Rule
Limit Definition of the Derivative
[pic]
OR
[pic] |Example
[pic]
| |
|Differentiation Rule |Example |
|Simple/General Power Rule | |
|[pic] |[pic] |
|Product Rule | |
|[pic] |[pic] |
|OR | |
|[pic] | |
|Differentiation Rule |Example |
|Quotient Rule | |
|[pic] |[pic] |
|OR | |
|[pic] | |
|Chain Rule | |
|[pic] |[pic] |
|OR | |
|[pic] | |
|Exponential Differentiation | |
|[pic] |[pic] |
|OR | |
|[pic] | |
|Logarithmic Differentiation | |
|[pic] |[pic] |
|OR | |
|[pic] | |
|Logarithmic Rewrites | |
|[pic] |[pic] |
Use the …
|Function |Purpose |
|Original Function |Find any y-coordinate on the graph |
|f(x) |Find any additional points for graphing |
| |Average Rate of Change |
| |[pic] on the interval [pic] |
| |Actual Change |
| |[pic] on the interval [pic] |
|First Derivative |Slope of the Tangent Line |
|f'(x) |{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 |
| |[pic] |
|Second Derivative |X-coordinates for the possible points of Inflection |
|f''(x) |{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) |
|[pic] OR [pic] |
|Elasticity of Demand |
|[pic] |Elasticity Conclusions |
| |[pic]then Demand is Elastic |
| |[pic]then Demand is Inelastic |
| |[pic]then Demand is Unitary |
|Compounding |
|Continuously |Non-Continuously |
|[pic] |[pic]= Amount at the end of the period of time |[pic] |[pic]= Amount at the end of the period of time |
| |[pic]= Amount at the start | |[pic]= Amount at the start |
| |[pic]= percentage rate (change to decimal) | |[pic]= percentage rate (change to decimal) |
| |[pic]= Amount of time in years | |[pic]= number of times in 1 year that the amount is |
| | | |compounded |
| | | |[pic]= Amount of time in years |
| | |
|Used to determine… |Used to determine… |
|Amount of a lump sum investment compounded continuously over a period of |Amount of a lump sum investment compounded a specific amount of times in 1 year, |
|time. |over a period of time. |
| |E.g. Annually n = 1 |
| |Semi-annually n = 2 |
| |Quarterly n = 4 |
| |Monthly n = 12 |
| |Bi-Monthly n = 24 |
| |Weekly n = 52 |
| |Daily n = 365 |
|Continuous Money Flow (Net Present Value) |
|[pic] |[pic]= Present Value |
| |[pic]= Amount that flows uniformly |
| |[pic]= percentage rate (change to decimal) |
| |[pic]= number of years |
Finding Asymptotes
|Vertical Asymptotes |Example 1 |
|Vertical asymptotes are found in the denominator of a rational function. |[pic] |
|Simplify the rational function by factoring then cancelling. Set whatever is| |
|left in the denominator equal to zero and solve. |Example 2 |
| |[pic] |
|Anything that remains in the denominator after cancelling is a vertical | |
|asymptote and is also a non-removable discontinuity. |The [pic] cancels in the equation and becomes a hole in the graph at[pic]. It is |
| |still excluded from the functions domain. |
|Anything that cancels in the denominator is a hole in the graph and is also a| |
|removable discontinuity. | |
Finding Asymptotes (continued)
|Horizontal Asymptote (Three Conditions) | |
|When the numerator’s highest exponent is larger than the denominator’s |[pic] |
|highest exponent, there is no horizontal asymptote. | |
| | |
|OR | |
| | |
|TOP BIGGER => NONE | |
|2. When the denominator’s highest exponent is larger then the numerator’s |[pic] |
|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,|[pic] |
|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 [pic]OR[pic]of the function. |
Approximate Area under a Curve using Rectangles
|Left-End Point |[pic] |
|[pic] | |
|Right-End Point | |
|[pic] | |
|Mid-Point | |
|[pic] | |
|Riemann Sum Formulas |
|Formula |Example |
|[pic] |[pic] |
|[pic] |[pic] |
|[pic] |[pic] |
|[pic] |[pic] |
|Integration |
|Formula |Example |
|Let [pic]be a constant. | |
|[pic] |[pic] |
|Let [pic]be an exponent [pic]. | |
|[pic] |[pic] |
| | |
|[pic] |[pic] |
|[pic] |[pic] |
|[pic] |[pic] |
| | |
| |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 u must also exist in the integral * |
|Example 1 |Example 2 |
|[pic] |{This look like the chain rule} |[pic] |{This looks like the quotient rule} |
|[pic] | |[pic] | |
|[pic] | |[pic] | |
|[pic] | |[pic] | |
|[pic] | |[pic] | |
|[pic] | |[pic] | |
|[pic] | |[pic] | |
|[pic] | | | |
|[pic] | | | |
|Example 3 |Example 4 |
|[pic] |{This look like the exponential rule} |[pic] |{This looks like a product rule} |
|[pic][pic] | |[pic] | |
| | |[pic] | |
|[pic] | |[pic] | |
|[pic] | |[pic] | |
|[pic] | |[pic] | |
|[pic] | |[pic] |
|Average Value |
|[pic] | |
|Consumer and Producer Surplus |
|[pic] is the point of equilibrium. Set the Demand and Supply equations equal to each other and solve for[pic]. Plug this value into either the Demand or |
|Supply equation to solve for[pic]. |
|Consumer Surplus = [pic] |Producer Surplus = [pic] |
|OR |OR |
|[pic] |[pic] |
|Area Between 2 Curves |
|Find the area bound between [pic] |
|If [pic]between [pic]then |If [pic]between [pic]then |
|[pic] |[pic] |
................
................
In order to avoid copyright disputes, this page is only a partial summary.
To fulfill the demand for quickly locating and searching documents.
It is intelligent file search solution for home and business.