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.
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":
To find the derivatives and graph the functions, we'll take the following steps:
i) Ln(1+e^x):
Step 1: Find the derivative of f(x) = Ln(1+e^x)
To find the derivative, we can use the chain rule. The derivative of Ln(u) is 1/u multiplied by the derivative of u. So, we have:
f'(x) = (1/(1+e^x)) * (e^x)
ii) (x+sqrt(1+x^2))/2:
Step 1: Find the derivative of f(x) = (x+sqrt(1+x^2))/2
To find the derivative, we'll use the quotient rule. The quotient rule states that for a function g(x) = f(x)/h(x), the derivative can be calculated as:
g'(x) = (f'(x) * h(x) - f(x) * h'(x))/(h(x))^2
So, let's calculate the derivative:
f'(x) = [(1+2x)/2 * (2) - (x + sqrt(1+x^2))/2 * (2x)] / (2)^2
= (1 + 2x - 2x - 2x^2) / 4
= (1 - 2x^2) / 4
iii) x*exp(a*x)/(1+exp(a*x)), with a = 2:
Step 1: Find the derivative of f(x) = x*exp(a*x)/(1+exp(a*x))
To find the derivative, we'll use the quotient rule again. Let's calculate the derivative with respect to x:
f'(x) = [(1+exp(a*x)) * (x * d(exp(a*x))/dx) - x * exp(a*x) * d(1+exp(a*x))/dx] / (1+exp(a*x))^2
Since d(exp(a*x))/dx = a * exp(a*x), and d(1+exp(a*x))/dx = a * exp(a*x), we can simplify:
f'(x) = [(1+exp(a*x)) * (x * a * exp(a*x)) - x * exp(a*x) * a * exp(a*x)] / (1+exp(a*x))^2
= (a * x * exp(a*x) * (1+exp(a*x)) - a * x * exp(a*x)^2) / (1+exp(a*x))^2
= (a * x * exp(a*x) + a * x * exp(a*x)^2 - a * x * exp(a*x)^2) / (1+exp(a*x))^2
= a * x * exp(a*x) / (1+exp(a*x))^2
iv) Graphing the original functions and their derivatives:
To graph the original functions and their derivatives, we can use graphing tools like Desmos. Set the x-axis domain to [-3,3]. Plot the functions Ln(1+e^x), (x+sqrt(1+x^2))/2, and x*exp(a*x)/(1+exp(a*x)) with a = 2 on the same graph.
QB: "The Pencil on String question":
i) To write an implicit equation for the situation where the sum of the distances from the pencil point to the two thumbtacks is always 8 cm, we can consider the coordinates of the thumbtacks as (-3cm, 0) and (3cm, 0), and the pencil point coordinates as (x, y).
The distance from the pencil point to the first thumbtack can be calculated using the distance formula:
√((x - (-3))^2 + (y - 0)^2)
The distance from the pencil point to the second thumbtack can also be calculated using the distance formula:
√((x - 3)^2 + (y - 0)^2)
So, the implicit equation is:
√((x - (-3))^2 + (y - 0)^2) + √((x - 3)^2 + (y - 0)^2) = 8
ii) The geometric figure you get from this situation is an ellipse.
iii) If the difference of the distances is always 2 cm, the equation can be written as:
|√((x - (-3))^2 + (y - 0)^2) - √((x - 3)^2 + (y - 0)^2)| = 2
iv) The geometric figure you get from this situation is a lemniscate.
QD: "Derivative of 6000-4000*cos(2*pi*(t-3)/24)":
To find the derivative and graph the function, follow these steps:
Step 1: Find the derivative of the function f(t) = 6000-4000*cos(2*pi*(t-3)/24)
To find the derivative, we can use the chain rule. The derivative of cos(u) is -sin(u), and the derivative of (t-3)/24 is 1/24.
So, we have:
f'(t) = -4000 * -sin(2*pi*(t-3)/24) * (2*pi/24)
= 4000 * sin(2*pi*(t-3)/24) * (2*pi/24)
= 4000 * sin((1/12)*pi*(t-3))
Step 2: Determine a good x window and y window for the graph
Based on the function, we can assume that the frequency is 1/12*pi and the amplitude is 4000. To ensure a good visual representation of the graph, we can set the x window to [-6, 12] (covering a full cycle) and the y window to [-5000, 7000] (encompassing the maximum and minimum values of the function).
Step 3: Plot the original function and its derivative
Use a graphing tool like Desmos and plot the function 6000-4000*cos(2*pi*(t-3)/24) and its derivative 4000*sin((1/12)*pi*(t-3)) using the defined x and y windows.
By following these steps, you should be able to find the derivatives and graph the given functions accurately.