4 Derivative-Based Analysis Free Online FlashCards

What does a derivative measure?

The derivative measures the instantaneous rate of change of a function and equals the slope of its tangent line at a point.

What units does a derivative have?

A derivative has units of output divided by input. For example, if position is in meters and time is in seconds, velocity is measured in meters per second.

How do average and instantaneous rates of change differ?

The average rate of change over $[a,b]$ is $\frac{f(b)-f(a)}{b-a}$, while $f'(a)$ is the instantaneous rate of change at the single input $a$.

What is the power rule?

The power rule is $\frac{d}{dx}(x^n)=nx^{n-1}$ for any real number $n$.

How do you differentiate a product of functions?

The product rule is $\frac{d}{dx}[f(x)g(x)]=f'(x)g(x)+f(x)g'(x)$.

What is the chain rule?

For $y=f(g(x))$, the chain rule gives $\frac{d}{dx}f(g(x))=f'(g(x))g'(x)$. Differentiate the outside function, then multiply by the inside derivative.

How does the sign of $f'$ determine behavior?

If $f'(x)>0$ throughout an interval, $f$ is increasing there; if $f'(x)<0$, $f$ is decreasing there.

What is a critical number?

A critical number is a domain input where $f'(x)=0$ or where $f'(x)$ is undefined.

What does marginal cost represent?

Marginal cost is $C'(q)$, the approximate additional cost of producing one more unit near production level $q$.

How can a derivative approximate a cost change?

For a small production change, cost changes approximately according to $\Delta C\approx C'(q)\Delta q$.

What sign change identifies a local maximum?

A change from positive to negative in $f'$ indicates a local maximum at the critical point.

Why is a critical point not always an extremum?

A critical point need not be an extremum. For $f(x)=x^3$, $f'(0)=0$, but the function increases on both sides of $x=0$.