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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 $$f'(1)?$$

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

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

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?

True or False?

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

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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