Chapter 5, Bracketing Methods Video Solutions, Introductory Differential Equations

Problem 1

Determine the real roots of $f(x)=-0.6 x^{2}+2.4 x+5.5:$
(a) Graphically.
(b) Using the quadratic formula.
(c) Using three iterations of the bisection method to determine the highest root. Employ initial guesses of $x_{1}=5$ and $x_{u}=10$ Compute the estimated error $\varepsilon_{a}$ and the true error $\varepsilon_{t}$ after each iteration.

Susan Hallstrom

Numerade Educator

07:17

Problem 2

Determine the real root of $f(x)=4 x^{3}-6 x^{2}+7 x-2.3:$
(a) Graphically.
(b) Using bisection to locate the root. Employ initial guesses of $x_{1}=0$ and $x_{u}=1$ and iterate until the estimated error $\varepsilon_{a}$ falls below a level of $\varepsilon_{s}=10 \%$.

Susan Hallstrom

Numerade Educator

07:17

Problem 3

Determine the real root of $f(x)=-26+85 x-91 x^{2}+$ $44 x^{3}-8 x^{4}+x^{5}:$
(a) Graphically.
(b) Using bisection to determine the root to $\varepsilon_{s}=10 \%$. Employ initial guesses of $x_{1}=0.5$ and $x_{u}=1.0$
(c) Perform the same computation as in
(b) but use the false-position method and $\varepsilon_{s}=0.2 \%$.

Susan Hallstrom

Numerade Educator

04:08

Problem 4

(a) Determine the roots of $f(x)=-13-20 x+19 x^{2}-3 x^{3}$ graphically. In addition, determine the first root of the function with
(b) bisection, and (c) false position. For (b) and (c) use initial guesses of $x_{1}=-1$ and $x_{u}=0,$ and a stopping criterion of $1 \%$.

James Kiss

Numerade Educator

01:58

Problem 5

Locate the first nontrivial root of $\sin x=x^{3},$ where $x$ is in radians. Use a graphical technique and bisection with the initial interval from 0.5 to $1 .$ Perform the computation until $\varepsilon_{a}$ is less than $\varepsilon_{s}=2 \% .$ Also perform an error check by substituting your final answer into the original equation.

Bahar Tehranipoor

Numerade Educator

17:36

Problem 6

Determine the positive real root of $\ln \left(x^{4}\right)=0.7$ (a) graphically, (b) using three iterations of the bisection method, with initial guesses of $x_{1}=0.5$ and $x_{u}=2,$ and (c) using three iterations of the false-position method, with the same initial guesses as in (b).

Susan Hallstrom

Numerade Educator

13:55

Problem 7

5.7 Determine the real root of $f(x)=(0.8-0.3 x) / x:$
(a) Analytically.
(b) Graphically.
(c) Using three iterations of the false-position method and initial guesses of 1 and 3 . Compute the approximate error $\varepsilon_{a}$ and the true error $\varepsilon_{t}$ after each iteration. Is there a problem with the result?

Susan Hallstrom

Numerade Educator

03:55

Problem 8

Find the positive square root of 18 using the false-position method to within $\varepsilon_{s}=0.5 \%$. Employ initial guesses of $x_{l}=4$ and $x_{u}=5$.

Susan Hallstrom

Numerade Educator

01:34

Problem 9

Find the smallest positive root of the function ( $x$ is in radians) $x^{2}|\cos \sqrt{x}|=5$ using the false-position method. To locate the region in which the root lies, first plot this function for values of $x$ between 0 and $5 .$ Perform the computation until $\varepsilon_{a}$ falls below $\varepsilon_{s}=1 \% .$ Check your final answer by substituting it into the original function.

Carson Merrill

Numerade Educator

07:17

Problem 10

Find the positive real root of $f(x)=x^{4}-8 x^{3}-35 x^{2}+$ $450 x-1001$ using the false-position method. Use initial guesses of $x_{l}=4.5$ and $x_{u}=6$ and performs five iterations. Compute both the true and approximate errors based on the fact that the root is $5.60979 .$ Use a plot to explain your results and perform the computation to within $\varepsilon_{s}=1.0 \%$.

Susan Hallstrom

Numerade Educator

08:33

Problem 11

Determine the real root of $x^{3.5}=80:$ (a) analytically, and (b) with the false-position method to within $\varepsilon_{s}=2.5 \%$. Use initial guesses of 2.0 and 5.0.

Susan Hallstrom

Numerade Educator

06:19

Problem 12

Given \[f(x)=-2 x^{6}-1.6 x^{4}+12 x+1\]
Use bisection to determine the maximum of this function. Employ initial guesses of $x_{1}=0$ and $x_{u}=1,$ and perform iterations until the approximate relative error falls below $5 \%$.

Carlos Pinilla

Numerade Educator

00:01

Problem 13

The velocity $v$ of a falling parachutist is given by \[v=\frac{g m}{c}\left(1-e^{-(c / m) t}\right)\]
where $g=9.8 \mathrm{m} / \mathrm{s}^{2} .$ For a parachutist with a drag coefficient $c=15 \mathrm{kg} / \mathrm{s},$ compute the mass $m$ so that the velocity is $v=35 \mathrm{m} / \mathrm{s}$ at $t=9$ s. Use the false-position method to determine $m$ to a level of $\varepsilon_{s}=0.1 \%$.

Linda Hand

Numerade Educator

02:17

Problem 14

Use bisection to determine the drag coefficient needed so that an 80 -kg parachutist has a velocity of $36 \mathrm{m} / \mathrm{s}$ after $4 \mathrm{s}$ of free fall. Note: The acceleration of gravity is $9.81 \mathrm{m} / \mathrm{s}^{2}$. Start with initial guesses of $x_{1}=0.1$ and $x_{u}=0.2$ and iterate until the approximate relative error falls below $2 \%$.

Willis James

Numerade Educator

03:13

Problem 15

A beam is loaded as shown in Fig. P5.15. Use the bisection method to solve for the position inside the beam where there is no moment.

James Kiss

Numerade Educator

01:47

Problem 16

Water is flowing in a trapezoidal channel at a rate of $Q=$ $20 \mathrm{m}^{3} / \mathrm{s} .$ The critical depth $y$ for such a channel must satisfy the equation \[0=1-\frac{Q^{2}}{g A_{c}^{3}} B\]
where $g=9.81 \mathrm{m} / \mathrm{s}^{2}, A_{c}=$ the cross-sectional area $\left(\mathrm{m}^{2}\right),$ and $B=$ the width of the channel at the surface (m). For this case, the width and the cross-sectional area can be related to depth $y$ by \[B=3+y \quad \text { and } \quad A_{c}=3 y+\frac{y^{2}}{2}\]
Solve for the critical depth using (a) the graphical method, (b) bisection, and (c) false position. For (b) and (c) use initial guesses of $x_{1}=0.5$ and $x_{u}=2.5,$ and iterate until the approximate error falls below $1 \%$ or the number of iterations exceeds $10 .$ Discuss your results.

Dominador Tan

Numerade Educator

13:49

Problem 17

You are designing a spherical tank (Fig. $P 5.17$ ) to hold water for a small village in a developing country. The volume of liquid it can hold can be computed as \[V=\pi h^{2} \frac{[3 R-h]}{3}\]
where $V=$ volume $\left[\mathrm{m}^{3}\right], h=$ depth of water in $\operatorname{tank}[\mathrm{m}],$ and $R=$ the tank radius [m].
If $R=3 \mathrm{m},$ to what depth must the tank be filled so that it holds $30 \mathrm{m}^{3} ?$ Use three iterations of the false-position method to determine your answer. Determine the approximate relative error after each iteration. Employ initial guesses of 0 and $R$.

Susan Hallstrom

Numerade Educator

02:01

Problem 18

The saturation concentration of dissolved oxygen in freshwater can be calculated with the equation (APHA, 1992 ) $$\begin{aligned}
\ln o_{s f}=&-139.34411+\frac{1.575701 \times 10^{5}}{T_{a}}-\frac{6.642308 \times 10^{7}}{T_{a}^{2}} \\
&+\frac{1.243800 \times 10^{10}}{T_{a}^{3}}-\frac{8.621949 \times 10^{11}}{T_{a}^{4}}
\end{aligned}$$
where $o_{s f}=$ the saturation concentration of dissolved oxygen in freshwater at 1 atm $(\mathrm{mg} / \mathrm{L})$ and $T_{a}=$ absolute temperature $(\mathrm{K})$ Remember that $T_{a}=T+273.15,$ where $T=$ temperature $\left(^{\circ} \mathrm{C}\right)$ According to this equation, saturation decreases with increasing temperature. For typical natural waters in temperate climates, the equation can be used to determine that oxygen concentration ranges from $14.621 \mathrm{mg} / \mathrm{L}$ at $0^{\circ} \mathrm{C}$ to $6.413 \mathrm{mg} / \mathrm{L}$ at $40^{\circ} \mathrm{C}$. Given a
value of oxygen concentration, this formula and the bisection method can be used to solve for temperature in $^{\circ} \mathrm{C}$
(a) If the initial guesses are set as 0 and $40^{\circ} \mathrm{C}$, how many bisection iterations would be required to determine temperature to an absolute error of $0.05^{\circ} \mathrm{C} ?$
(b) Develop and test a bisection program to determine $T$ as a function of a given oxygen concentration to a prespecified absolute error as in (a). Given initial guesses of 0 and $40^{\circ} \mathrm{C}$, test your program for an absolute error $=0.05^{\circ} \mathrm{C}$ and the following cases: $o_{s f}=8,10$ and $12 \mathrm{mg} / \mathrm{L}$. Check your results.

Lottie Adams

Numerade Educator

07:25

Problem 19

A reversible chemical reaction $2 \mathrm{A}+\mathrm{B} \longrightarrow \mathrm{C}$ can be characterized by the equilibrium relationship
\[
K=\frac{c_{c}}{c_{a}^{2} c_{b}}
\] where the nomenclature $c_{i}$ represents the concentration of constituent $i$ Suppose that we define a variable $x$ as representing the number of moles of $\mathrm{C}$ that are produced. Conservation of mass can be used to reformulate the equilibrium relationship as
\[
K=\frac{\left(c_{c, 0}+x\right)}{\left(c_{a, 0}-2 x\right)^{2}\left(c_{b, 0}-x\right)}
\]
where the subscript 0 designates the initial concentration of each constituent. If $K=0.016, c_{a, 0}=42, c_{b, 0}=28,$ and $c_{c, 0}=4$ determine the value of $x$. (a) Obtain the solution graphically. (b) On the basis of (a), solve for the root with initial guesses of $x_{1}=0$ and $x_{u}=20$ to $\varepsilon_{s}=0.5 \% .$ Choose either bisection or false position to obtain your solution. Justify your choice.

Henry He

Numerade Educator

02:13

Problem 20

Figure $\mathrm{P} 5.20$ a shows a uniform beam subject to a linearly increasing distributed load. The equation for the resulting elastic curve is (see Fig. P5.20 $b$ ) $$y=\frac{w_{0}}{120 E I L}\left(-x^{5}+2 L^{2} x^{3}-L^{4} x\right)$$
Use bisection to determine the point of maximum deflection (that is, the value of $x$ where $d y / d x=0$ ). Then substitute this value into Eq. (P5.20) to determine the value of the maximum deflection. Use the following parameter values in your computation: $L=600 \mathrm{cm}$ $E=50,000 \mathrm{kN} / \mathrm{cm}^{2}, I=30,000 \mathrm{cm}^{4},$ and $w_{0}=2.5 \mathrm{kN} / \mathrm{cm}$

Surendra Kumar

Numerade Educator

02:18

Problem 21

You buy a $\$ 25,000$ piece of equipment for nothing down and $\$ 5,500$ per year for 6 years. What interest rate are you paying? The formula relating present worth $P,$ annual payments $A,$ number of years $n,$ and interest rate $i$ is
\[
A=P \frac{i(1+i)^{n}}{(1+i)^{n}-1}
\]

Laura Skalaski

Numerade Educator

04:32

Problem 22

Many fields of engineering require accurate population estimates. For example, transportation engineers might find it necessary to determine separately the population growth trends of a city and adjacent suburb. The population of the urban area is declining with time according to
\[P_{u}(t)=P_{u, \max } e^{-k_{u} t}+P_{u, \min }\] while the suburban population is growing, as in
$$P_{s}(t)=\frac{P_{s, \max }}{1+\left[P_{s, \max } / P_{0}-1\right] e^{-k_{s} t}}$$
where $P_{u, \max }, k_{u}, P_{s, \max }, P_{0},$ and $k_{s}=$ empirically derived parameters. Determine the time and corresponding values of $P_{u}(t)$ and $P_{s}(t)$ when the suburbs are $20 \%$ larger than the city. The parameter values are $P_{u, \max }=75,000, k_{u}=0.045 / \mathrm{yr}, \quad P_{u, \min }=100,000$
people, $P_{s, \max }=300,000$ people, $P_{0}=10,000$ people, $k_{s}=$ $0.08 /$ yr. To obtain your solutions, use (a) graphical and (b) false position methods.

Subhadeepta Sahoo

Numerade Educator

02:40

Problem 23

Integrate the algorithm outlined in Fig. 5.10 into a complete, user-friendly bisection subprogram. Among other things:
(a) Place documentation statements throughout the subprogram to identify what each section is intended to accomplish.
(b) Label the input and output.
(c) Add an answer check that substitutes the root estimate into the original function to verify whether the final result is close to zero.
(d) Test the subprogram by duplicating the computations from Examples 5.3 and 5.4.

Yaw Asomani

Numerade Educator

01:28

Problem 24

Develop a subprogram for the bisection method that minimizes function evaluations based on the pseudocode from Fig. 5.11 Determine the number of function evaluations $(n)$ per total iterations. Test the program by duplicating Example 5.6.

Adriano Chikande

Numerade Educator

02:36

Problem 25

Develop a user-friendly program for the false-position method. The structure of your program should be similar to the bisection algorithm outlined in Fig. $5.10 .$ Test the program by duplicating Example 5.5.

Morgan Cheatham

Numerade Educator

01:26

Problem 26

Develop a subprogram for the false-position method that minimizes function evaluations in a fashion similar to Fig. 5.11 Determine the number of function evaluations (n) per total iterations. Test the program by duplicating Example 5.6.

Adriano Chikande

Numerade Educator

08:41

Problem 27

Develop a user-friendly subprogram for the modified false-position method based on Fig. $5.15 .$ Test the program by determining the root of the function described in Example $5.6 .$ Perform a number of runs until the true percent relative error falls below $0.01 \%$. Plot the true and approximate percent relative errors versus number of iterations on semilog paper. Interpret your results.

Elham Kordzadeh

Numerade Educator

04:00

Problem 28

Develop a function for bisection in a similar fashion to Fig. $5.10 .$ However, rather than using the maximum iterations and Eq. $(5.2),$ employ Eq. (5.5) as your stopping criterion. Make sure to round the result of Eq. (5.5) up to the next highest integer. Test your function by solving Example 5.3 using $E_{a, d}=0.0001$.

Olivia Demers

Numerade Educator