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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 Exercises
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...
Concave Up
We say a function f is concave up if it curves upward like a rightside up spoon:It's also possible to have only part of the spoon. Both of these functions are concave up:"f is concave up" means ex...
Concave Down
We say a function f is concave down if it curves downward like an upsidedown spoon (concave side down):It's also fine to have only part of the bowl. Both of these functions are concave down:"f is...
No Concavity
If f " is positive, then f is concave up. If f " is negative, then f is concave down. If f " is zero, we say that the function f has no concavity. It's flat. Pancakes can survive in...
Critical Points
We say x = c is a critical point of the function f if f (c) exists and f'(c) = 0 or is undefined. It's generally a peak or valley in the curve. It's where the slopes becomes interesting. When...
Points of Inflection
A point of inflection or inflection point, abbreviated IP, is an xvalue at which the concavity of the function changes. In other words, an IP is an xvalue where the sign of the second derivative...
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...
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