Section Week 5 (9/22 - 9/26)
This week we finish section 2.1 and 2.2. This will give us lots of functions that we can take the derivative of.
Handout Monday 9/22
Last time we learned how to take derivatives of power functions, like \(f(x) = x^4\text{,}\) but also \(f(x) = \frac{1}{x^2} = x^{-2}\) and \(f(x) = \sqrt[3]{x} = x^{-1/3}\text{.}\) The rule is that for any constant \(n\text{,}\) if
\begin{equation*}
f(x) = x^n
\end{equation*}
then
\begin{equation*}
f'(x) = nx^{n-1}\text{.}
\end{equation*}
We also looked at how to combine functions: \(f(x) = g_1(x) +g_2(x)\) has derivative \(f'(x) = g_1'(x) + g_2'(x)\text{.}\) And if \(f(x)\) is a constant multiple of another function, say \(f(x) = 5g(x)\text{,}\) then \(f'(x) = 5g'(x)\text{.}\) For example, \(f(x) = 7x^3\) will have derivative \(f'(x) = 7\cdot(3x^2) = 21x^2\text{.}\)
Today, let’s practice these derivatives. Write down a bunch of functions that we can take derivatives of and do it. Include some that need to be simplified first.
What if we have \(f(x) = 2^x\text{?}\) Is the derivative \(f'(x) = x2^{x-1}\text{?}\) No! We can verify this by graphing both. So actually we do want to graph these exponential functions and numerically compute the derivatives. It turns out that for any constant \(a\text{,}\) the function \(f(x) = a^x\) has derivative \(f'(x) = a^x \ln(a)\text{.}\)
In other words, the derivative if an exponential function is proportional to itself, with constant of proportionality \(\ln(a)\) where \(a\) is the base. Note that since \(\ln(e) = 1\text{,}\) this says that \(\frac{d}{dx}e^x = e^x\text{.}\)
Handout Tuesday 9/23
The goal of section 2.2 is to develop rules for the derivatives of \(\sin(x)\) and \(\cos(x)\text{.}\) We could think of these rules as arbitrary, but to help us remember and make sense of them, it will be useful to briefly review some trigonometry.
Perhaps you learned trigonometry as a method for finding the lengths of legs of right triangles. What does \(\sin(x)\) and \(\cos(x)\) tell you there?
We want to extend this meaning to a larger domain of angles, not just between 0 and 90 degrees.
Now we want the triangle with base on the \(x\)-axis and angle at the origin to still obey the usual relationships for \(\sin(\theta)\) and \(\cos(\theta)\text{.}\) So this means that \(\sin(\theta)\) will be the \(y\)-value of the circle, and \(\cos(\theta)\) is the \(x\)-value.
Now continuing this, it makes sense to ask for \(\sin(3\pi/4)\text{.}\) We just need to know what the \(y\) value is on the circle at that point.
Notice that we are using a circle with radius 1 here, so while \(\sin(\theta) = y/r\text{,}\) we really get \(y\text{.}\)
Handout Wednesday 9/24
Let’s pick up where we left off with our review of basic trigonometry.
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First we want to switch to using radians. What do those means? 1 radian is the angle that gives you an arc of the circle of length equal to the radius of the circle.
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The key thing though is that the circumference of a circle is \(\2\pi r\text{,}\) so the full angle of a circle gives an arc length of \(2\pi\) times the radius. Thus a full circle (\(360^\circ\)) is \(2\pi\) radians.
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From this, we can divide a circle into smaller chunks. Half a circle is \(\pi\) radians, etc.
Now we can think of both \(\sin(x)\) and \(\cos(x)\) as being functions of \(x\text{.}\) But here \(x\) is the angle and \(y = f(x)\) is the value of the function (which is an \(x\) or \(y\) value on the unit circle). Let’s graph those functions.
Now what about the rate of change of those functions? Think about what increasing from 0 to 1 radian does to the \(x\) and \(y\) values on the unit circle!
We can also look at this graphically, by considering slopes of the sine and cosine functions.
Now let’s see how to combine all the rules we have so far. Work through the activity.
Handout Friday 9/26
We used today as an opportunity to catch up on learning target quizzes. No new content was discussed.
