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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 right-side 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 upside-down 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 x-value at which the concavity of the function changes. In other words, an IP is an x-value 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 stay-cations. 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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