graph the original function and the derivative to make sure you got it right. Spend some time trying to figure out a good x window and y window, based on what you can tell from the frequency and the amplitude and vertical shift. Don’t just plot it in the default window on Desmos then zoom out.
1 answer
1 answer