Differential Calculus Flashcards
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))
Parallelepiped with maximum volume
cube
Open Square container with maximum volume
l = w = 2h
w = sqrt(SA/3)
Location of single stake at ground level to minimize length of wire
x = dh1/ (h1+h2)
x - smaller distance from point to shorter tower
d - distance between towers
Least amount of material to be used for a square base rectangular parallelepiped
l = w = 2h
w = (2Volume)^(1/3)
Least amount of material to be used for an open top cylindrical tank
r = h
Minimum cost for a given volume
r = (V/2π)^(1/3)
Ratio of the weight of heaviest cylinder, Wc, to the weight of the circumscribing sphere, Ws.
Wc/Ws = 1/sqrt(3)
Least amount of material for a given volume
r = h/sqrt(2)
Maximum volume of cone with a given slant height
h = s/sqrt(3) θ = arctan(sqrt(2))
Volume of largest cone, Vc that can be inscribed in a hemisphere
V = 1/2(Vhemisphere)
Largest cylinder that can be inscribed in a cone
height of cylinder = 1/3 height of cone
Maximum volume of right circular cylinder inscribed in a sphere of radius R
V = (4/sqrt(27))πR^3 h = (2sqrt(3)/3)R
Time rates
If average value is asked, do not differentiate
use:
(y2-y1)/(x2-x1) = Δy/Δx
If instantaneous:
differentiate @ t=something
- Form equations for each variable wrt time (if possible)
- Implicitly differentiate wrt time (resulting into δx/δt and δy/δt)
- Use the property of parametric equations and play with the equation to find for what is asked
