1 Functions, Rates of Change, and the Idea of a Limit Free Online FlashCards

What is a function?

A function assigns exactly one output to each input in its domain.

What are a function’s domain and range?

The domain is the set of permitted inputs; the range is the set of resulting outputs.

What are the domain and range of $f(x)=\sqrt{x-1}$?

For $f(x)=\sqrt{x-1}$, the domain is $[1,\infty)$ and the range is $[0,\infty)$.

What is the average rate-of-change formula?

The average rate of change from $x=a$ to $x=b$ is $\frac{f(b)-f(a)}{b-a}$, where $b\ne a$.

What does a secant line represent?

A secant line passes through two points on a curve, so its slope gives an average rate of change.

What is the average velocity of $s(t)=t^2+2t$ from $t=1$ to $t=3$?

For $s(t)=t^2+2t$, the average velocity from $t=1$ to $t=3$ is $\frac{15-3}{3-1}=6$ meters per second.

What is the difference quotient?

The difference quotient is $\frac{f(a+h)-f(a)}{h}$, with $h\ne 0$. It gives the average rate of change from $a$ to $a+h$.

What does instantaneous rate of change mean?

The instantaneous rate of change at $x=a$ is the limiting value of average rates over intervals that shrink toward $a$.

How is the derivative defined from a difference quotient?

The derivative at $a$ is $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$, when this limit exists.

What does a tangent line represent?

A tangent line describes the local direction of a curve at one point; its slope is the instantaneous rate of change there.

What is the tangent line to $f(x)=x^2$ at $x=2$?

For $f(x)=x^2$ at $x=2$, the tangent slope is $4$, and the tangent line is $y-4=4(x-2)$, or $y=4x-4$.

What does $\lim_{x\to a}f(x)=L$ mean?

$\lim_{x\to a}f(x)=L$ means that function values approach $L$ as $x$ approaches $a$. The value at $a$ itself may differ or be undefined.