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<mattextxml>
<div class="exercise-statement">
<p>
<strong>Zfinal.</strong>
</p>
<p> For each of the following scenarios, provide a reasonable IVP model. Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t" alt="t" title="t" data-latex="t"/> represent time, let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?z" alt="z" title="z" data-latex="z"/> represent the value given in the scenario, and use uppercase letters for positive constants. Then label each term and initial value of each IVP to describe what it represents from the scenario. </p>
<ul>
<li> An object is thrown upward from above the ground. Let z measure its altitude at a given time. Assume linear air resistance. </li>
<li> A glass of sweet tea is left outside on a summer day. After some time, ice is added to the glass, chilling the drink at a constant rate. Let z measure the temperature of the tea over time, assuming the outside temperature is kept constant. </li>
<li> A circuit includes a battery providing constant voltage, a resistor, and an inductor. Assume some initial current is flowing, and let z measure the current throughout this circuit at a given time. </li>
<li> A mass is attached to a spring. The mass is compressed inward from the spring's natural position, then released from rest. After some time, a hammer instantly strikes the mass inward. Assume the presence of friction. Let z measure the outward displacement of the mass from its natural position on the spring. </li>
</ul>
<p> Then, show that your model is equivalent to one of the following IVP models by relabeling constants and using algebra as needed. (It's possible that more than one IVP may be used for a scenario. If so, choose any of them.) </p>
<ol type="a">
<li>
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-J z^{2} - {z'} = -B z + S \delta\left(-C + t\right) \hspace{2em}z(0)= Q" alt="-J z^{2} - {z'} = -B z + S \delta\left(-C + t\right) \hspace{2em}z(0)= Q" title="-J z^{2} - {z'} = -B z + S \delta\left(-C + t\right) \hspace{2em}z(0)= Q" data-latex="-J z^{2} - {z'} = -B z + S \delta\left(-C + t\right) \hspace{2em}z(0)= Q"/>
</li>
<li>
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?A {z'} = -V Y - V {z''} \hspace{2em}z(0)= E ,z'(0)= W" alt="A {z'} = -V Y - V {z''} \hspace{2em}z(0)= E ,z'(0)= W" title="A {z'} = -V Y - V {z''} \hspace{2em}z(0)= E ,z'(0)= W" data-latex="A {z'} = -V Y - V {z''} \hspace{2em}z(0)= E ,z'(0)= W"/>
</li>
<li>
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-A z - R {z'} + L = 0 \hspace{2em}z(0)= W" alt="-A z - R {z'} + L = 0 \hspace{2em}z(0)= W" title="-A z - R {z'} + L = 0 \hspace{2em}z(0)= W" data-latex="-A z - R {z'} + L = 0 \hspace{2em}z(0)= W"/>
</li>
<li>
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-X z - Y {z''} - R \delta\left(-D + t\right) = N {z'} \hspace{2em}z(0)=- V ,z'(0)=0" alt="-X z - Y {z''} - R \delta\left(-D + t\right) = N {z'} \hspace{2em}z(0)=- V ,z'(0)=0" title="-X z - Y {z''} - R \delta\left(-D + t\right) = N {z'} \hspace{2em}z(0)=- V ,z'(0)=0" data-latex="-X z - Y {z''} - R \delta\left(-D + t\right) = N {z'} \hspace{2em}z(0)=- V ,z'(0)=0"/>
</li>
<li>
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -P {\left(R - V\right)} \mathrm{u}\left(-M + t\right) - P V + \frac{P z}{L} + {z'} \hspace{2em}z(0)= J" alt="0 = -P {\left(R - V\right)} \mathrm{u}\left(-M + t\right) - P V + \frac{P z}{L} + {z'} \hspace{2em}z(0)= J" title="0 = -P {\left(R - V\right)} \mathrm{u}\left(-M + t\right) - P V + \frac{P z}{L} + {z'} \hspace{2em}z(0)= J" data-latex="0 = -P {\left(R - V\right)} \mathrm{u}\left(-M + t\right) - P V + \frac{P z}{L} + {z'} \hspace{2em}z(0)= J"/>
</li>
<li>
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-J \mathrm{u}\left(-V + t\right) = -K L + L z + {z'} \hspace{2em}z(0)= D" alt="-J \mathrm{u}\left(-V + t\right) = -K L + L z + {z'} \hspace{2em}z(0)= D" title="-J \mathrm{u}\left(-V + t\right) = -K L + L z + {z'} \hspace{2em}z(0)= D" data-latex="-J \mathrm{u}\left(-V + t\right) = -K L + L z + {z'} \hspace{2em}z(0)= D"/>
</li>
</ol>
</div>
</mattextxml>
<mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>Zfinal.</strong>
</p>
<p> For each of the following scenarios, provide a reasonable IVP model. Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t" alt="t" title="t" data-latex="t"> represent time, let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?z" alt="z" title="z" data-latex="z"> represent the value given in the scenario, and use uppercase letters for positive constants. Then label each term and initial value of each IVP to describe what it represents from the scenario. </p>
<ul>
<li> An object is thrown upward from above the ground. Let z measure its altitude at a given time. Assume linear air resistance. </li>
<li> A glass of sweet tea is left outside on a summer day. After some time, ice is added to the glass, chilling the drink at a constant rate. Let z measure the temperature of the tea over time, assuming the outside temperature is kept constant. </li>
<li> A circuit includes a battery providing constant voltage, a resistor, and an inductor. Assume some initial current is flowing, and let z measure the current throughout this circuit at a given time. </li>
<li> A mass is attached to a spring. The mass is compressed inward from the spring's natural position, then released from rest. After some time, a hammer instantly strikes the mass inward. Assume the presence of friction. Let z measure the outward displacement of the mass from its natural position on the spring. </li>
</ul>
<p> Then, show that your model is equivalent to one of the following IVP models by relabeling constants and using algebra as needed. (It's possible that more than one IVP may be used for a scenario. If so, choose any of them.) </p>
<ol type="a">
<li>
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-J%20z%5E%7B2%7D%20-%20%7Bz'%7D%20=%20-B%20z%20+%20S%20%5Cdelta%5Cleft(-C%20+%20t%5Cright)%20%5Chspace%7B2em%7Dz(0)=%20Q" alt="-J z^{2} - {z'} = -B z + S \delta\left(-C + t\right) \hspace{2em}z(0)= Q" title="-J z^{2} - {z'} = -B z + S \delta\left(-C + t\right) \hspace{2em}z(0)= Q" data-latex="-J z^{2} - {z'} = -B z + S \delta\left(-C + t\right) \hspace{2em}z(0)= Q">
</li>
<li>
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?A%20%7Bz'%7D%20=%20-V%20Y%20-%20V%20%7Bz''%7D%20%5Chspace%7B2em%7Dz(0)=%20E%20,z'(0)=%20W" alt="A {z'} = -V Y - V {z''} \hspace{2em}z(0)= E ,z'(0)= W" title="A {z'} = -V Y - V {z''} \hspace{2em}z(0)= E ,z'(0)= W" data-latex="A {z'} = -V Y - V {z''} \hspace{2em}z(0)= E ,z'(0)= W">
</li>
<li>
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-A%20z%20-%20R%20%7Bz'%7D%20+%20L%20=%200%20%5Chspace%7B2em%7Dz(0)=%20W" alt="-A z - R {z'} + L = 0 \hspace{2em}z(0)= W" title="-A z - R {z'} + L = 0 \hspace{2em}z(0)= W" data-latex="-A z - R {z'} + L = 0 \hspace{2em}z(0)= W">
</li>
<li>
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-X%20z%20-%20Y%20%7Bz''%7D%20-%20R%20%5Cdelta%5Cleft(-D%20+%20t%5Cright)%20=%20N%20%7Bz'%7D%20%5Chspace%7B2em%7Dz(0)=-%20V%20,z'(0)=0" alt="-X z - Y {z''} - R \delta\left(-D + t\right) = N {z'} \hspace{2em}z(0)=- V ,z'(0)=0" title="-X z - Y {z''} - R \delta\left(-D + t\right) = N {z'} \hspace{2em}z(0)=- V ,z'(0)=0" data-latex="-X z - Y {z''} - R \delta\left(-D + t\right) = N {z'} \hspace{2em}z(0)=- V ,z'(0)=0">
</li>
<li>
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-P%20%7B%5Cleft(R%20-%20V%5Cright)%7D%20%5Cmathrm%7Bu%7D%5Cleft(-M%20+%20t%5Cright)%20-%20P%20V%20+%20%5Cfrac%7BP%20z%7D%7BL%7D%20+%20%7Bz'%7D%20%5Chspace%7B2em%7Dz(0)=%20J" alt="0 = -P {\left(R - V\right)} \mathrm{u}\left(-M + t\right) - P V + \frac{P z}{L} + {z'} \hspace{2em}z(0)= J" title="0 = -P {\left(R - V\right)} \mathrm{u}\left(-M + t\right) - P V + \frac{P z}{L} + {z'} \hspace{2em}z(0)= J" data-latex="0 = -P {\left(R - V\right)} \mathrm{u}\left(-M + t\right) - P V + \frac{P z}{L} + {z'} \hspace{2em}z(0)= J">
</li>
<li>
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-J%20%5Cmathrm%7Bu%7D%5Cleft(-V%20+%20t%5Cright)%20=%20-K%20L%20+%20L%20z%20+%20%7Bz'%7D%20%5Chspace%7B2em%7Dz(0)=%20D" alt="-J \mathrm{u}\left(-V + t\right) = -K L + L z + {z'} \hspace{2em}z(0)= D" title="-J \mathrm{u}\left(-V + t\right) = -K L + L z + {z'} \hspace{2em}z(0)= D" data-latex="-J \mathrm{u}\left(-V + t\right) = -K L + L z + {z'} \hspace{2em}z(0)= D">
</li>
</ol>
</div>
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<div class="exercise-answer">
<h4>Partial Answer:</h4>
<ul>
<li> An object is thrown upward from above the ground. Let z measure its altitude at a given time. Assume linear air resistance. <ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?A {z'} = -V Y - V {z''} \hspace{2em}z(0)= E ,z'(0)= W" alt="A {z'} = -V Y - V {z''} \hspace{2em}z(0)= E ,z'(0)= W" title="A {z'} = -V Y - V {z''} \hspace{2em}z(0)= E ,z'(0)= W" data-latex="A {z'} = -V Y - V {z''} \hspace{2em}z(0)= E ,z'(0)= W"/></li></ul></li>
<li> A solution of salt water is pumped into a tank of less salty water, while mixed water is pumped out at the same rate. After a certain amount of time, the concentration of salt flowing into the tank is instantly increased. Let z measure the mass of salt in the tank at a given time. <ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -P {\left(R - V\right)} \mathrm{u}\left(-M + t\right) - P V + \frac{P z}{L} + {z'} \hspace{2em}z(0)= J" alt="0 = -P {\left(R - V\right)} \mathrm{u}\left(-M + t\right) - P V + \frac{P z}{L} + {z'} \hspace{2em}z(0)= J" title="0 = -P {\left(R - V\right)} \mathrm{u}\left(-M + t\right) - P V + \frac{P z}{L} + {z'} \hspace{2em}z(0)= J" data-latex="0 = -P {\left(R - V\right)} \mathrm{u}\left(-M + t\right) - P V + \frac{P z}{L} + {z'} \hspace{2em}z(0)= J"/></li></ul></li>
<li> Suppose the population of a species generally follows the logistical model: births are linear based on population, and deaths are quadratic based on population. However, at some point in the model, a natural disaster instantly wipes out a fraction of the population. Let z measure the population of this species at a given time. <ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-J z^{2} - {z'} = -B z + S \delta\left(-C + t\right) \hspace{2em}z(0)= Q" alt="-J z^{2} - {z'} = -B z + S \delta\left(-C + t\right) \hspace{2em}z(0)= Q" title="-J z^{2} - {z'} = -B z + S \delta\left(-C + t\right) \hspace{2em}z(0)= Q" data-latex="-J z^{2} - {z'} = -B z + S \delta\left(-C + t\right) \hspace{2em}z(0)= Q"/></li></ul></li>
<li> A mass is attached to a spring. The mass is compressed inward from the spring's natural position, then released from rest. After some time, a hammer instantly strikes the mass inward. Assume the presence of friction. Let z measure the outward displacement of the mass from its natural position on the spring. <ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-X z - Y {z''} - R \delta\left(-D + t\right) = N {z'} \hspace{2em}z(0)=- V ,z'(0)=0" alt="-X z - Y {z''} - R \delta\left(-D + t\right) = N {z'} \hspace{2em}z(0)=- V ,z'(0)=0" title="-X z - Y {z''} - R \delta\left(-D + t\right) = N {z'} \hspace{2em}z(0)=- V ,z'(0)=0" data-latex="-X z - Y {z''} - R \delta\left(-D + t\right) = N {z'} \hspace{2em}z(0)=- V ,z'(0)=0"/></li></ul></li>
<li> A glass of sweet tea is left outside on a summer day. After some time, ice is added to the glass, chilling the drink at a constant rate. Let z measure the temperature of the tea over time, assuming the outside temperature is kept constant. <ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-J \mathrm{u}\left(-V + t\right) = -K L + L z + {z'} \hspace{2em}z(0)= D" alt="-J \mathrm{u}\left(-V + t\right) = -K L + L z + {z'} \hspace{2em}z(0)= D" title="-J \mathrm{u}\left(-V + t\right) = -K L + L z + {z'} \hspace{2em}z(0)= D" data-latex="-J \mathrm{u}\left(-V + t\right) = -K L + L z + {z'} \hspace{2em}z(0)= D"/></li></ul></li>
<li> A circuit includes a battery providing constant voltage, a resistor, and an inductor. Assume some initial current is flowing, and let z measure the current throughout this circuit at a given time. <ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-A z - R {z'} + L = 0 \hspace{2em}z(0)= W" alt="-A z - R {z'} + L = 0 \hspace{2em}z(0)= W" title="-A z - R {z'} + L = 0 \hspace{2em}z(0)= W" data-latex="-A z - R {z'} + L = 0 \hspace{2em}z(0)= W"/></li></ul></li>
</ul>
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<mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<ul>
<li> An object is thrown upward from above the ground. Let z measure its altitude at a given time. Assume linear air resistance. <ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?A%20%7Bz'%7D%20=%20-V%20Y%20-%20V%20%7Bz''%7D%20%5Chspace%7B2em%7Dz(0)=%20E%20,z'(0)=%20W" alt="A {z'} = -V Y - V {z''} \hspace{2em}z(0)= E ,z'(0)= W" title="A {z'} = -V Y - V {z''} \hspace{2em}z(0)= E ,z'(0)= W" data-latex="A {z'} = -V Y - V {z''} \hspace{2em}z(0)= E ,z'(0)= W"></li></ul>
</li>
<li> A solution of salt water is pumped into a tank of less salty water, while mixed water is pumped out at the same rate. After a certain amount of time, the concentration of salt flowing into the tank is instantly increased. Let z measure the mass of salt in the tank at a given time. <ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-P%20%7B%5Cleft(R%20-%20V%5Cright)%7D%20%5Cmathrm%7Bu%7D%5Cleft(-M%20+%20t%5Cright)%20-%20P%20V%20+%20%5Cfrac%7BP%20z%7D%7BL%7D%20+%20%7Bz'%7D%20%5Chspace%7B2em%7Dz(0)=%20J" alt="0 = -P {\left(R - V\right)} \mathrm{u}\left(-M + t\right) - P V + \frac{P z}{L} + {z'} \hspace{2em}z(0)= J" title="0 = -P {\left(R - V\right)} \mathrm{u}\left(-M + t\right) - P V + \frac{P z}{L} + {z'} \hspace{2em}z(0)= J" data-latex="0 = -P {\left(R - V\right)} \mathrm{u}\left(-M + t\right) - P V + \frac{P z}{L} + {z'} \hspace{2em}z(0)= J"></li></ul>
</li>
<li> Suppose the population of a species generally follows the logistical model: births are linear based on population, and deaths are quadratic based on population. However, at some point in the model, a natural disaster instantly wipes out a fraction of the population. Let z measure the population of this species at a given time. <ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-J%20z%5E%7B2%7D%20-%20%7Bz'%7D%20=%20-B%20z%20+%20S%20%5Cdelta%5Cleft(-C%20+%20t%5Cright)%20%5Chspace%7B2em%7Dz(0)=%20Q" alt="-J z^{2} - {z'} = -B z + S \delta\left(-C + t\right) \hspace{2em}z(0)= Q" title="-J z^{2} - {z'} = -B z + S \delta\left(-C + t\right) \hspace{2em}z(0)= Q" data-latex="-J z^{2} - {z'} = -B z + S \delta\left(-C + t\right) \hspace{2em}z(0)= Q"></li></ul>
</li>
<li> A mass is attached to a spring. The mass is compressed inward from the spring's natural position, then released from rest. After some time, a hammer instantly strikes the mass inward. Assume the presence of friction. Let z measure the outward displacement of the mass from its natural position on the spring. <ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-X%20z%20-%20Y%20%7Bz''%7D%20-%20R%20%5Cdelta%5Cleft(-D%20+%20t%5Cright)%20=%20N%20%7Bz'%7D%20%5Chspace%7B2em%7Dz(0)=-%20V%20,z'(0)=0" alt="-X z - Y {z''} - R \delta\left(-D + t\right) = N {z'} \hspace{2em}z(0)=- V ,z'(0)=0" title="-X z - Y {z''} - R \delta\left(-D + t\right) = N {z'} \hspace{2em}z(0)=- V ,z'(0)=0" data-latex="-X z - Y {z''} - R \delta\left(-D + t\right) = N {z'} \hspace{2em}z(0)=- V ,z'(0)=0"></li></ul>
</li>
<li> A glass of sweet tea is left outside on a summer day. After some time, ice is added to the glass, chilling the drink at a constant rate. Let z measure the temperature of the tea over time, assuming the outside temperature is kept constant. <ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-J%20%5Cmathrm%7Bu%7D%5Cleft(-V%20+%20t%5Cright)%20=%20-K%20L%20+%20L%20z%20+%20%7Bz'%7D%20%5Chspace%7B2em%7Dz(0)=%20D" alt="-J \mathrm{u}\left(-V + t\right) = -K L + L z + {z'} \hspace{2em}z(0)= D" title="-J \mathrm{u}\left(-V + t\right) = -K L + L z + {z'} \hspace{2em}z(0)= D" data-latex="-J \mathrm{u}\left(-V + t\right) = -K L + L z + {z'} \hspace{2em}z(0)= D"></li></ul>
</li>
<li> A circuit includes a battery providing constant voltage, a resistor, and an inductor. Assume some initial current is flowing, and let z measure the current throughout this circuit at a given time. <ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-A%20z%20-%20R%20%7Bz'%7D%20+%20L%20=%200%20%5Chspace%7B2em%7Dz(0)=%20W" alt="-A z - R {z'} + L = 0 \hspace{2em}z(0)= W" title="-A z - R {z'} + L = 0 \hspace{2em}z(0)= W" data-latex="-A z - R {z'} + L = 0 \hspace{2em}z(0)= W"></li></ul>
</li>
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