Practice Problems

1. At which points is the function in Fig. 14 discontinuous?

3. Find at least one point at which each function is not continuous and state which of the 3 conditions in the definition of continuity is violated at that point.

(a) $\frac{x+5}{x-3}$

(b) $\frac{x^{2}+x-6}{x-2}$

(c) $\sqrt{\cos (\mathrm{x})}$

(d) $\operatorname{INT}\left(\mathrm{x}^{2}\right)$

(e) $\frac{x}{\sin (x)}$

(f) $\frac{\mathrm{x}}{\mathrm{x}}$

(g) $\ln \left(x^{2}\right)$

(h) $\frac{\pi}{x^{2}-6 x+9}$

(i) $\tan (\mathrm{x})$

5. A continuous function $\mathrm{f}$ has the values given below:

$\begin{array}{l|l|l|l|l|l|l}\mathrm{x} & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline \mathrm{f}(\mathrm{x}) & 5 & 3 & -2 & -1 & 3 & -2\end{array}$

(a) $\mathrm{f}$ has at least ______ roots between 0 and 5.

(b) $f(x)=4$ at least ______ times between 0 and 5.

(c) $\mathrm{f}(\mathrm{x})=2$ at least ______ times between 0 and 5.

(d) $\mathrm{f}(\mathrm{x})=3$ at least ______ times between 0 and 5.

7. This problem asks you to verify that the Intermediate Value Theorem is true for some particular functions, intervals and intermediate values. In each problem you are given a function $\mathrm{f}$, an interval $[\mathrm{a}, \mathrm{b}]$ and a value $\mathrm{V}$. Verify that $\mathrm{V}$ is between $\mathrm{f}(\mathrm{a})$ and $\mathrm{f}(\mathrm{b})$ and find a value of $\mathrm{c}$ in the interval so that $\mathrm{f}(\mathrm{c})=\mathrm{V}$

(a) $f(x)=x^{2}$ on $[0,3], V=2$.

(b) $\mathrm{f}(\mathrm{x})=\mathrm{x}^{2}$ on $[-1,2], \mathrm{V}=3$.

(c) $\mathrm{f}(\mathrm{x})=\sin (\mathrm{x})$ on $[0, \pi / 2], \mathrm{V}=1 / 2$.

(d) $\mathrm{f}(\mathrm{x})=\mathrm{x}$ on $[0,1], \mathrm{V}=1 / 3$

(e) $f(x)=x^{2}-x$ on $[2,5], V=4$

(f) $\mathrm{f}(\mathrm{x})=\ln (\mathrm{x})$ on $[1,10], \mathrm{V}=2$.

9. Two students claim that they both started with the points $\mathrm{x}=0$ and $\mathrm{x}=5$ and applied the Bisection Algorithm to the function in Fig. 17. The first student says that the algorithm converged to the root labeled $\mathrm{A}$, but the second claims that the algorithm will converge to the root labeled B. Who is right?

11. If you apply the Bisection Algorithm to the function in Fig. 19 and use the given starting points, which root does the algorithm find?

(a) starting points 3 and 7.

(b) starting points 4 and 6.

(c) starting points 1 and 6.

In problems 13 – 17, use the Intermediate Value Theorem to verify that each function has a root in the given interval(s). Then use the Bisection Algorithm to narrow the location of that root to an interval of length less than or equal to 0.1.

13. $\mathrm{g}(\mathrm{x})=\mathrm{x}^{3}-3 \mathrm{x}^{2}+3$ on $[-1,0],[1,2],[2,4]$.

15. $r(x)=5-2^{x}$ on $[1,3]$.

17. $\mathrm{p}(\mathrm{t})=\mathrm{t}^{3}+3 \mathrm{t}+1$ on $[-1,1]$

19. Each of the following statements is false for some functions. For each statement, sketch the graph of a counterexample.

a) If $\mathrm{f}(3)=5$ and $\mathrm{f}(7)=-3$, then $\mathrm{f}$ has a root between $\mathrm{x}=3$ and $\mathrm{x}=7$.

b) If $\mathrm{f}$ has a root between $\mathrm{x}=2$ and $\mathrm{x}=5$, then $\mathrm{f}(2)$ and $\mathrm{f}(5)$ have opposite signs.

c) If the graph of a function has a sharp corner, then the function is not continuous there.

21. Define $A(x)$ to be the area bounded by the $x$ and y axes, the curve $\mathrm{y}=\mathrm{f}(\mathrm{x})$, and the vertical line at $\mathrm{x}$ (Fig. 21).

a) Shade the part of the graph represented by $\mathrm{A}(2.1)-\mathrm{A}(2)$ and estimate the value of $\frac{\mathrm{A}(2.1)-\mathrm{A}(2)}{0.1}$.

b) Shade the part of the graph represented by $\mathrm{A}(4.1)-\mathrm{A}(4)$ and estimate the value of $\frac{\mathrm{A}(4.1)-\mathrm{A}(4)}{0.1}$.

23. A piece of string is tied in a loop and tossed onto quadrant I enclosing a single region (Fig. 22).

(a) Is it always possible to find a line $L$ which goes through the origin so that $\mathrm{L}$ divides the region into two equal areas? (Justify your answer).

(b) Is it always possible to find a line $\mathrm{L}$ which is parallel to the $\mathrm{x}$-axis so that $\mathrm{L}$ divides the region into two equal areas?

(Justify your answer).

(c) Is it always possible to find 2 lines, $\mathrm{L}$ parallel to the $\mathrm{x}$-axis and

M parallel to the y-axis, so $\mathrm{L}$ and $\mathrm{M}$ divide the region into 4 equal areas? (Justify your answer).

Source: Dave Hoffman, https://s3.amazonaws.com/saylordotorg-resources/wwwresources/site/wp-content/uploads/2012/12/MA005-2.4-Continuous-Functions.pdf
This work is licensed under a Creative Commons Attribution 3.0 License.

1. Discontinuous at $\mathrm{1}$, $\mathrm{3}$, and $\mathrm{4}$.

3. (a) Discontinuous at $\mathrm{x}=3$. Fails condition (i) there.

(b) Discontinuous at $\mathrm{x}=2$. Fails condition (i) there.

(c) Discontinuous where $\cos (x)$ is negative, (e.g., at $x=\pi$). Fails condition (i) there.

(d) Discontinuous where $\mathrm{x}^{2}$ is an integer (e.g., at $\mathrm{x}=1$ or $\sqrt{2}$). Fails condition (ii) there.

(e) Discontinuous where $\sin (x)=0$ (e.g., at $x=0, \pm \pi, \pm 2 \pi, \ldots)$. Fails condition (i) there.

(f) Discontinuous at $\mathrm{x}=0$. Fails condition (i) there.

(g) Discontinuous at $\mathrm{x}=0$. Fails condition (i) there.

(h) Discontinuous at $\mathrm{x}=3$. Fails condition (i) there.

(i) Discontinuous at $\mathrm{x}=\pi / 2$. Fails condition (i) there.

5. (a) $\mathrm{f}(\mathrm{x})=0$ for at least 3 values of $\mathrm{x}, 0 \leq \mathrm{x} \leq 5$.

(b) $\mathrm{1}$

(c) $\mathrm{3}$

(d) $\mathrm{3}$

(e) Yes. It does not have to happen, but it is possible.

7. (a) $\mathrm{f}(0)=0, \mathrm{f}(3)=9$ and $0 \leq 2 \leq 9. \mathrm{c}=\sqrt{2} \approx 1.414$

(b) $f(-1)=1, f(2)=4$ and $1 \leq 3 \leq 4. c=\sqrt{3} \approx 1.732$

(c) $\mathrm{f}(0)=0, \mathrm{f}(\pi / 2)=1$ and $0 \leq 1 / 2 \leq 1. \mathrm{c}=($ inverse sine of $1 / 2) \approx 0.524$

(d) $f(0)=0, f(1)=1$ and $0 \leq 1 / 3 \leq 1. c=1 / 3$

(e) $f(2)=2, f(5)=20$ and $2 \leq 4 \leq 20. c=(1+\sqrt{17}) / 2 \approx 2.561$.

(f) $f(1)=0, f(10) \approx 2.30$ and $0 \leq 2 \leq 2.30. c=$ (inverse of $\ln (2))=e^{2} \approx 7.389$.

9. Neither student is correct. The bisection algorithm converges to the root labeled $\mathrm{C}$.

11. (a) D

(b) $\mathrm{D}$

(c) hits B

13. $[-0.9375,-0.875], \approx-0.879$

$[1.3125,1.375], \approx 1.347$

$[2.5,2.5625], \approx 2.532$

15. $[2.3125,2.375], \approx 2.32$.

17. $[-0.375,-0.3125], \approx-0.32$.

19. See the three graphs in Fig. 1.3P19.

21. (a) $\mathrm{A}(2.1)-\mathrm{A}(2)$ is the area of the region bounded below by the $\mathrm{x}$-axis, above by the graph of $\mathrm{f}$, on the left by the vertical line $\mathrm{x}=2$, and on the right by the vertical line $\mathrm{x}=2.1$.

$\frac{\mathrm{A}(2.1)-\mathrm{A}(2)}{0.1} \approx \mathrm{f}(2) \text { or } \mathrm{f}(2.1) \text { so } \frac{\mathrm{A}(2.1)-\mathrm{A}(2)}{0.1} \approx 1$

(b) $\mathrm{A}(4.1)-\mathrm{A}(4)$ is the area of the region bounded below by the $\mathrm{x}$-axis, above by the graph of $\mathrm{f}$, on the left by the vertical line $x=4$, and on the right by the vertical line $x=4.1 . \frac{A(4.1)-A(4)}{0.1} \approx f(4) \approx 2$.

23. (a) Yes. You supply the justification.

(b) Yes

(c) Try it.