Basics of Calculus
True or False: Calculus is used to study functions on the infinitesimal level (ex: instantaneous rate of change, exact area under a curve)
What is true? Oh I’ll tell you, this statement is true!
True or False: Calculus plays a role in almost every part of everyday life
True or False: Calculus is used today
True
Calculus was invented (mainly) by which 17th century scientist?
Isaac Newton
What is the derivative (instantaneous rate of change) of the function f(x)=0 at the value x = 0
It is zero
Derivative
What are 2 methods used to find volumes of solids generated by revolving functions around coordinate (x and y) axes?
True or False: Calculus probably helped with the invention of the steam engine, thus giving rise to the Industrial Revolution
True
Isaac Newton invented calculus as a means to support his work in what branch of science?
Physics
Find the slope of the tangent line to f(x)=x^2 at x=32
It is 64
You can either use the power rule or set up a limit as h=> 0 to get that the instantaneous rate of change for the function f(x)=x^2 at any value of x is given by the function f’(x)=2x. Then plug in x=32 to get f’(32) = 2 x 32 = 64
You can also set up a limit as h=> 0 of the quantity (f(32+h)-f(32))/h
This type of calculus is used to find the area under a curve
Integral Calculus
True or False: Calculus is very important even in the production of regularly-shaped goods (ex: cubes, boxes, balls)
True
Although we don’t need to rely on calculus to find the volume / surface area of these items, we still rely on calculus to help with other aspects of production (such as design. Ex: We might need to optimize the volume of the box with a certain surface area)
True or False: Copernicus’ heliocentric model of the universe was built in part with the help of calculus
False
Copernicus’ heliocentric theory was out before Galileo proved it (prior to early 1600s), and Newton was not even born until 1641 (IE no calculus). Thus the heliocentric theory had to be modeled without calculus
In which century was calculus developed?
17th century (1600s)
Find the area under the curve of f(x)=x from x=0 to x=100
(Hint: In this case, “the area under the curve” forms a very basic shape)
5000
The area under the curve of f(x)=x simply forms a right triangle with a base and a heightIn this case, both the base and the height of this triangle are 100 units each. Plugging b=100, h=100 into area(triangle)=0.5bh, we get that the area of this triangle is 0.5 x 100 x 100 = 5000 square units
T or F: Calculus is all about limits
True
True or False: Calculus was necessary for Galileo to invent his first telescope
False
Galileo invented / used his telescope in 1609, well before calculus was officially invented by Isaac Newton (around 1665)
Calculus played a fundamental role in the Industrial Revolution. Give one example of how calculus was used in shaping the Industrial Revolution
Hint: Production of goods
Anything that makes sense. One example would be something along the lines of:
“Calculus allowed us to optimize the efficiency at which we produced all kinds of goods”
Another example would be the fact that the use of calculus clearly helped to invent the steam engine
How old was Isaac Newton when he finished inventing calculus?
24 years old
Sarthak opens up a bank account at the local Barbie Bank. His initial investment in this bank is $100; the Barbie Bank offers continuous compound interest at a rate of 4.17% annually. What is Sarthak’s account balance at the end of 417 years? (Round down to the nearest cent)
Hint: r = 0.0417, t = 417
$3,563,717,882.90 (also accept $3,563,717,882.91)
The equation for account balance is in the form P x e^(rt), for P being the principle value ($100 in this case), r being the rate of continuous interest (4.17% equates to 0.0417), and t being the time since the initial investment was made (417 years in this case). Plugging that all in, P x e^(rt) = $100 x e^(417x0.0417) = $100 x e^(17.3889) ≈ $3,563,717,882.906 (It’s 90 cents after rounding down. You also could have rounded to the nearest cent by mistake, but I’ll spare you)
True or False: Calculus can always find the exact areas / volumes of irregular shapes
False
Calculus uses antiderivatives to find areas of irregular shapes / solids. However, not every function always has an elementary antiderivative
Scientists always use functions to describe their data. Almost all growth/decay functions involve a certain constant. What is the name of this constant?
Hint: It is a single letter of the English alphabet. It is also used when working with continuous interest
The name is Euler’s Constant, abbreviated as the letter e. (Accept either of these answers)
How did calculus play a role in helping to start the Enlightenment?
The invention of calculus gave humans a new confidence in their ability to understand, and thus control, the world around them.
This lesser-known mathematician invented calculus separately from Newton, though he did not go as deep into its applications
Leibniz (also accept Gottfried, Gottfried Wilhelm Leibniz, etc.)
Mr. Pettersen has to produce a number of barbie dolls so as to yield a maximum profit from selling them The profit yielded (in $) by selling x Barbie dolls is given by the function p(x) = 200x - 100x^2. Given that n is the critical x-value that maximizes the value of p(x), find n (rounded to the nearest integer if necessary) and the profit associated with selling n dolls
Hint: You can also use vertex form of a quadratic
1 doll for $100 profit
You could use calculus (that’s a way bigger explanation though), but for simplicity I will use parabola vertex form (you will probably need to use calculus to analyze higher degree polynomials though)
P(x)=200x-100x^2 = -100(x^2-2x) = -100((x-1)^2 - 1) = -100(x-1)^2 + 100 => The vertex is a maximum value at (1, 100) => 1 Barbie doll for $100 profit