Things to Remember for MTH 2003, 2205, and 2207

(Last Updated 04/27/2009)

Limit Definition of the Derivative

OR
/ Example

Differentiation Rule / Example
Simple/General Power Rule

/
Product Rule

OR
/
Differentiation Rule / Example
Quotient Rule

OR
/
Chain Rule

OR

/
Exponential Differentiation

OR

/
Logarithmic Differentiation

OR

/
Logarithmic Rewrites

/

Use the …

Function / Purpose
Original Function
f(x) /
  • Find any y-coordinate on the graph
  • Find any additional points for graphing
  • Average Rate of Change
    on the interval
  • Actual Change
    on the interval

First Derivative
f'(x) /
  • Slope of the Tangent Line
    {Substitute any x-value into the first derivative}
  • Instantaneous Rate of Change
    {Find the derivative and substitute the given value into it}
  • Critical Numbers
    {Set the first derivative equal to zero and solve for x}
  • Intervals of Increasing/Decreasing
    {Use the critical numbers}
  • Relative Extrema
    {Based on the direction of the Increasing and Decreasing}
  • Marginal Equations
  • Differential Equations


Second Derivative
f''(x) /
  • X-coordinates for the possible points of Inflection
    {Use the original function to get the y-coordinates.}
  • Points of Diminishing Return (a.k.a inflection points)
  • Intervals of Concavity
    {Use the x-coordinates of the possible Inflection Points.}
  • Relative Extrema
    {Substitute the critical numbers from the first derivative into the second derivative. The concavity will tell you which extrema you have, if any.}

Epsilon (DO NOT GET THIS MIXED UP WITH ELASTICITY)
OR
Elasticity of Demand
/ Elasticity Conclusions
  1. then Demand is Elastic
  2. then Demand is Inelastic
  3. then Demand is Unitary

Compounding
Continuously / Non-Continuously
/ =Amount at the end of the period of time
=Amount at the start
=percentage rate (change to decimal)
=Amount of time in years / / =Amount at the end of the period of time
=Amount at the start
=percentage rate (change to decimal)
=number of times in 1 year that the amount is compounded
=Amount of time in years
Used to determine…
Amount of a lump sum investment compounded continuously over a period of time. / Used to determine…
Amount of a lump sum investment compounded a specific amount of times in 1 year, over a period of time.
E.g.Annuallyn = 1
Semi-annuallyn = 2
Quarterlyn = 4
Monthlyn = 12
Bi-Monthlyn = 24
Weeklyn = 52
Dailyn = 365
Continuous Money Flow (Net Present Value)
/ =Present Value
=Amount that flows uniformly
=percentage rate (change to decimal)
=number of years

Finding Asymptotes

Vertical Asymptotes
Vertical asymptotes are found in the denominator of a rational function. Simplify the rational function by factoring then cancelling. Set whatever is left in the denominator equal to zero and solve.
Anything that remains in the denominator after cancelling is a vertical asymptote and is also a non-removable discontinuity.
Anything that cancels in the denominator is a hole in the graph and is also a removable discontinuity. / Example 1

Example 2

The cancels in the equation and becomes a hole in the graph at.It is still excluded from the functions domain.

Finding Asymptotes (continued)

Horizontal Asymptote (Three Conditions)
  1. When the numerator’s highest exponent is larger than the denominator’s highest exponent, there is no horizontal asymptote.
    OR
    TOP BIGGER => NONE
/
2.When the denominator’s highest exponent is larger then the numerator’s highest exponent, the horizontal asymptote is y=0.
OR
BOTTOM BIGGER => ZERO
/
3.When the highest exponents in the numerator and the denominator are equal, the horizontal asymptote is the ratio of the leading coefficients.
OR
SAME => FRACTION of Leading Coefficients
/
Note:The horizontal asymptote can also be found by finding either the ORof the function.

Approximate Area under a Curve using Rectangles

Left-End Point
/
Right-End Point

Mid-Point

Riemann Sum Formulas
Formula / Example
Integration
Formula / Example
Let be a constant.
/
Let be an exponent .

/

/
Note: If the power is something other than x, u-substitution will be used.
u-Substitution for Integration
Use u-substitution when the function you are trying to integrate does not look like any of the forms listed on
page 6.
Another way to determine if you should use u-substitution is to look at the function as if you were taking the derivative. If you have to use the product rule, quotient rule, chain rule, or an exponential e then the integral is a good candidate for u-substitution.
* Remember that whatever you select as your u values, the derivative of that umust also exist in the integral *
Example 1 / Example 2
/ {This look like the chain rule} / / {This looks like the quotient rule}
Example 3 / Example 4
/ {This look like the exponential rule} / / {This looks like a product rule}
Average Value
Consumer and Producer Surplus
is the point of equilibrium. Set the Demand and Supply equations equal to each other and solve for. Plug this value into either the Demand or Supply equation to solve for.
Consumer Surplus =
OR
/ Producer Surplus =
OR

Area Between 2 Curves
Find the area bound between
If between then
/ If between then

File Name: Math 2003 2205 2207 Things to Remember / Page 1 of 8