Section 1
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ln x^y
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Last updated
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Date created
Mar 1, 2020
Cards (53)
Section 1
(50 cards)
ln x^y
y* lnx
Infinite discontinuity
Y approaches (+ or -) infinity as x approaches c from left and right
Minimum distance
d^2 = (x2-x1)^2 + (y2-y1)^2; d^2 = Q; use derivative of Q; feasible domain; EVT
Derivative of e^x
e^x because e^h = 1 + h
ln x/y
ln x - ln y
dy/dx arccos f(x)
-f'(x)/ sqrt(1-f(x)^2)
percent change
change in y / f(x0) *100 [dy for estimate]
dy/dx arcsin f(x)
f'(x)/ sqrt(1-f(x)^2)
Limits at Infinity
As the limit approaches +/- infinity then k/bx^a=0 where a>0
Optimization
find primary and secondary equations; use derivative of the primary plug in secondary; feasible domain; EVT
Average Function value
integral from a to b of f(x) dx / b-a
Rolle's Theorem
Let f be continuous on [a,b] and differentiable on (a,b) and if f(a)=f(b) then there is at least one number c on (a,b) such that f'(c)=0 (If the slope of the secant is 0, the derivative must = 0 somewhere in the interval).
Integration
used to find the antiderivative (don't forget + C)
Inscribed Area
ln e
1
First Derivative Test
Used to determine where a function's graph has a min/max and is increasing or decreasing.
dy/dx arctan f(x)
f'(x) / f(x)^2 + 1
e^(lnx)
x
Power rule
d/dx[x^n]=nx^(n-1)
concavity
f is concave up when f' is increasing; f is concave down when f' is decreasing
dy/dx arcsec f(x)
f'(x)/ |f(x)| * sqrt( f(x)^2 -1 )
Rectangular Approximation Method
RRAM , LRAM, MRAM
dy/dx b^f(x)
lnb b^f(x)f'(x)
Definition of Derivative
lim h->0 f(x+h)-f(x)/h
Sum and Difference Rule
d/dx [f(x) +/- g(x)] = d/dx [f(x)] +/- d/dx [g(x)]
Mean Value Theorem
if f(x) is continuous and differentiable, slope of tangent line equals slope of secant line at least once in the interval (a, b) f '(c) = [f(b) - f(a)]/(b - a)
Intermediate Value Theorem (IVT)
If f is continuous on the closed interval [a,b] and k is any number between f(a) and f(b), then there is at least one number c in [a,b] such that f(c)=k
dy/dx arccot f(x)
-f'(x) / f(x)^2 + 1
Chain rule
d/dx f(g(x)) = f'(g(x)) g'(x)
dy/dx arccsc f(x)
-f'(x)/ |f(x)| * sqrt( f(x)^2 -1 )
Constant Rule
d/dx (c) = 0
ln 1
0
Second Derivative Test
Used to determine on what intervals a function is concave up/concave down and the points of inflections.
Quotient Rule
low d high - high d low / low^2
Product rule
1st d 2nd + 2nd d 1st
ln xy
ln(x)+ln(y)
Points of Inflection
If f has a tangent line at ( c, f(c) ) then c is a point of inflection if concavity of f is changing from up to down (vice versa)
Extreme Value Theorem
If f is continuous on [a,b] then f has an absolute maximum and an absolute minimum on [a,b]. The global extrema occur at critical points in the interval or at endpoints of the interval.
ln e^x
x
To find total distance, integrate the ____________ function.
velocity
Secant Line
a line that connects or passes through 2 points of a function's curve
Constant Multiple Rule
d/dx[cf(x)]=cf'(x)
MVT for Integrals
integral from a to b of f(x) dx / b-a = f(c)
f is continuous at a number c if
1. f(c) is defined 2. the limit as x approaches c exist 3. the limit as x approaches c = f(c)
Removable discontinuity
not continuous when x=c; a "hole"
dy/dx logb f(x)
f'(x)/ lnb * f(x)
Area by definite integrals
Relative Change
change in y / f(x0) [dy for estimate]
Newton's Method
x2=x1-f(x1)/f'(x1)
Non-removable discontinuity
y-values approach different values at x=c; a "jump"
Section 2
(3 cards)