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How do you do a first derivative test (find the behavior of a function analytically without plotting it)?
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Definition
1.) Take the Derivative of the Function
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2.) Set the Derivative Function to Equal Zero and solve it for "x"
2a.) Factor it further to where you have multiple values of "x"
or
2b.) Use the Quadratic Formula to Obtain Multiple Values For "x"
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3.) Use the multiple values of "x" you have found, and set a test value between -infinity and your first value, your first value and your second value, and your second value and positive infinity. Represent this information as a chart.
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4.) Substitute your test values into the derivative in place of "x" and mark whether the solution to each value is positive or negative and whether the function is increasing or decreasing on your chart
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5.) Your x-values are the critical points, mark whether they are maximums or minimums depending on whether or not the slope right before the function is positive or negative
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Definition
A'(x)B(x)+B'(x)A(x)
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(A(x)/B(x))'=
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Definition
A'(x)B(x)-B'(x)A(x)/B(x)^2
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What is Quadratic Formula?
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Definition
x= (-b(+or-)(sqrt(b^2-4ac)))/2a
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What is the Integral Notation Used for a number not included?
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Definition
An open circle on a numberline
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Parenthesis when writing the point
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Parenthesis on a number line
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What is the Integral Notation Used for a number is included?
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Definition
A closed circle on a numberline
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Bracket when writing the point
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Bracket on a number line
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What are higher derivatives?
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Definition
When you keep taking the derivative of a function.
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A normal derivative is
f(x)'
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The Derivative of a derivative is
f(x)''
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The Derivative of a Derivative's Derivative is
f(x)'''
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When given the "x" value for a function, how can you solve for the "y" value of the function?
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Definition
Plug the "x" value into the function and solve for the "y" value of the function
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Local Maximum means the same thing as what?
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Definition
Concavity Downwards
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Local Minimum means the same thing as what?
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Definition
Concavity Upwards
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The second derivative of distance =
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Definition
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The first derivative of distance =
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Definition
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How do you do a second derivative test (find the behavior of a function analytically without plotting it)?
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Definition
1.) Take the first derivative of the function
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2.) Set the Derivative Equal to Zero and factor it
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3.)Set each factor equal to zero, and solve for critical points
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4.)Take the Second Derivative of the Original Function
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5.) Plug each critical point into the second derivative in place of "x", and record whether each one yields a positive solution or a negative solution. If a solution is negative, then that critical point is a maximum. If a solution is positive, then that critical point is a minimum.
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(x^2+A^2)
Can be factored into....
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Definition
Two sets of conjugates:
(x+A)(x-A)
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How do you find the point of infliction (aka the point of diminishing returns)?
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Definition
1.)f(x)''
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2.)Set f(x)''=0
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3.)Solve f(x)''=0 for "x"
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If the second derivative test yields only a zero, and zero is not positive or negative, then...
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Definition
The second derivative test has failed
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