Throughout Chapter 2, we will develop shortcut derivative rules to help us bypass the limit definition and quickly compute \(f'(x)\) from a formula for \(f(x)\text{.}\) In Section 2.5, we stated the rule for power functions,
Use the limit definition of the derivative to estimate \(g'(0)\) by using small values of \(h\text{,}\) and compare the result to your visual estimate for the slope of the tangent line to \(y = g(x)\) at \(x = 0\) in (b).
Write at least one sentence that explains why it is reasonable to think that \(g'(x) = cg(x)\text{,}\) where \(c\) is a constant. In addition, calculate \(\ln(2)\text{,}\) and then discuss how this value, combined with your work above, reasonably suggests that \(g'(x) = 2^x \ln(2)\text{.}\)
The sine and cosine functions are among the most important functions in all of mathematics. Sometimes called the circular functions due to their definition on the unit circle (Section 1.6), these periodic functions play a key role in modeling repeating phenomena such as tidal elevations, the behavior of an oscillating mass attached to a spring, or weather patterns. Like polynomial and exponential functions, the sine and cosine functions are considered basic functions, ones that are often used in building more complicated functions. As such, we would like to know formulas for \(\frac{d}{dx} [\sin(x)]\) and \(\frac{d}{dx} [\cos(x)]\text{,}\) and the next two activities lead us to that end.
Consider the function \(f(x) = \sin(x)\text{,}\) which is graphed in Figure 2.6.2 below. Note carefully that the grid in the diagram does not have boxes that are \(1 \times 1\text{,}\) but rather approximately \(1.57 \times 1\text{,}\) as the horizontal scale of the grid is \(\pi/2\) units per box.
At each of \(x = -2\pi, -\frac{3\pi}{2}, -\pi, -\frac{\pi}{2}, 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi\text{,}\) use a straightedge to sketch an accurate tangent line to \(y = f(x)\text{.}\)
Use the limit definition of the derivative to estimate \(f'(0)\) by using small values of \(h\text{,}\) and compare the result to your visual estimate for the slope of the tangent line to \(y = f(x)\) at \(x = 0\) in (b). Using periodicity, what does this result suggest about \(f'(2\pi)\text{?}\) about \(f'(-2\pi)\text{?}\)
Consider the function \(g(x) = \cos(x)\text{,}\) which is graphed in Figure 2.6.3 below. Note carefully that the grid in the diagram does not have boxes that are \(1 \times 1\text{,}\) but rather approximately \(1.57 \times 1\text{,}\) as the horizontal scale of the grid is \(\pi/2\) units per box.
At each of \(x = -2\pi, -\frac{3\pi}{2}, -\pi, -\frac{\pi}{2}, 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi\text{,}\) use a straightedge to sketch an accurate tangent line to \(y = g(x)\text{.}\)
Use the limit definition of the derivative to estimate \(g'(\frac{\pi}{2})\) by using small values of \(h\text{,}\) and compare the result to your visual estimate for the slope of the tangent line to \(y = g(x)\) at \(x = \frac{\pi}{2}\) in (b). Using periodicity, what does this result suggest about \(g'(-\frac{3\pi}{2})\text{?}\) can symmetry on the graph help you estimate other slopes easily?
The results of the two preceding activities suggest that the sine and cosine functions not only have beautiful connections such as the identities \(\sin^2(x) + \cos^2(x) = 1\) and \(\cos(x - \frac{\pi}{2}) = \sin(x)\text{,}\) but that they are even further linked through calculus, as the derivative of each involves the other. The following rules summarize the results of the activities 1
These two rules may be formally proved using the limit definition of the derivative and the expansion identities for \(\sin(x+h)\) and \(\cos(x+h)\text{.}\)
We have now added the sine and cosine functions to our library of basic functions whose derivatives we know. The constant multiple and sum rules still hold, of course, as well as all of the inherent meaning of the derivative.
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.
For an exponential function \(f(x) = b^x\)\((b \gt 1)\text{,}\) the graph of \(f'(x)\) appears to be a scaled version of the original function. In particular, careful analysis of the graph of\(f(x) = 2^x\text{,}\) suggests that \(\frac{d}{dx}[2^x] = 2^x \ln(2)\text{,}\) which is a special case of the rule we stated in Section 2.5.
By carefully analyzing the graphs of \(y = \sin(x)\) and \(y = \cos(x)\text{,}\) and by using the limit definition of the derivative at select points, we found that \(\frac{d}{dx} [\sin(x)] = \cos(x)\) and \(\frac{d}{dx} [\cos(x)] = -\sin(x)\text{.}\)
We note that all previously encountered derivative rules still hold, but now may also be applied to functions involving the sine and cosine. All of the established meaning of the derivative applies to these trigonometric functions as well.
Suppose that \(V(t) = 24 \cdot 1.07^t + 6 \sin(t)\) represents the value of a person’s investment portfolio in thousands of dollars in year \(t\text{,}\) where \(t = 0\) corresponds to January 1, 2010.
At what instantaneous rate is the portfolio’s value changing on January 1, 2012? Include units on your answer.
On the interval \(0 \le t \le 20\text{,}\) use technology to graph the function \(V(t) = 24 \cdot 1.07^t + 6 \sin(t)\) and describe its behavior in the context of the problem. Then, compare the graphs of the functions \(A(t) = 24 \cdot 1.07^t\) and \(V(t) = 24 \cdot 1.07^t + 6 \sin(t)\text{,}\) as well as the graphs of their derivatives \(A'(t)\) and \(V'(t)\text{.}\) What is the impact of the term \(6 \sin(t)\) on the behavior of the function \(V(t)\text{?}\)
At the point where \(a = \frac{3\pi}{2}\text{,}\) does the tangent line to \(y = f(x)\) lie above the curve, below the curve, or neither? How can you answer this question without even graphing the function or the tangent line?
Let \(s(\theta)= \sin\left(\theta + \dfrac{\pi}{2} \right)\text{.}\)
Explain why none of the derivative rules we have developed up to this point tell us how to compute \(s'(\theta)\) (HINT: We will develop a rule to help us differentiate functions in this form in Section 2.8). Even so, give your best guess as to what you think \(s'(\theta)\) equals and explain your thought process.
