Activity 24.
Each of the following questions asks you to use derivatives to answer key questions about functions. Be sure to think carefully about each question and to use proper notation in your responses.
(a)
Find the slope of the tangent line to \(h(z) = \sqrt{z} + \frac{1}{z}\) at the point where \(z = 4\text{.}\)
Hint.
How would \(h'(z)\) help you answer the question?
Answer.
\(h'(4) = \frac{3}{16}\text{.}\)
Solution.
Note that since \(h(z) = z^{1/2} + z^{-1}\text{,}\) we have \(h'(z) = \frac{1}{2}z^{-1/2} - z^{-2}\text{.}\) Thus, \(h'(4) = \frac{1}{2(4)^{1/2}} - \frac{1}{4^2} = \frac{1}{4} - \frac{1}{16} = \frac{3}{16}\text{.}\) Thus, the slope of the tangent line to \(h(z)\) at the point where \(z = 4\) is \(\frac{3}{16}\text{.}\)
(b)
A population of cells is growing in such a way that its total number in millions is given by the function \(P(t) = 2(1.37)^t + 32\text{,}\) where \(t\) is measured in days.
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Determine the instantaneous rate at which the population is growing on day 4, and include units on your answer.
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Is the population growing at an increasing rate or growing at a decreasing rate on day 4? Explain.
Hint.
Think about finding both \(P'(t)\) and \(P''(t)\text{.}\)
Answer.
(i.)\(P'(4) = 2(1.37)^4 \ln(1.37) \approx 2.218\) million cells per day; (ii.) the population is growing at an increasing rate.
Solution.
For (i.), note that \(P'(t) = 2(1.37)^t \ln(1.37)\text{,}\) and therefore \(P'(4) = 2(1.37)^4 \ln(1.37) \approx 2.218\) million cells per day. For (ii.), we can compute \(P''(t)\) and find that \(P''(t) = 2(1.37)^t \ln(1.37) \ln(1.37)\) and therefore find that \(P''(4) \approx 0.69825\text{.}\) Since \(P''(4)\) is positive, this tells us that the instantaneous rate of change \(P'(t)\) is increasing at \(t = 4\text{,}\) and thus the population is growing at an increasing rate.
(c)
Find an equation for the tangent line to the curve \(p(a) = 3a^4 - 2a^3 + 7a^2 - a + 12\) at the point where \(a=-1\text{.}\)
Hint.
What two important pieces of information do you need to know to determine the equation of a line?
Answer.
\(y - 25 = -33(a+1)\text{.}\)
Solution.
Given \(p(a) = 3a^4 - 2a^3 + 7a^2 - a + 12\text{,}\) first observe that \(p(-1) = 3 + 2 + 7 + 1 + 12 = 25\text{,}\) so the tangent line will pass through \((-1,25)\text{.}\) Further, since \(p'(a) = 12a^3 - 6a^2 + 14a - 1\text{,}\) we have \(p'(-1) = -12 - 6 - 14 - 1 = -33\text{,}\) which is the slope of the tangent line. The equation of the tangent line is therefore \(y - 25 = -33(a+1)\text{.}\)
(d)
What is the difference between being asked to find the slope of the tangent line (asked in (a)) and the equation of the tangent line (asked in (c))?
Hint.
What information do you find in both (a) and (c)?
Answer.
The slope is a number, while the equation is, well, an equation.
Solution.
The biggest difference is that (a) asks for the slope of the tangent line, while (c) asks for the equation of the tangent line. The latter requires the slope of the tangent line, but the slope and equation are two different entities.
