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Applied Derivatives, Integration, Derivatives, Functions & Graphs,…
Applied Derivatives
Related Rates
Rate of change within lines, shapes, objects, distance etc.
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Extrema
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Found when f'(x) = 0, or undefined
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Optimization
Can be used to minimize area, time, material used, etc.
know the variable you are going to minimize and find an equation for it with the other given variables
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Can be used to maximize area, time, material used, etc.
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Rules & Methods
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Linear Approximation
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L(x) = f(a) + f'(a)(x-a) 
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Integration
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Substitution
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Example:
du is the derivative of u
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Sometimes we have to use our u equation to substitute something that wasn't included with the derivative of du "sub sub method"
Example
Take antiderivative here
Definite Integral Steps:
- Take the derivative of du and see if it matches with the unsubstituted part
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- Determine the correct U to substitute [try choosing one where the derivative will match the unsubstituted part.]
- Once you have your U & Du in, take the anti-derivative. if you alternated the equation do forget what you altered and include it with du & u. For example, the 1/2 shown above
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- evaluate the end points with u(x) and change them
- Use FTC to evaluate equation. Don't return U to its function. Keep it.
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Exponents & Logarithms
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When evaluating definite integrals, they follow the same steps with substitution.
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Sigma notation
i formulas
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n^2=
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approximate area/Reimann sums
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Net Change
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Total distance
Steps:
separate those the negative velocity and positive
using integral properties and their intervals. (if any negative velocity)
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Derivatives
rules
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Derivatives can be undefined as well (vertical tangent, no limit, etc.)
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Original Function
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Lines
TANGENT LINE
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Exponents & Logarithms
Formulas/Applied rules:
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E formulas
where
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Example:
>>> Remember that ln(b) or log(b) is constant therefore use the constant rule if necessary
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Functions & Graphs
Graphs
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To find range of a function, solve for the x alone and see what restricts y
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Rules
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Used to find zero's if factoring poly's don't work
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Functions
To find Y-intercept, make x equal to 0 then solve
To find x-intercept(s)(zero's),
make y equal to 0 then solve
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Absolute Value takes the equation and multiplies it by a negative anywhere where y = negative number. Usually within an interval.
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Limits
Rules
The limit is as x approaches a certain number but never touches it (super close), how does the graph react towards?
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Notation :
One way to find the limit is to sub in the limit adjusted slightly in f(x)... like 0.00001 if limit is 0
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Direct Substitution
when q(a) =! 0
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Lines
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limit is still 3 here and not 4 because its approaching 3
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