Differential Calculus Flashcards
(37 cards)
Calculus was derived from the Latin Word “Calx” and Greek Word “Chalis” meaning
Calx - Stone
Chalis - Limestone
Mindblown
Founders of Calculus
Gottfried Wilhelm von Leibniz
Isaac Newton
Theorem of Limits
Cal Tech:
instead of using the exact values, use a value approximately equal to the limit
Input Equation
CALC
limit +/- 1x10^-5
Continuity
Cal Tech:
instead of using the exact values, use a value approximately equal to the limit, (one at higher end and one at lower end) and compare if equal
Input Equation
CALC
limit + 1x10^-5
compare to:
Input Equation
CALC
limit - 1x10^-5
if equal then it is continuous
Explicit Derivatives
Cal Tech
L’Hospitals Rule
If Limit of f(x)/g(x) is Indeterminate
then try f’(x)/g’(x)
Maxima/Minima
Cal Tech
Mode Table
Form an equation describing the problem then input limits and increments in the Table Mode to determine the maximum or minimum values from the table’s output column
Change in Concavity
Inflection Point
Largest Rectangle inscribed in a circle
A square with diagonal equal to the diameter
Largest Rectangle inscribed in a semicircle
A rectangle with length equal to the twice the width
Largest Rectangle inscribed in a isosceles / right triangle with corner at the 90 degree corner of the rght triangle
A rectangle with length equal to half the base and width equal to half the height
Largest Rectangle inscribed in an ellipse
A rectangle with base and height equal to sqrt(2) of the semi-major and semi-minor axis of the ellipse respectively
Largest area of triangle w/ given perimeter
Equilateral Triangle
Minimum perimeter of sector given area
r = sqrt(A)
θ=2rad
Rectangle with given area with minimum perimeter
square
Rectangle with given area and minimum perimeter to be fenced along 3 sides only
x=2y
Right triangle with maximum perimeter or area
45-45-90 triangle
Stiffest beam that can be cut from a circular section of radius r
y = x sqrt(3)
Strongest beam that can be cut from an elliptical section
x = 2b sqrt(1/3) y = 2a sqrt(2/3)
Largest rectangle that can be inscribed in a given ellipse
A ellipse / A rectangle
π / 2
Most efficient Trapezoidal Section
smaller base = the two non parallel sides
larger base = twice the smaller base
Length of rigid beam that can pass a perpendicular hallway
L = (a^(2/3) + b^(2/3))^(2/3)
Minimum length of ladder/rod to be extended from ground to a wall with an intervening fence
L = (a^(2/3) + b^(2/3))^(2/3)
Best possible view of a picture or a clock
distance = sqrt((height from bottom)(height from top))
