1 Functions, Rates of Change, and the Idea of a Limit Online Quiz Questions

True or false: A function must assign exactly one output to each input in its domain.

A car's position is modeled by $s(t)=t^2+2t$, where position is measured in meters and time in seconds. What is its average velocity from $t=1$ to $t=3$? Enter the value in meters per second.

Select all correct statements about limits.

For $f(x)=x^2-3x+2$, what is the average rate of change from $x=1$ to $x=4$?

True or false: A limit can exist even when the function is not defined at the point being approached.

For $f(x)=3x^2$, use the difference-quotient idea to find the instantaneous rate of change at $x=2$. Enter the numerical value.

Select all statements that correctly interpret features of a function's graph.

Suppose $\lim_{x\to a^-}f(x)=3$ and $\lim_{x\to a^+}f(x)=5$. What can be concluded about $\lim_{x\to a}f(x)$?

Explain how the average rate of change over an interval leads to the instantaneous rate of change at a point. Include the relevant formula, the roles of secant and tangent lines, and the limiting process.

For $x\ne 1$, which expression is equivalent to $\frac{x^2-1}{x-1}$?

Which statement correctly describes differential calculus?

For $f(x)=x^2-3x+2$, what is $f(4)$?

A car's position is modeled by $s(t)=t^2+2t$, where $s$ is measured in meters and $t$ in seconds. What is its average velocity from $t=1$ to $t=3$?

What is $\displaystyle\lim_{x\to 1}\frac{x^2-1}{x-1}$?

For $f(x)=x^2$, what does the difference quotient $\frac{f(2+h)-f(2)}{h}$ simplify to, assuming $h\ne 0$?

What is the mathematical term for the set of permitted input values of a function?

True or false: When it exists, $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$ represents the instantaneous rate of change at $x=a$.

What is the standard name for the expression $\frac{f(a+h)-f(a)}{h}$, where $h\ne 0$?