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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"
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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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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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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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How do you sketch the curve of a function?
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Definition
1.) Find the x and y intercepts of the function
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2.) Find the Behavior of the function Using Either the First or Second Derivative Test: which will tell you of a curve's local maximum and local minimum
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3.) Use the information you have gathered from the first two steps, to sketch the curve
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Optional:
You can find the points of infliction, and graph those as well.
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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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The first derivative of distance =
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Definition
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The second derivative of distance =
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Definition
Acceleration
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