4 Applications of Derivatives Free Online FlashCards

What is the standard procedure for a related-rates problem?

Differentiate the relationship with respect to time, substitute known values afterward, solve for the requested rate, and include appropriate units.

What rate equation relates a sphere’s volume and radius?

For a sphere, $V=\frac{4}{3}\pi r^3$, so differentiating with respect to time gives $\frac{dV}{dt}=4\pi r^2\frac{dr}{dt}$.

What is the linearization of $f$ at $x=a$?

The linearization is $L(x)=f(a)+f'(a)(x-a)$, and for values near $a$, $f(x)\approx L(x)$.

How is the differential $dy$ defined?

If $y=f(x)$, then $dy=f'(x)\,dx$. The differential estimates the output change corresponding to a small input change.

What is a critical point?

A critical point occurs at a domain number $c$ where $f'(c)=0$ or where $f'(c)$ does not exist.

How does the First Derivative Test classify a critical point?

A change in $f'$ from positive to negative indicates a local maximum; a change from negative to positive indicates a local minimum.

What does the Second Derivative Test conclude at $f'(c)=0$?

If $f'(c)=0$, $f''(c)>0$ implies a local minimum and $f''(c)<0$ implies a local maximum. If $f''(c)=0$, the test is inconclusive.

How are absolute extrema found on a closed interval?

For a continuous function on $[a,b]$, evaluate the function at every critical point in $(a,b)$ and at both endpoints, then compare all values.

What does the Mean Value Theorem guarantee?

If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, some $c\in(a,b)$ satisfies $f'(c)=\frac{f(b)-f(a)}{b-a}$.

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

Wherever $f'(x)>0$, $f$ is increasing; wherever $f'(x)<0$, $f$ is decreasing.

What does the sign of $f''$ reveal about a graph?

If $f''(x)>0$, the graph is concave up; if $f''(x)<0$, it is concave down. An inflection point requires a change in concavity.

What is an objective function in optimization?

The objective function represents the quantity to maximize or minimize, such as area, cost, profit, distance, surface area, or volume.