Functions can be used to model real life scenarios
- The reasonable domain is
. - The average rate of change from t = 0 to 2 is 20 persons per week
- The instantaneous rate of change is
. - The slope of the tangent line at point (2,V(20) is 10
- The rate of infection at the maximum point is 8.79 people per week
The function is given as:
![\mathbf{V(t) = -t^3 + t^2 + 12t}](https://tex.z-dn.net/?f=%5Cmathbf%7BV%28t%29%20%3D%20-t%5E3%20%2B%20t%5E2%20%2B%2012t%7D)
<u>(a) Sketch V(t)</u>
See attachment for the graph of ![\mathbf{V(t) = -t^3 + t^2 + 12t}](https://tex.z-dn.net/?f=%5Cmathbf%7BV%28t%29%20%3D%20-t%5E3%20%2B%20t%5E2%20%2B%2012t%7D)
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<u>(b) The reasonable domain</u>
t represents the number of weeks.
This means that: <em>t cannot be negative.</em>
So, the reasonable domain is: ![\mathbf{[0,\infty)}](https://tex.z-dn.net/?f=%5Cmathbf%7B%5B0%2C%5Cinfty%29%7D)
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<u>(c) Average rate of change from t = 0 to 2</u>
This is calculated as:
![\mathbf{m = \frac{V(a) - V(b)}{a - b}}](https://tex.z-dn.net/?f=%5Cmathbf%7Bm%20%3D%20%5Cfrac%7BV%28a%29%20-%20V%28b%29%7D%7Ba%20-%20b%7D%7D)
So, we have:
![\mathbf{m = \frac{V(2) - V(0)}{2 - 0}}](https://tex.z-dn.net/?f=%5Cmathbf%7Bm%20%3D%20%5Cfrac%7BV%282%29%20-%20V%280%29%7D%7B2%20-%200%7D%7D)
![\mathbf{m = \frac{V(2) - V(0)}{2}}](https://tex.z-dn.net/?f=%5Cmathbf%7Bm%20%3D%20%5Cfrac%7BV%282%29%20-%20V%280%29%7D%7B2%7D%7D)
Calculate <em>V(2) and V(0)</em>
![\mathbf{V(2) = (-2)^3 + (2)^2 + 12 \times 2 = 20}](https://tex.z-dn.net/?f=%5Cmathbf%7BV%282%29%20%3D%20%28-2%29%5E3%20%2B%20%282%29%5E2%20%2B%2012%20%5Ctimes%202%20%3D%2020%7D)
![\mathbf{V(0) = (0)^3 + (0)^2 + 12 \times 0 = 0}](https://tex.z-dn.net/?f=%5Cmathbf%7BV%280%29%20%3D%20%280%29%5E3%20%2B%20%280%29%5E2%20%2B%2012%20%5Ctimes%200%20%3D%200%7D)
So, we have:
![\mathbf{m = \frac{20 - 0}{2}}](https://tex.z-dn.net/?f=%5Cmathbf%7Bm%20%3D%20%5Cfrac%7B20%20-%200%7D%7B2%7D%7D)
![\mathbf{m = \frac{20}{2}}](https://tex.z-dn.net/?f=%5Cmathbf%7Bm%20%3D%20%5Cfrac%7B20%7D%7B2%7D%7D)
![\mathbf{m = 10}](https://tex.z-dn.net/?f=%5Cmathbf%7Bm%20%3D%2010%7D)
Hence, the average rate of change from t = 0 to 2 is 20
<u>(d) The instantaneous rate of change using limits</u>
![\mathbf{V(t) = -t^3 + t^2 + 12t}](https://tex.z-dn.net/?f=%5Cmathbf%7BV%28t%29%20%3D%20-t%5E3%20%2B%20t%5E2%20%2B%2012t%7D)
The instantaneous rate of change is calculated as:
![\mathbf{V'(t) = \lim_{h \to \infty} \frac{V(t + h) - V(t)}{h}}](https://tex.z-dn.net/?f=%5Cmathbf%7BV%27%28t%29%20%3D%20%5Clim_%7Bh%20%5Cto%20%5Cinfty%7D%20%5Cfrac%7BV%28t%20%2B%20h%29%20-%20V%28t%29%7D%7Bh%7D%7D)
So, we have:
![\mathbf{V(t + h) = (-(t + h))^3 + (t + h)^2 + 12(t + h)}](https://tex.z-dn.net/?f=%5Cmathbf%7BV%28t%20%2B%20h%29%20%3D%20%28-%28t%20%2B%20h%29%29%5E3%20%2B%20%28t%20%2B%20h%29%5E2%20%2B%2012%28t%20%2B%20h%29%7D)
![\mathbf{V(t + h) = (-t - h)^3 + (t + h)^2 + 12(t + h)}](https://tex.z-dn.net/?f=%5Cmathbf%7BV%28t%20%2B%20h%29%20%3D%20%28-t%20-%20h%29%5E3%20%2B%20%28t%20%2B%20h%29%5E2%20%2B%2012%28t%20%2B%20h%29%7D)
Expand
![\mathbf{V(t + h) = (-t)^3 +3(-t)^2(-h) +3(-t)(-h)^2 + (-h)^3 + t^2 + 2th+ h^2 + 12t + 12h}](https://tex.z-dn.net/?f=%5Cmathbf%7BV%28t%20%2B%20h%29%20%3D%20%28-t%29%5E3%20%2B3%28-t%29%5E2%28-h%29%20%2B3%28-t%29%28-h%29%5E2%20%2B%20%28-h%29%5E3%20%2B%20t%5E2%20%2B%202th%2B%20h%5E2%20%2B%2012t%20%2B%2012h%7D)
![\mathbf{V(t + h) = -t^3 -3t^2h -3th^2 - h^3 + t^2 + 2th+ h^2 + 12t + 12h}](https://tex.z-dn.net/?f=%5Cmathbf%7BV%28t%20%2B%20h%29%20%3D%20-t%5E3%20-3t%5E2h%20-3th%5E2%20-%20h%5E3%20%2B%20t%5E2%20%2B%202th%2B%20h%5E2%20%2B%2012t%20%2B%2012h%7D)
Subtract V(t) from both sides
![\mathbf{V(t + h) - V(t)= -t^3 -3t^2h -3th^2 - h^3 + t^2 + 2th+ h^2 + 12t + 12h - V(t)}](https://tex.z-dn.net/?f=%5Cmathbf%7BV%28t%20%2B%20h%29%20-%20V%28t%29%3D%20-t%5E3%20-3t%5E2h%20-3th%5E2%20-%20h%5E3%20%2B%20t%5E2%20%2B%202th%2B%20h%5E2%20%2B%2012t%20%2B%2012h%20-%20V%28t%29%7D)
Substitute ![\mathbf{V(t) = -t^3 + t^2 + 12t}](https://tex.z-dn.net/?f=%5Cmathbf%7BV%28t%29%20%3D%20-t%5E3%20%2B%20t%5E2%20%2B%2012t%7D)
![\mathbf{V(t + h) - V(t)= -t^3 -3t^2h -3th^2 - h^3 + t^2 + 2th+ h^2 + 12t + 12h +t^3 - t^2 - 12t}](https://tex.z-dn.net/?f=%5Cmathbf%7BV%28t%20%2B%20h%29%20-%20V%28t%29%3D%20-t%5E3%20-3t%5E2h%20-3th%5E2%20-%20h%5E3%20%2B%20t%5E2%20%2B%202th%2B%20h%5E2%20%2B%2012t%20%2B%2012h%20%2Bt%5E3%20-%20t%5E2%20-%2012t%7D)
Cancel out common terms
![\mathbf{V(t + h) - V(t)= -3t^2h -3th^2 - h^3 + 2th+ h^2 + 12h}](https://tex.z-dn.net/?f=%5Cmathbf%7BV%28t%20%2B%20h%29%20-%20V%28t%29%3D%20-3t%5E2h%20-3th%5E2%20-%20h%5E3%20%20%2B%202th%2B%20h%5E2%20%20%2B%2012h%7D)
becomes
![\mathbf{V'(t) = \lim_{h \to \infty} \frac{ -3t^2h -3th^2 - h^3 + 2th+ h^2 + 12h}{h}}](https://tex.z-dn.net/?f=%5Cmathbf%7BV%27%28t%29%20%3D%20%5Clim_%7Bh%20%5Cto%20%5Cinfty%7D%20%5Cfrac%7B%20-3t%5E2h%20-3th%5E2%20-%20h%5E3%20%20%2B%202th%2B%20h%5E2%20%20%2B%2012h%7D%7Bh%7D%7D)
![\mathbf{V'(t) = \lim_{h \to \infty} -3t^2 -3th - h^2 + 2t+ h + 12}](https://tex.z-dn.net/?f=%5Cmathbf%7BV%27%28t%29%20%3D%20%5Clim_%7Bh%20%5Cto%20%5Cinfty%7D%20-3t%5E2%20-3th%20-%20h%5E2%20%20%2B%202t%2B%20h%20%20%2B%2012%7D)
Limit h to 0
![\mathbf{V'(t) = -3t^2 -3t\times 0 - 0^2 + 2t+ 0 + 12}](https://tex.z-dn.net/?f=%5Cmathbf%7BV%27%28t%29%20%3D%20-3t%5E2%20-3t%5Ctimes%200%20-%200%5E2%20%20%2B%202t%2B%200%20%20%2B%2012%7D)
![\mathbf{V'(t) = -3t^2 + 2t + 12}](https://tex.z-dn.net/?f=%5Cmathbf%7BV%27%28t%29%20%3D%20-3t%5E2%20%2B%202t%20%2B%2012%7D)
<u>(e) V(2) and V'(2)</u>
Substitute 2 for t in V(t) and V'(t)
So, we have:
![\mathbf{V(2) = (-2)^3 + (2)^2 + 12 \times 2 = 20}](https://tex.z-dn.net/?f=%5Cmathbf%7BV%282%29%20%3D%20%28-2%29%5E3%20%2B%20%282%29%5E2%20%2B%2012%20%5Ctimes%202%20%3D%2020%7D)
![\mathbf{V'(2) = -3 \times 2^2 + 2 \times 2 + 12 = 4}](https://tex.z-dn.net/?f=%5Cmathbf%7BV%27%282%29%20%3D%20-3%20%5Ctimes%202%5E2%20%2B%202%20%5Ctimes%202%20%2B%2012%20%3D%204%7D)
<em>Interpretation</em>
V(2) means that, 20 people were infected after 2 weeks of the virus spread
V'(2) means that, the rate of infection of the virus after 2 weeks is 4 people per week
<u>(f) Sketch the tangent line at (2,V(2))</u>
See attachment for the tangent line
The slope of this line is:
![\mathbf{m = \frac{V(2)}{2}}](https://tex.z-dn.net/?f=%5Cmathbf%7Bm%20%3D%20%5Cfrac%7BV%282%29%7D%7B2%7D%7D)
![\mathbf{m = \frac{20}{2}}](https://tex.z-dn.net/?f=%5Cmathbf%7Bm%20%3D%20%5Cfrac%7B20%7D%7B2%7D%7D)
![\mathbf{m = 10}](https://tex.z-dn.net/?f=%5Cmathbf%7Bm%20%3D%2010%7D)
The slope of the tangent line is 10
<u>(g) Estimate V(2.1)</u>
The <em>value of 2.1 </em>is
![\mathbf{V(2.1) = (-2.1)^3 + (2.1)^2 + 12 \times 2.1}](https://tex.z-dn.net/?f=%5Cmathbf%7BV%282.1%29%20%3D%20%28-2.1%29%5E3%20%2B%20%282.1%29%5E2%20%2B%2012%20%5Ctimes%202.1%7D)
![\mathbf{V(2.1) = 20.35}](https://tex.z-dn.net/?f=%5Cmathbf%7BV%282.1%29%20%3D%2020.35%7D)
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<u>(h) The maximum number of people infected at the same time</u>
Using the graph, the maximum point on the graph is:
![\mathbf{(t,V(t) = (2.361,20.745)}](https://tex.z-dn.net/?f=%5Cmathbf%7B%28t%2CV%28t%29%20%3D%20%282.361%2C20.745%29%7D)
This means that:
The maximum number of people infected at the same time is approximately 21.
The rate of infection at this point is:
![\mathbf{m = \frac{V(t)}{t}}](https://tex.z-dn.net/?f=%5Cmathbf%7Bm%20%3D%20%5Cfrac%7BV%28t%29%7D%7Bt%7D%7D)
![\mathbf{m = \frac{20.745}{2.361}}](https://tex.z-dn.net/?f=%5Cmathbf%7Bm%20%3D%20%5Cfrac%7B20.745%7D%7B2.361%7D%7D)
![\mathbf{m = 8.79}](https://tex.z-dn.net/?f=%5Cmathbf%7Bm%20%3D%208.79%7D)
The rate of infection is <em>8.79 people per week</em>
Read more about graphs and functions at:
brainly.com/question/18806107