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Derivatives

A derivative is simply a rate of change. Whether you're modeling the movement of a particle or a supply/demand model, this is a key instrument of Calculus.

Derivatives Warmup

         

Let \[f(x) = x^3 + x^2 + x + 1.\]

What is the value of \(f'(1)?\)

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Let \(n\) be a real number and \[f(x) = x^n + x^{-n}.\]

What is the value of \(f'(1)?\)

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Cameron is out running for an hour. His distance (in kilometers) from home \(t\) hours after leaving is given by the formula \[s(t) = 12(t - t^2),\text{ for } 0 \leq t \leq 1.\]

His average velocity for the hour-long run is 0 km./h., since

\[\frac{s(1) - s(0)}{1-0} = \frac{0-0}{1-0} = 0.\]

What is Cameron's average velocity (in km./h.) during the first half-hour of his trip?

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True or False?

For every quadratic function \(f(x) = ax^2 + bx + c,\) there is a real number \(x\) such that \[f(x)=f'(x).\]

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True or False?

If \(n\) is an odd integer and \(f(x) = x^n,\) then \(f'(x)\) is an even function, i.e. \(f'(x) = f'(-x)\) for all \(x\) in the domain of \( f' .\)

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