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Problem 1
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<p>
To check your answers put them in the appropriate box and click the &#8216;Check' button. The checker can do arithmetic. So, 3^2, [mathjaxinline]5/3[/mathjaxinline] and [mathjaxinline]3*4[/mathjaxinline] are okay. </p>
<p>
Unless otherwise stated the checker expects numerical answers to 3 significant digits. </p>
<p>
For formulas you should use Matlab notation. Remember to use a * for multiplication, i.e. [mathjaxinline]3*x[/mathjaxinline] <b class="bf">NOT </b> [mathjaxinline]3x[/mathjaxinline]. </p>
<p>
The checker knows many of the standard functions. Use exp(x) and ln(x) for the exponential and natural log functions. </p>
<p>
The formula checker expects a specific letter for the variables, so be sure to use the one specified. </p>
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<b class="bf">1(a)</b>
<em>No answer checker given.</em>
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<p><b class="bf">1(b)</b> Using [mathjaxinline]m[/mathjaxinline], [mathjaxinline]b[/mathjaxinline], [mathjaxinline]t[/mathjaxinline], give the formula for the interest rate [mathjaxinline]a(t)[/mathjaxinline], </p>
<p>
<p style="display:inline">[mathjaxinline]a(t) =[/mathjaxinline]</p>
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Problem 2a
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<p><b class="bf">2(a)</b> Your solution should be in three cases. Give each case using the variables [mathjaxinline]t[/mathjaxinline] and [mathjaxinline]k[/mathjaxinline]. </p>
<p>
<p style="display:inline">For [mathjaxinline]0&lt;t&lt;0.5[/mathjaxinline], [mathjaxinline]x(t) =[/mathjaxinline]</p>
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<p>
<p style="display:inline">For [mathjaxinline]0.5&lt;t&lt;1[/mathjaxinline], [mathjaxinline]x(t) =[/mathjaxinline]</p>
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<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<p id="answer_ps1-2a-check_3_1" class="answer"/>
<div id="display_ps1-2a-check_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps1-2a-check_3_1_dynamath" name="input_ps1-2a-check_3_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<p style="display:inline">For [mathjaxinline]1&lt;t[/mathjaxinline], [mathjaxinline]x(t) =[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_ps1-2a-check_4_1" class="text-input-dynamath capa_inputtype inline textline">
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<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<p id="answer_ps1-2a-check_4_1" class="answer"/>
<div id="display_ps1-2a-check_4_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps1-2a-check_4_1_dynamath" name="input_ps1-2a-check_4_1_dynamath"/>
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<b class="bf">2(b)</b>
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Problem 3a
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<b class="bf">3a(i)</b>
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<p><b class="bf">3a(ii)</b> Give your answer correct to 3 significant digits. (On your paper, don't forget to supply the units.) </p>
<p>
<p style="display:inline">[mathjaxinline]k =[/mathjaxinline]</p>
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<p><b class="bf">3a(iii)</b> Give your solution, use the symbols <tt class="tt">t, k, t_0, x_0</tt>. </p>
<p>
(Even though you have a numerical value for [mathjaxinline]k[/mathjaxinline], use the symbol so the checker does not have to worry about rounding errors.) </p>
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<p style="display:inline">[mathjaxinline]x(t) =[/mathjaxinline]</p>
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Problem 3b
</h3>
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<p>
<b class="bf">3b(i)</b>
<em>No answer checker given.</em>
</p>
<p><b class="bf">3b(ii)</b> Give your solution using the symbols <tt class="tt">t, k, r, t_0, x_0</tt>. </p>
<p>
(Even though you have a numerical value for [mathjaxinline]k[/mathjaxinline], use the symbol so the checker does not have to worry about rounding errors.) </p>
<p>
<p style="display:inline">[mathjaxinline]x(t) =[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps1-3b-check_2_1" class="text-input-dynamath capa_inputtype inline textline">
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Problem 3c
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<p><b class="bf">3(c)</b> Give your answer for the number of lemmings as a whole number. </p>
<p>
<p style="display:inline"/>
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<textarea style="display:none" id="input_ps1-3c-check_2_1_dynamath" name="input_ps1-3c-check_2_1_dynamath"/>
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Problem 4a
</h3>
<div class="problem-progress" id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps1-4a-check-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bf">4a(i)</b> Give the DE for the given family. The function given should be in terms of [mathjaxinline]x[/mathjaxinline] and [mathjaxinline]y[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]\frac{dy}{dx} =[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps1-4a-check_2_1" class="text-input-dynamath capa_inputtype inline textline">
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<input type="text" name="input_ps1-4a-check_2_1" id="input_ps1-4a-check_2_1" aria-describedby="status_ps1-4a-check_2_1" value="" class="math" size="50"/>
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<div id="display_ps1-4a-check_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps1-4a-check_2_1_dynamath" name="input_ps1-4a-check_2_1_dynamath"/>
</div>
</div></div>
</p>
<p><b class="bf">4a(ii)</b> Give the DE for the orthogonal family. The function given should be in terms of [mathjaxinline]x[/mathjaxinline] and [mathjaxinline]y[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]\frac{dy}{dx} =[/mathjaxinline]</p>
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<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps1-4a-check_3_1" class="answer"/>
<div id="display_ps1-4a-check_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps1-4a-check_3_1_dynamath" name="input_ps1-4a-check_3_1_dynamath"/>
</div>
</div></div>
</p>
<p>
The problem asks for the orthogonal family. It's tricky to check this answer automatically because different people might have different signs on the constant of integration. So, find the curve in the orthogonal family that goes through the point (0,2). Write [mathjaxinline]y[/mathjaxinline] as a function of [mathjaxinline]x[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]y(x) =[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_ps1-4a-check_4_1" class="text-input-dynamath capa_inputtype inline textline">
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<div id="display_ps1-4a-check_4_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps1-4a-check_4_1_dynamath" name="input_ps1-4a-check_4_1_dynamath"/>
</div>
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</p>
<p>
<b class="bf">4a(iii)</b>
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Problem 4b
</h3>
<div class="problem-progress" id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps1-4b-check-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bf">4b(i)</b> Give the DE for the given family. The function given should be in terms of [mathjaxinline]x[/mathjaxinline] and [mathjaxinline]y[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]\frac{dy}{dx} =[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps1-4b-check_2_1" class="text-input-dynamath capa_inputtype inline textline">
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<p><b class="bf">4b(ii)</b> Give the DE for the orthogonal family. The function given should be in terms of [mathjaxinline]x[/mathjaxinline] and [mathjaxinline]y[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]\frac{dy}{dx} =[/mathjaxinline]</p>
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The problem asks for the orthogonal family. It's tricky to check this answer automatically because different people might have different signs on the constant of integration. So, find the curve in the orthogonal family that goes through the point (1,2). Write [mathjaxinline]y[/mathjaxinline] as a function of [mathjaxinline]x[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]y =[/mathjaxinline]</p>
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<b class="bf">4b(iii)</b>
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