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Basic Derivative (Power Rule)
f(x^n)= nX^(n-1)
Derivative of sin(x)
= cosx
Derivative of cos(x)
= -sinx
Derivative of tan(x)
= sec^2(x)
Derivative of cot(x)
= -csc^2(x)
Derivative of sec(x)
= secxtanx
Derivative of csc(x)
= -cscxcotx
Derivative of ln(u)
= 1/u u'
Derivative of e^x
= e^x
Chain Rule
d/dx f(g(x)) = f'(g(x)) g'(x)
product rule
f'(x)g(x)+f(x)g'(x)
Quotient Rule
g(x)f'(x)-f(x)g'(x)/g(x)^2
Intermediate Value Theorem
If f is continuous on [a,b] and y is a number between f(a) and f(b), then there exists at least one number x=c in the open interval (a,b) such that f(c)=y
Mean Value Theorem
If f(x) is continuous on [a,b] AND the first derivative exists on the interval (a, b), then there is at least one number x=c in (a,b) such that
f '(c) = [f(b) - f(a)]/(b - a)
average value of a function
1/b-a integral from a to b f(x)dx
Extreme Value Theorem
If f is continuous on [a,b] then f has an absolute maximum and an absolute minimum on [a,b].
Derivative of an Inverse Function
g'(x) = 1/ f'(g(x)) where g(x) is the inverse of f(x)
Average Rate of change
f(b)-f(a)/b-a
Critical Point
dy/dx=0 or undefined
Local Minimum
dy/dx goes (-,+) or second derivative > 0
Local Maximum
dy/dx goes (+,-) or second derivative < 0
Point of Inflection
second derivative goes from (+,-) or (-,+)
f'(x) < 0
f(x) is decreasing
f'(x) > 0
f(x) is increasing
f'(x) = 0 or DNE
Critical Values at x
Relative Maximum
f'(x) = 0 or DNE AND sign of f'(x) changes from + to -
Relative Minimum
f'(x) = 0 or DNE AND sign of f'(x) changes from - to +
f''(x) > 0
concave up
f''(x) < 0
concave down
f'(x) = 0 and sign change of f''(x) changes then there is a
point of inflection at x
Horizontal Asymptotes
If the largest exponent in the numerator is < largest exponent in the numerator in the denominator then the lim as x approaches positive and negative infinity of f(x)=0
If the largest exponent in the numerator is > largest exponent in the numerator in the denominator then the lim as x approaches positive and negative infinity of f(x)=DNE
If the largest exponent in the numerator is = largest exponent in the numerator in the denominator then the lim as x approaches positive and negative infinity of f(x)=a/b
x(t) - physics problem
= position function
v(t) - physics problem
= velocity function
a(t) - physics problem
= acceleration function
The derivative of position is
velocity
The derivative of velocity is
acceleration
The integral of acceleration is
velocity
The integral of velocity is
position
Speed is
absolute value of velocity
If acceleration and velocity have the SAME SIGN, then
the speed is INCREASING, particle is moving right
If acceleration and velocity have DIFFERENT SIGNS, then
the speed is DECREASING, particle is moving left
Displacement
∫ v(t)
Distance
∫ absolute value of position
Average Velocity
= final position - initial position / total time
ln e
= 1
ln 1
= 0
ln(MN)
= lnM + lnN
ln(M/N)
= lnM - lnN
ln(M)^P
= P ln(M)
The Fundamental Theorem of Calculus
from a to b ∫ f(x) dx = F(b) - F(a)
Corollary to FTC
from a to g(u) ∫ f(t) dt = f(g(u)) du/dx
sin(0)
= 0
sin(π/6)
= 1/2
sin(π/4)
= √(2)/2
sin(π/3)
= √(3)/2
sin(π/2)
= 1
sin(π)
= 0
cos(0)
= 1
cos(π/6)
= √(3)/2
cos(π/4)
= √(2)/2
cos(π/3)
= 1/2
cos(π/2)
= 0
cos(π)
= -1
tan(0)
= 0
tan(π/6)
= √(3)/3
tan(π/4)
= 1
tan(π/3)
= √(3)
tan(π/2)
undefined
tan(π)
= 0
Riemann Sum
Approximating Area under the curve using rectangles
sin^2x+cos^2x
= 1
tan(x)
= sinx/cosx
cot(x)
= cosx/sinx
csc(x)
= 1/sinx
sec(x)
= 1/cosx
∫ u^n du
= (u^(n+1))/(n+1) +C
∫ 1/u du
= ln |u| + C
∫ e^u du
= e^u + C
∫ sinu du
= -cosu + C
∫ cosu du
= sinu + C
∫ tanu du
= -ln |cosu + C|
∫ cotu du
= ln |sinu| + C
∫ secu du
= ln |secu + tanu| + C
∫ cscu du
= -ln |cscu + cotu| + C
∫ sec^2 du
= tanu + C
∫ csc^2 dy
= -cotu + C
∫ secu tanu du
= secu + C
∫ cscu cotu du
= -cscu + C
Derivative of inverse sin
Derivative of inverse tan
Derivative of inverse of sec
Integral of du/sqrt(a^2-u^2)
sin^-1(u/a) + C
integral of du/a^2+u^2
(1/a)arctan(u/a) + C
Integral of du/u √u^2-a^2
(1/a)arcsec(|u|/a) + C
Note 
Flashcard (80)