Studied by 9 peopleDefinition of derivatives
f’( c ) = lim x→ c of ( f ( c ) - f (c ) ) / ( x-c )
Intermediate value theorem
If the function f(x) is continuous on[ a,b ] and y is a number between f(a) and f(b) then their exist at at least one number x = C in the interval, such that f( c ) = y
Mean value theorem
If the function 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)
Rolle’s theorem
If the function f(x) is continuous on [a,b], and the first derivative exists on the interval, (a,b) and f(b) = f( a), then there is at least one number x =c in (a,b) such that f’( c) = O
Extreme value theorem
If the function f(x) is continuous on [ a , b ], then the function is guaranteed to have an absolute maximum and an absolute minimum on the interval.
Derivative of X^n
nx^(n-1)
Derivative of sinx
cosx
Derivative of cosx
-sinx
Derivative of tanx
sec²x
Derivative of cotx
-csc²x
Derivative of secx
secx*tanx
Derivative of cscx
-cscx*cotx
Derivative of ln(u)
1/u * du/dx
Derivative of e^u
e^u du/dx
Chain rule
d/dx [f(u)] = f’(u) * du/dx
Product rule
d/dx(uv) = u*dv/dx + v*du/dx
Quotient rule
d/dx (u/v) = (v du/dx - u dv/dx)/ v²
f’(x) > 0
function is increasing
f’(x) < 0
function is decreasing
f’(x) = 0 or DNF
Critical value
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
function is concave up
f’’(x)<0
function is concave down
f’’(x) = 0 and sign of f’’(x) changes
point of inflection
f’’(x)<0
relative maximum
f’’(x) > 0
relative minimum
Average rate of change
[f(b) - f(a)]/[b-a]
Instantaneous rate of change
f’(x)
Equation of a tangent line at a point
y2 -y1 = m(x2 - x1)
critical point
dy/dx = 0 or undefinded
If the largest exponent in the numerator is < largest exponent in the denominator then
lim [ x→ ±infinity] f(x) = 0.
Horizontal Asymptotes
If the largest exponent in the numerator is > the largest exponent in the denominator then
lim [x→ ±infinity] f(x) = DNE
Horizontal Asymptotes
If the largest exponent in the numerator is = to the largest exponent in the denominator then the quotient of the leading coefficients is the asymptote.
lim [x→±infinity] f(x) = a/b
Horizontal Asymptotes
ln e =
1
ln 1 =
0
ln(MN) =
lnM + lnN
ln(M/N) =
lnM - lnN
p* lnM =
lnM^p
s(t)
position function
v(t)
velocity function
a(t)
acceleration function
Derivative of position (ft)
velocity (ft/sec)
Derivative of velocity (ft/sec)
acceleration (ft/sec2).
Integral of acceleration (ft/sec2)
velocity (ft/sec)
Integral of velocity (ft/sec)
position (ft).
Speed =
| velocity |
If acceleration and velocity have the same signs
The speed is increasing
If the acceleration and velocity have different signs
The speed is increasing
The particle is moving right
velocity is positive
The particle is moving left
velocity is negative
Displacement =
integral from t0 to t1 v(t) dt
Total Distance =
integral from initial time to final time |v(t)| dt
Average Velocity =
(final position - initial position)/total time
Accumulation =
x(0)+ integral from t0 to t1 v(t)dt
Exponential growth and decay
y = Ce^kt
“y is a differentiable function of t such that y > 0 and y’ = ky “ think
y = Ce^kt
“the rate of change of y is proportional to y”
y = Ce^kt
steps to solve differential equations
1. Separate variables first
2. Integrate
3. Add +C to one side
4. Use initial conditions to find “C”
5. Write the equation if the form of =f(x)
Fundamental theorem of calculus
integral from a to b f(x) = F(b) - F(a), Where F’(x) = f(x)
d/dx integral g(u) to a of f(t)dt =
f(g(u))du/dx
Mean Value Theorem for Integrals: The Average Value
If the function f(x) is continuous on [ a , b ] and the first derivative exists on the interval ( a , b ), then there exists a number = on (a , b ) such that f average = 1/(b-a) integral a to b f(x) dx
Riemann Sums
A rectangular approximation: add up the areas of the rectangles.
Trapezoidal rule
A = ½ h [b1 + b2]
sin²x + cos²x =
1
Note
Flashcard