Activity 2.2.4.
Respond to each of the following prompts. Where a derivative is requested, be sure to label the derivative function with its name using proper notation.
(a)
Determine the derivative of \(h(t) = 3\cos(t) - 4\sin(t)\text{.}\)
(b)
Find the exact slope of the tangent line to \(y = f(x) = 2x + \frac{\sin(x)}{2}\) at the point where \(x = \frac{\pi}{6}\text{.}\)
(c)
Find the equation of the tangent line to \(y = g(x) = x^2 + 2\cos(x)\) at the point where \(x = \frac{\pi}{2}\text{.}\)
(d)
Determine the derivative of \(p(z) = z^4 + 4^z + 4\cos(z) - \sin(\frac{\pi}{2})\text{.}\)
(e)
The function \(P(t) = 24 + 8\sin(t)\) represents a population of a particular kind of animal that lives on a small island, where \(P\) is measured in hundreds and \(t\) is measured in decades since January 1, 2010. What is the instantaneous rate of change of \(P\) on January 1, 2030? What are the units of this quantity? Write a sentence in everyday language that explains how the population is behaving at this point in time.
