Differential Calculus Flashcards
Who are the people who devised the differential calculus?
Isaac Newton and G.W Leibniz
concerned with the problem of finding the rate of change of a function
Differential Calculus
_____ is the value the function gets arbitrarily close to as the function approaches an arbitrary value, a.
limits of function
f(x) negative side of zero
-0.00001
f(x) positive side of zero
0.00001
value of limit as x approaches negative infinity
-99999
value of limit as x approaches positive infinity
99999
lines that continually approach a given curve but does not meet at any finite distance
asymptote
What is the condition in getting the vertical asymptote?
equate denominator to zero to get the x eqn
What are the condition in getting the horizontal asymptote
If:
m=n ; y= an/bm
m<n ; no horizontal asymptote
m>n ; y=0
What is the condition in getting the inclined asymptote
when m<n , or when there is no horizontal asympote
How to get the eqn of the inclined asymptote
using limit approach; use eqn y=mx+c
where:
m= lim as x approaches positive or negative infinity= f(x)/x
c= lim as x approaches positive or negative infinity= f(x)-mx
then substitute m and c to y=mx+c
d/dx (u^n)
n . u^n-1
d/dx (uv)
vdu+udv
d/dx (u/v)
(vdu-udv)/v^2
d/dx (e^u)
e^u. du
d/dx (ln u)
du/u
d/dx (sin X)
cos x
d/dx (cos x)
-sin x
d/dx (tan x)
sec^2 x
d/dx (cot x)
csc^2 x
d/dx (sec x)
sec x tan x
d/dx (csc x)
-csc x cot x
how to solve implicit differentiation
differentiate x and y but add y’ to y, then isolate y’ to left side to get the eqn
implicit differentiation caltech: format / formula
B-A / 1X10-5
implicit differentiation caltech steps
- input the eqn in the calculator, do not forget to put [,Y], then let x= 0.1
- Store value to A
- input the eqn in the calculator, do not forget to put [,Y], then let x= 0.1+1x10-5
- Store answer to B
- Apply format/formula
- For checking: input the choices in the calcu and let y be the stored value Y
Higher derivatives caltech
use d/dx function on calcu, let x= -1 and -1+1x10-5
apply to format/formula
B-A / 1X10-5
Motion problems: What is the relationship between s, v , and a
v= ds/dt
a= dv/dt
Tangent Line eqn
- use implicit differentiation to make gen eqn to std eqntp get m
- use shortcut in creating the eqn (see notebook for full explanation)
Normal Line eqn
- Do the steps in getting the tangent line eqn first
- Apply the shortcut for normal line eqn (see notebook for full explanation)
Slope of the curve
- use implicit differentiation, then substitute P(x,y)
Critical points of the curve
- Take the first derivative of the eqn
- get x1 and x2 using the quadratic eqn
- substitute x1 and x2 to the original eqn to get y1 and y2
Inflection point
- take the second derivative of the eqn
- get x1 and x2 using the quadratic eqn
- substitute x1 and x2 to the original eqn to get y1 and y2
what is the order of differentiation in getting the slope of the curve?
1st order
what is the order of differentiation in getting the inflection point
2nd order
what is the order of differentiation in getting the critical point
1st order
