Calculus
# Derivatives

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 in a straight path 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 $f' .$