Calculus
Derivative Tables
Integration Tables
Riemann Sums
Given
Midpoint Riemann Sum
We can also do away with the index notation and simplify things.
Trapezoidal Riemann Sum
Simpson's Rule
Infinite Sequences
Infinite Series
Infinite Series
Note that the limit of every convergent series is equal to zero. But the inverse isn't always true. If the limit is equal to zero, it may not be convergent.
For example, does diverge; but it's limit is equal to zero.
If the limit is equal to zero; the test is inconclusive.
Geometric Series
Given
Alternatively
Tests
Furthermore
The Integral Test
Constraints on
- Continuous
- Positive
- Decreasing （i.e. use derivative test）
P-Series -or- Harmonic Series
Note: the Harmonic series is the special case where
Comparison Test
Limit Comparison Test
Warning
- If , this only means that the limit comparison test can be used. You still need to determine if either or converges or diverges.
- Therefore, this does not apply to any arbitrary rational function.
Notes
- For many series, we find a suitable comparison, , by keeping only the highest powers in the numerator and denominator of .
Estimating Infinite Series
Differential Equations
Separable Differential Equations
Growth and Decay Models
The above states that all solutions for are of the form .
Where
Exponential growth occurs when , and exponential decay occurs when .
The Law of Natural Growth:
The Logistic Model of Population Growth:
Where
Solving the Logistic Equation
Via partial fraction decomposition
Rewriting the differential equation
Second Order Homogeneous Linear Differential Equations with Constant Coefficients
Properties
If and are solutions; then is also a solution. Therefore, the most general solution to some second order homogeneous linear differential equations with constant coefficients would be .
This is why, when you find two solutions to the characteristic equation and respectively, we write it like so.
Given some:
We can presume that is of the form , and therefore:
Substituting this back into the original equation, we have:
Where:
So therefore:
Where the general solution is of the form:
Parametric Equations
First Derivative Formula
To find the derivative of a given function defined parametrically by the equations and .
Second Derivative Formula
To find the second derivative of a given function defined parametrically by the equations and .
Given
Therefore
The above shows different ways of representing . （I.e. it doesn't correspond to some final solution.）
Arc Length
Formula for the arc length of a parametric curve over the interval .