AP Calculus BC Quiz: Confirming Continuity Over An Interval

What this quiz covers

This quiz focuses on Confirming Continuity Over An Interval, giving you a quick way to practice the rules, question types, and explanations that matter most for AP Calculus BC.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. Ensure $p(0)=0$ so the numerator vanishes at the midpoint.
2. Ensure $p(2)=p(-2)=0$ so the denominator's zeros are removable. (correct answer)
3. Ensure $p$ is differentiable on $[-3,3]$.
4. Ensure $p(x)$ has degree at least $2$.
5. No additional condition; all rational functions are continuous on closed intervals.

Question 2

1. Show $|x-3|$ is continuous on $[1,5]$ and adding a constant $k$ preserves continuity; endpoints are defined. (correct answer)
2. Show $p$ is differentiable at $x=3$ for all $k$ and continuous elsewhere.
3. Show $p(1)=p(5)$ and $\lim_{x\to 3}p(x)=0$.
4. Show $\lim_{x\to 1}p(x)$ and $\lim_{x\to 5}p(x)$ exist; interior continuity is not required.
5. Show $p$ is continuous only at $x=3$, since absolute value can be discontinuous elsewhere.

Question 3

1. $s$ is continuous on $[-1,3]$ because absolute value and rational functions are continuous where defined (correct answer)
2. $s$ is not continuous on $[-1,3]$ because $|x-1|$ has a cusp at $x=1$
3. $s$ is not continuous on $[-1,3]$ because $\dfrac{1}{x+2}$ is undefined at $x=-2$
4. $s$ is continuous on $[-1,3]$ only if $s'(1)$ exists
5. $s$ is not continuous on $[-1,3]$ because $\lim_{x\to-1}s(x)$ does not exist

Question 4

1. Confirm $q$ is defined for all $x\in[-1,1]$ and rational functions are continuous on their domains. (correct answer)
2. Confirm $q$ is defined for all real $x$ and has no vertical asymptotes.
3. Confirm $\lim_{x\to-1}q(x)$ and $\lim_{x\to1}q(x)$ both equal $0$.
4. Confirm $q(-1)=q(1)$ and $q$ is increasing on $[-1,1]$.
5. Confirm $q$ is continuous at $x=0$ only.

Question 5

1. Set $c=0$ so $r(0)=0$ and $\sin 0=0$.
2. Set $c=1$ so $\lim_{x\to0}\dfrac{\sin x}{x}=r(0)$ and $r$ is defined on $[-1,1]$. (correct answer)
3. Any $c$ works because $\dfrac{\sin x}{x}$ is continuous for $x\ne0$.
4. No $c$ works because $\dfrac{\sin x}{x}$ has an infinite discontinuity at $0$.
5. Set $c=\sin(1)$ so endpoint behavior matches at $x=1$.

Question 6

1. Require $s(2)$ is defined; then $s$ is continuous on $[0,5]$.
2. Require $\lim_{x\to2}s(x)$ exists; then $s$ is continuous on $[0,5]$.
3. Require $\lim_{x\to2^-}s(x)=\lim_{x\to2^+}s(x)=s(2)$. (correct answer)
4. Require $s'(2)$ exists; then $s$ is continuous on $[0,5]$.
5. Require $s(0)=s(5)$ to avoid an endpoint discontinuity.

Question 7

1. On $(-\infty,2)\cup(2,\infty)$, since only $\frac{1}{x-2}$ is discontinuous at $2$. (correct answer)
2. On $(-\infty,\infty)$, since integrals always produce continuous functions.
3. On $[1,\infty)$ only, since the integral starts at $1$.
4. On $(-\infty,2]\cup(2,\infty)$, since continuity can include $x=2$ by definition.
5. Only on $(2,\infty)$, since the integrand is undefined for $t<0$.

Question 8

1. $(-\infty,\infty)$, since absolute value is continuous.
2. $(-\infty,-1)\cup(-1,\infty)$, excluding where the denominator is zero. (correct answer)
3. $(-\infty,4)\cup(4,\infty)$, excluding the corner at $x=4$.
4. $[-1,\infty)$, because continuity fails only to the left of $-1$.
5. $(-\infty,-1]$, because continuity can include $x=-1$.

Question 9

1. $c=0$, because the numerator is $0$ when $x=0$.
2. $c=\tfrac{1}{4}$, because rationalizing gives $\lim_{x\to0}y(x)=\tfrac{1}{4}$. (correct answer)
3. $c=\tfrac{1}{2}$, because $\sqrt{x+4}$ has derivative $\tfrac{1}{2}$ at $x=0$.
4. $c=2$, because continuity requires matching the square root value at $0$.
5. Any real $c$, because $y$ is continuous for all $x\ne0$.

Question 10

1. $h$ is continuous on $[2,5]$ because logarithms are continuous for all real inputs.
2. $h$ is not continuous on $[2,5]$ because it is undefined at $x=2$, so it cannot be continuous on the interval. (correct answer)
3. $h$ is continuous on $[2,5]$ because it is continuous on $(2,5]$ and endpoints do not matter.
4. $h$ is not continuous on $[2,5]$ because $\ln$ has a removable discontinuity at $x=2$.
5. $h$ is continuous on $[2,5]$ if $\lim_{x\to2^+}h(x)=0$.

Question 11

1. Ensure $F(2)=5$ and $F$ is defined for every $x\in[1,4]$. (correct answer)
2. Ensure $F(2)\ne5$ so the limit is not equal to the function value.
3. Ensure $F$ is differentiable at $x=2$; continuity then follows automatically.
4. Ensure only that $F(1)$ and $F(4)$ are defined; interior points need not be checked.
5. Ensure $\lim_{x\to2^-}F(x)=5$; the right-hand limit is unnecessary.

Question 12

1. Any real $c$, since $\ln(x-1)$ is continuous for $x>1$ (correct answer)
2. $c$ must satisfy $g(2)=0$
3. $c$ must satisfy $g(6)=0$
4. $c$ must be positive
5. $c$ must be $0$ because endpoints require $c=0$

Question 13

1. $\left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right]$
2. $\left[-\dfrac{\pi}{4},\dfrac{\pi}{4}\right]$ (correct answer)
3. $\left[0,\pi\right]$
4. $\left[\dfrac{\pi}{3},\dfrac{2\pi}{3}\right]$
5. $\left[-\pi,\pi\right]$

Question 14

1. Continuous on $[0,5]$ for all real $m$ (correct answer)
2. Continuous on $[0,5]$ only if $m\ge0$
3. Discontinuous at $x=2$ unless $m=0$
4. Discontinuous at $x=0$ unless $m=2$
5. Discontinuous at $x=5$ unless $m=3$

Question 15

1. $a+b=0$
2. $1+a=3+b$ (correct answer)
3. $a=3$
4. $b=0$
5. $a=b$

Question 16

1. Set $a=3$ so $g(1^-)=g(1)=g(1^+)$ and $g$ is defined on $[-2,3]$. (correct answer)
2. Set $a=1$ so both pieces are polynomials and therefore continuous everywhere.
3. Any $a$ works because each piece is continuous on its own subinterval.
4. Set $a=0$ so $g(1)=0$ and the one-sided limits at $1$ must differ.
5. No value of $a$ works because continuity on $[-2,3]$ requires differentiability at $x=1$.

Question 17

1. Verify $\lim_{x\to 0}h(x)=h(0)$ and that $h$ is continuous on $[-1,0)$ and $(0,1]$, with $h(-1)$ and $h(1)$ defined. (correct answer)
2. Verify $h'(0)$ exists and $h$ is bounded on $[-1,1]$.
3. Verify $\lim_{x\to 0}h(x)=0$ and $h(0)=0$.
4. Verify $\lim_{x\to -1}h(x)=h(-1)$ only, since interior points are automatically continuous.
5. Verify $\lim_{x\to 0^-}h(x)=\lim_{x\to 0^+}h(x)$ only.

Question 18

1. $a=0$, since $g(3)$ must equal the numerator at $x=3$.
2. $a=3$, since the factor $x-3$ cancels leaving $x+3$.
3. $a=6$, since $\lim_{x\to3}g(x)=6$. (correct answer)
4. $a=9$, since $\lim_{x\to3}g(x)=9$.
5. Any real $a$, since redefining one point never affects continuity.

Question 19

1. $k=0$
2. $k=3$
3. $k=6$ (correct answer)
4. $k=9$
5. $k$ can be any real number

Question 20

1. $u$ is continuous on the interval because $\cos x\ne0$ for all $x$ in it (correct answer)
2. $u$ is not continuous because $\tan x$ is never continuous at $x=0$
3. $u$ is not continuous because the interval contains $x=\tfrac{\pi}{2}$
4. $u$ is continuous only if $u\left(\tfrac{\pi}{3}\right)=u\left(-\tfrac{\pi}{3}\right)$
5. $u$ is continuous only if $u'(x)$ exists at every point