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Second Derivatives and Beyond
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Calculus
Second Derivatives and Beyond
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Second Derivatives and Beyond Topics
Second Derivatives and Beyond explanations, examples, practice problems. Ready? Let’s do this.
Second Derivatives via Formulas
When we take the derivative of a differentiable function f, we end with a new function f '. If this new function f ' is differentiable, then we can take its derivative to find (f')', also...
Third Derivatives and Beyond
We're learning calculus, so, by definition, we're in the prime of our lives. Things are great. Things are awesome. We can take third derivatives, fourth derivatives, and so on. Writing f " for...
Concavity
The slope or derivative of a function f describes whether f is increasing, decreasing, or constant. The concavity of a function f describes whether f is curving up, curving down, or not curving at...
Special Points
Special points are those random points awarded to the guy who invited us over to play board games. Blast you, house rules. They never seem to work in our favor. In derivative land, if we have a fun...
Extreme Points and How to Find Them
Sample ProblemThe maximum value of the function f (x) = x2 – 1 is y = 1:Sample ProblemThe maximum value of the function f (x) = cos x is y = 1:Extreme points, also called extrema...
Local vs. Global Points
Sometimes we take vacations, sometimes we take staycations. Second derivatives can be used to determine if the function will be traveling somewhere extreme or if it will travel somewhere more subd...
Using Derivatives to Draw Graphs
The goal of this section is to be able to go from a formula of a function to an accurate graph of that function. We use the first and second derivatives to help find exact points on the graph and t...
In the Real World
Calculators good at graphing functions. However, calculators can only show a useful graph if a useful domain and range are specified. Sample ProblemPlay with the function f (x) = xln xIf we gr...
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