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Thoughts on Theorems
People of Math
Some things never change
Give me a Sign
Meaning what exactly?
100
The way to show that f(x) dominates g(x) as x grows without bound.
What is L'hopital's rule?
100
Even though an apple didn't fall on his head, this Russian mathematician deserves some credit for founding calculus.
Who is Leibniz?
100
A cylindrical piece of clay is sitting on a base of radius. It is slowly smashed downward towards the table, causing it to spread out. We can use the equation V=pi r^2 h to model this scenario these letters are variables.
What are r and h?
100
The value of the integral from -2 to 3 of sin(x) with respect to x.
What is positive?
100
This is the meaning of the quantity pi*.05^2*10 m^3 in the context where water flows out of a hose of radius 5cm at a rate of 5m/s.
What is the volume of water that came out of the hose after 2 seconds?
200
The hypothesis of the Fundamental Theorem of Calculus
What is "Suppose f is a continuous function on the interval [a,b] and F'=f"
200
He might have said "Sure.. I'll give you 100 francs for that theorem."
Who is L'hopital?
200
A ladder leaning against a wall begins to slide down the wall. We can model this situation by using the pythagorean theorem x^2+y^2=z^2 in which this letter represents a constant.
What is z?
200
The velocity of a car that is traveling forwards but slowing down.
What is positive?
200
Suppose C(x) is the cost per sweater when x sweaters are sold in a year. Then this is the meaning of 1/400 times the integral from 200 to 600 of C(x) with respect to x.
What is the average cost of a sweater over the period when the number of sweaters sold grew from 200 per year to 600 per year?
300
An underestimate of the definite integral of an increasing function
What is a left hand Riemann Sum?
300
We remember this German mathematician as a sumbody in calculus, and an entire area of geometry is named after him.
Who is Riemann?
300
Two people run away from the same spot along two different straight lines. The distance between them can be related to the distances they have run by the law of cosines: a^2+b^2-2abcos(x)=c^2 where this letter represents a constant.
What is x?
300
A car's acceleration when it is in the midst of switching directions.
What is zero?
300
Suppose F(x) is the amount of force (in newtons) being exerted on an object when the object is x meters from its starting point. This is the meaning of the integral from 0 to 3 of F(x) with respect to x.
What is the work done to move the object 3 meters from its starting point?
400
This is the name AND statement of the basic idea necessary to prove L'hopital's rule.
What is Local Linearization? In other words, given a differentiable function f and a number a in the domain of f, if x is near a, then f(x) is approximately f'(a)(x-a)+f(a).
400
This blind mathematician who had eight children and wrote volumes of mathematics is responsible for a method for solving differential equations that you may have used on your project. Also, you might mispronounce his name, which has more vowels than consonants.
Who is Euler?
400
The relationship between current (I), voltage (V) and resistance (R) is given by I=V/R. Suppose 2 amps of current flows through a resistor. At the moment when the resistance is 5 ohms and is changing at 2 ohms/minute, what is the rate of change of voltage?
0=R(t)V'(t)-V(t)R'(t), R(t)=5, V(t)=10, R'(t)=2 So 5V'(t)=10(2). What is 4 volts/min?
400
Suppose P is a differentiable function about which we know the following: P is decreasing on the interval [0,5]. This is the sign of the integral of P' over the interval [0,4].
What is negative?
400
Suppose R(t)=10e^-3t is the amount of radiation being emitted per hour after t hours. The integral from 0 to 10 of R(t) with respect to t measures this.
What is the total radiation emitted over a ten hour period?
500
Suppose f(x) = sin(pi/2 x) on the domain [0, 2]. This definite integral is the area between the horizontal line y=1/2 and the graph of f.
What is the integral from 1/3 to 5/3 of sin(pi/2 x) - 1/2 with respect to x?
500
This mathematician made clear the idea of different types of infinity, proving that the rational numbers are countable. Although his discoveries helped put the ideas of calculus on firm logical ground, he eventually went crazy.
Who is Cantor?
500
The relationship between current (I), voltage (V) and resistance (R) is given by I=V/R. Suppose we have a 3 ohm resistor. At the moment when current is changing at 2 amps per minute, how fast is the voltage changing?
What is 6 volts per minute?
500
Suppose f(x)=-(x^2+1)e^x The slope of a line tangent to the antiderivative of f.
What is negative?
500
Suppose a(t) is the acceleration (in m/s^2) of a car t seconds after the car starts. This is the meaning of x where the integral from 0 to x of a(t) with respect to t is 10.
What is the time at which the velocity is 10m/s?