Continuity (College Board AP® Calculus BC): Free Response Questions

Let $f$ be the function defined by

$f\left(x\right)=\{\begin{array}{l}\frac{\pi \left(x+2\right)}{2}\mathrm{for}x<-1\\ \arcsin \left(x^2\right)\mathrm{for}-1\le x\le 1\end{array}$

Is $f$ continuous at $x=-1$? Use the definition of continuity to explain your answer.

Let $f$ be the function defined by $f\left(x\right)=\frac{x^2-3x}{x^2-2x-3}$.

Show that $f$ has a removable discontinuity at $x=3$.

Explain how the discontinuity identified in part (a) can be removed.

Let $f$ be the function defined by $f\left(x\right)=\{\begin{array}{l}3+\frac{7}{x-4}\mathrm{for}x<4\\ x-2\mathrm{for}x\ge 4\end{array}$.

Show that $f$ has an essential discontinuity at $x=4$.

The value of a car manufactured by Warm Wheels is modeled by the continuous function $W$ and the value of a car manufactured by Tommy Thunder is modeled by the function $T$. $W\left(t\right)$ and $T\left(t\right)$ are measured in US dollars and $t$ is measured in years for $0\le t\le 6$.

Initially, the value of the car manufactured by Warm Wheels is $40,000 and after 6 years it is worth $15,500.The function $T$ is given by $T\left(t\right)=35000·{\left(\frac{10}{11}\right)}^t$.

Let $D\left(t\right)=W\left(t\right)-T\left(t\right)$. Apply the Intermediate Value Theorem to the function $D$ on the interval $0\le t\le 6$ to justify that there exists a time $t$, $0<t<6$, at which both cars have the same value.

An animal is released into the wild. During the time interval $0\le t\le 24$ hours, the animal's distance from the place where it was released, measured in feet, is given by the differentiable function $s$. The table above shows selected values of this function.

For $0<t<24$, must there be a time $t$ when the animal is exactly 55 feet from the place where it was released? Justify your answer.

Functions $f$ and $g$ are differentiable functions with $f\left(1\right)=g\left(1\right)=7$. It is known that $f\left(x\right)\le g\left(x\right)$ for $0<x<2$.

Let $h$ be a function satisfying $f\left(x\right)\le h\left(x\right)\le g\left(x\right)$ for $0<x<2$. Is $h$ continuous at $x=1$? Justify your answer.

A balloon is inflated with air using a machine. The surface area of the balloon is increasing at a rate modeled by $A\left(t\right)=4e^{\raisebox{1ex}{$t$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$ square inches per second for $0\le t\le 10$. A second balloon is inflated with air by a human. The surface area of the second balloon is increasing at a rate modeled by $B\left(t\right)$ square inches per second, where $B$ is differentiable on $0\le t\le 10$. Selected values of $B\left(t\right)$ are shown in the table above.

For $0\le t\le 10$, is there a time $t$ when the rate at which the surface areas of both balloons are increasing at the same rate? Explain why or why not.

Let $f$ be the function defined by

$f\left(x\right)=\{\begin{array}{ll}2x+7& \mathrm{for}x\le a\\ \frac{x^3-x^2-3x-9}{x^2-4x+3}& \mathrm{for}a<x<3\\ x^b& \mathrm{for}x\ge 3\end{array}$.

Given that $f$ is continuous for all real values of $x$, find the value $a$ and the value of $b$.

The functions $f$ and $g$ are continuous functions. The table above gives values of the functions at selected values of $x$. The function $h$ is given by $h\left(x\right)=\left|f\left(g\left(x\right)\right)\right|$.

For $1<x<5$, what is the minimum number of solutions that there must be to the equation $h\left(x\right)=8$? Justify your answer.

Let $f$ be the function defined by

$f\left(x\right)=\{\begin{array}{ll}\ln \left(x^2-3\right)& \mathrm{for}x\le -2\\ \frac{2x^2-10x+8}{x^2-2x-8}& \mathrm{for}-2<x<4\\ \tan \left(\frac{\pi }{2}-\frac{\pi }{x}\right)& \mathrm{for}x>4\end{array}$.

Explain whether $f$ is continuous at $x=-2$. If $f$ has a discontinuity at this point, then describe the type of discontinuity.

Explain whether $f$ is continuous at $x=4$. If $f$ has a discontinuity at this point, then describe the type of discontinuity.