Inverse functions and their derivatives
e^x
What are ln(x), 1/x?
What is differentiating a relation between the quantities?
This "EVT" states that a continuous function on a closed interval attains a global maximum and global minimum.
What is the Extreme Value Theorem?
The extreme value theorem states that a function with this property defined on a closed interval attains a global maximum.
What is continuity?
While best known for a story about an apple, this physicist is regarded as the father of calculus.
Who is Isaac Newton?
x^2
What are sqrt(x), 1/(2sqrt(x))?
A rectangle with side lengths a(t), b(t) varying in time has change in area given by this expression.
What is a'(t)b(t) + a(t)b'(t)?
When optimizing a function over an interval, global extrema always occur at either critical points or one of these kinds of points.
What are endpoints?
Rolle's theorem states that a continuous function f on a closed interval [a,b] with f(a) = f(b) = 0 has a critical point, but only under this additional hypothesis.
What is differentiability on (a,b)?
Applications of Newton's calculus to physics were treated in the Philosophiæ Naturalis Principia Mathematica, considered one of the most important scientific publications in history. It was first published in this century.
What is the 17th century?
ln(x^2)
What are e^(sqrt(x)), e^(sqrt(x))/(2sqrt(x))?
A cylinder of constant volume V with radius r(t) varying in time has this expression for its change in surface area.
What is 4πrr' -2Vr'/r^2?
To optimize a function we need a bit more: this piece of information is crucial to determining existence of global extrema.
What is the domain?
This example shows that the hypothesis of 'closedness' cannot be removed from the domain when applying the Extreme Value Theorem.
What is f(x)=x on (0,1)? (Many answers possible!)
Regarded as an independent discoverer of calculus, this mathematician's name is attached to the notation df/dx for a derivative.
tan(x)
What are arctan(x), 1/(x^2+1)?
Two resistors of resistances R1, R2 are connected in parallel with total resistance R governed by Ohm's law, 1/R = 1/(R1) + 1/(R2). This expression gives R' in terms of R, R1, R2, R1', R2'.
What is R^2 (R1'/ (R1^2) + R2' / (R2^2))?
These are the locations and types of all local extrema of (x-2)^2 (x+2)^2 on the interval [-3, 3]
What are -3 (global maximum), -2 (global minimum), 0 (local maximum), 2 (global minimum), and 3 (global maximum)?
This example shows that failure of differentiability at even a single point in the domain may cause the conclusion of Rolle's theorem to fail.
What is |x|-1 on [-1,1]? (Many other answers possible!)
Daily Double
Who is Augustin-Louis Cauchy?
(x+1)/(x-1)
What are (x+1)/(x-1), -2/(x-1)^2?
Water is poured into a 10m tall cylinder of radius 1m at a rate of 1,000 L/min. The height of water in the cylinder increases at this rate.
(Note: 1 cubic metre of water = 1,000 litres of water)
What is 1/pi m/min?
The alpha curve y^2 = x^3 + x^2 attains its smallest x value at this point.
What is (-1,0)?
If f is differentiable on (-∞,∞) and f'(a) < K < f'(b), it is true that there exists c between a and b with f'(c) = K, but not because of this ostensibly applicable theorem.
What is the Intermediate Value Theorem?
Are continuous functions differentiable most of the time? So everyone thought, until this "father of modern analysis" K.W. proved otherwise.
Who is Karl Weierstrass?