Limits & Differentiation
Differentiate f(x) = e^(-3x)
-3e^(3x)
int x/(x^2 + 4) dx
(1/2)ln(x^2 + 4) + C
A farmer has 200 meters of fencing and wants to enclose a rectangular field. What shape gives the largest area?
What is a square?
Find the sum: sum from n = 1 to infinity
1/2^n
1
This mathematician is most often paired with Leibniz as a founder of calculus.
Who was Gottfried von Leibniz?
He is credited with the phrase: cogito ergo sum.
Who was Rene Descartes?
Find the limit:
lim_(x->0) sinx/x
1
int 1/sqrt(1 - x^2) dx
What is arcsinx?
A ball is thrown straight upward. At the highest point of its path, what is its instantaneous velocity?
0
A conditionally-convergent version of a p-series with a p-value of 1 is known as this.
What is the alternating harmonic series?
He is most associated with his famous sums.
Who is Bernhard Riemann?
This writer won the 2024 Nobel Prize in Literature.
Who is Han Kang?
Find the limit:
lim_(h->0)(ln(2+h)-ln(2))/h
1/2
int_1^e 1/(x(ln x)^2) dx
infinity
Sand is falling from an hourglass, making a conical pile. If the radius and height stay proportional, what quantity changes as the pile grows?
What is volume?
Find the interval of convergence of
sum_(n=1)^oo (x-1)^n/(n*5^n)
[-4,6)
This analytic wizard gave us a rigorous foundation for the epsilon-delate definition of a limit.
Who was Augustin Louis-Cauchy?
In the latter half of the 19th century, the port city of Incheon went by this name.
What is Jemulpo?
FREE SPACE!!
FREE SPACE!!
int_0^1 1/(1 + sqrt(x)) dx
2 - 2ln 2
A closed cylindrical can must hold a fixed volume. What relationship between the height and radius gives the least surface area?
What is h = 2r?
Lagrange's error bound assures us that the error in the remainder is always less than the value evaluated at this derivative.
What is the (n+1)th derivative?
His method of exhaustion was an early version of integration that we deployed in solving volume problems.
Who was Archimedes?
The author of "Blood Meridian" - the novel with, arguably, the most vile villain in all of literature.
Who was Cormac McCarthy?
Differentiate:
(ln x)^x
(ln x)^x(ln(ln x) + 1/ln x)
int_1^(e^4) 1/(x*sqrt(ln x)) dx
4
A spherical balloon is being inflated so that its volume increases at 12*pi cm^3/s. How fast is the radius increasing when the radius is 3 cm?
1/3 cm/s
Also known as the least upper bound...
What is the supremum?
Amazingly, he attempted to provide a series approach to defining the derivative without using limits, but he better known to us for his analysis of the remainder term in a Taylor series.
Who was Joseph-Louis Lagrange?
The modulus of 3 + 4i
5