CoCalc Public FilesPresentation.sagews
Author: Trevor Hebert
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print("Derivatives For Dummys")
print("            by Trevor Hebert")
print("(I wanted to make this bigger but I didn't know how so any help would be appreciated, thank you <3)")

Derivatives For Dummys by Trevor Hebert (I wanted to make this bigger but I didn't know how so any help would be appreciated, thank you <3)
#1 Given the function f(x) = x^3 + x^2 -x+1, determine where f'(x) crosses the x axis, the intervals of increasing and decreasing for f(x), and where f'(x) is greater above or below the x axis.
f(x) = x^3 + x^2 -x+1
plot(f(x), -3, 3)


#After plotting the function, we can see, without calculating the derivative, that it crosses the x axis at -1 and 1/3. We can also see that f(x) increases from (-infinity,-1),(1/3,infinity) and decreases from (-1,1/3). This would mean that f'(x) is below the x axis where it decreases, (-1,1/3), and above the x axis where it increases, (-infinity,-1),(1/3,infinity). Just to prove where the minimum and maximum are, we'll calculate them here.
f(x) = x^3 + x^2 -x+1
fp(x) = diff(f(x))
fp(-1)
fp(1/3)

0 0
#And then we'll graph the derivative to get a visual representation of what we just discussed.

f(x) = x^3 + x^2 -x+1
fp(x) = diff(f(x))
plot(fp(x), -5, 5)

#2 Make a cool effect like a wavy line in a flashlight's light by graphing, and also plot the three functions given below to determine whether each function is increasing or decreasing over the interval (0,2pi).
f(x)=x
g(x)=-x+1
h(x)=sin(x)
plot(f(x), 0, 2*pi, color="yellow")+ plot(g(x), 0, 2*pi, color="black")+ plot(sin(x), 0, 2*pi, color="blue")


#f(x) is the yellow line, g(x) is the black line, and h(x) is the blue line. f(x) is increasing in this interval, and in fact, will continuilly increase forever since you are just increasing the number. g(x) is decreasing in the interval, and again, will decrease forever because you are adding a constant number to a negative number that keeps decreasing. h(x) is doing both in this interval and shows why derivatives are important, it measures the steepness of a graph a.k.a. the slope.






#3 Mr. Henson drove from Maryland to New Jersey to give a puppeteering presentation, the function L represents the distance driven in miles after t hours. What is a good interpretation of the following statement: L'(3)=70

# L is the function of distance driven, which is a function of time. To find the derivative, you would do L/Lt, which makes sense because that is distance over time. L', being the slope of a graph, therefore represents an instaneous rate. L'(3) would mean that, 3 hours after he left, he drove at an instaneous rate of 70mph.

#4 Plot f(x) =2x^2+3x-1 and determine where the relative minima is and the intervals of increasing/decreasing based on the graph

f(x) = 2*x^2+3*x-1
plot(f(x), -2, 2)

f(x) = 2*x^2+3*x-1
fp(x) = diff(f(x),x)
solve(fp(x) == 0, x)
plot(f(x), -10, 10)
#Relative minima at (-.75, -2.125), the second graph that is a zoomed out version of the same function shows that the function is a parabola. increasing from (-.75,inf) and decreasing from (-inf,-.75).


[x == (-3/4)]
#5 The function A(x) = 2500-10x-.01x^2+.0002x^3 shows us the cost of making cans of Arizona Tea(c), where x is the number of cans made and A(x) is the cost of the xth can. Why is it not possible to substitute 200 for determining the cost of making the 200th can?

# Substituting 200 would give us the cost of making 200 cans and not the 200th can. We would need to take the derivative of the function and plug the number into that function. The derivative is -10-.02x+.0006x^2
f(x) = -10-.02*x+.0006*x^2
f(200)

10.0000000000000
f(x) = -10-.02*x+.0006*x^2
f(300)


38.0000000000000
f(x) = -10-.02*x+.0006*x^2
f(500)


130.000000000000
#As you can see, the cost of making Arizona Tea(c) gets higher with more cans being produced. This information is helpful for businesses when determining how to make a profit and not bankrupt themselves.

#Fun fact while researching Arizona Tea(c) and the correct copyright symbol to use, the company has a line of beverages called Shaq fu Punch based on the 1994 (often considered one of the worst games of all time) 2D fighting game Shaq Fu, which is of course based on  Shaquille "Shaq" O'Neal, obviously only known for his starring role in the hit 1996 movie Kazaam

#6 BONUS QUESTION Calculate the odds of Trevor switching his major after two years of math courses

print("89.6487%")

89.6487%