Question

Water temperature affects the growth rate of brook trout. The table shows the amount of weight gained by the brook trout after 24 days in various water temperatures:

Temperature (\(^\circ {\bf{C}}\))	15.5	17.7	20.0	22.4	24.4
Weight gained (g)	37.2	31.0	19.8	9.7	\( - {\bf{9}}.{\bf{8}}\)

If \(W\left( x \right)\) is the weight gain at temperaturex, construct a table of estimated values for \(W'\) and sketch its graph. What are the units for \(W'\left( x \right)\)?

Short answer

The table is shown below:

\(x\)	15.5	17.7	20.0	22.4	24.4
\(W'\left( x \right)\)	\( - 2.82\)	\( - 3.85\)	\( - 4.53\)	\( - 6.73\)	\( - 9.75\)

The graph is shown below:

The units for \(W'\left( x \right)\) are grams per degree.

Step-by-step solution

Find the values of \(W'\left( x \right)\)

Find the value of \(W'\left( x \right)\) using the definition of derivative:

\(W'\left( x \right) = \mathop {\lim }\limits_{h \to 0} \frac{{W\left( {x + h} \right) - W\left( x \right)}}{h}\)

For \(x = 15.5\),

\(\begin{array}{c}W'\left( {15.5} \right) &=& \frac{{W\left( {17.7} \right) - W\left( {15.5} \right)}}{h}\\ &=& \frac{{54 - 41}}{{2.2}}\\ &=& - 2.82\end{array}\)

For \(x = 17.7\), \(h = - 2.2\) and \(h = 2.3\). So, the final result is the average of the two results.

\(\begin{array}{c}W'\left( {17.7} \right) &=& \frac{{W\left( {20.0} \right) - W\left( {17.7} \right)}}{h}\\ &=& \frac{{19.8 - 31.0}}{{2.3}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {h = 2.3} \right)\\ &=& - 4.869\end{array}\)

And,

\(\begin{array}{c}W'\left( {17.7} \right) &=& \frac{{W\left( {15.5} \right) - W\left( {17.7} \right)}}{h}\\ &=& \frac{{37.2 - 31.0}}{{\left( { - 2.2} \right)}}\\ &=& - 2.82\end{array}\)

So, the value of \(W'\left( {17.7} \right)\) is \( - 3.87\).

The table below represents the values of \(W'\left( x \right)\).

\(x\)	15.5	17.7	20.0	22.4	24.4
\(W'\left( x \right)\)	\( - 2.82\)	\( - 3.85\)	\( - 4.53\)	\( - 6.73\)	\( - 9.75\)

 Step 2: Plot the graph of \(W'\left( x \right)\)

The figure below represents the graph of \(W'\left( x \right)\) with respect to x.

Thus, the graph is obtained. The units for \(W'\left( x \right)\) are grams per degree.