Section Week 9 (10/20-10/25)
This week we will begin looking at applications of the derivative, starting with Related Rates, discussed in Section 3.1.
Handout Monday 10/20
Calculus gives us tools to understand how quantities change. We have seen this appear primarily when we are given a function that describes the values of a quantity in terms of some independent variable. But there are also interesting examples of multiple quantities that change in tandem, both functions of some common variable. These related quantities with have related rates of change.
Let’s start with a classic example: a ladder is sliding down a wall. Previously, we assumed that the bottom of the ladder was moving away from the wall at a constant rate, and described the top of the ladder’s speed in terms of the distance from the wall to the base of the ladder.
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Now let’s assume that we don’t know about the speed of the bottom of the ladder. We just want to describe the relationship between the speed of the top of the ladder and the speed of the bottom of the ladder, both in inches per minute (so in terms of time, not height in terms of distance).
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The relationship between the distance from the wall to the base of the ladder \(x\text{,}\) and the height of the top of the ladder \(y\text{,}\) can be described using the Pythagorean Theorem.
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We have \(x^2 + y^2 = l^2\text{,}\) where \(l\) is the length of the ladder. Both \(x\) and \(y\) are functions of time, \(t\text{.}\) The length of the ladder is presumably constant, let’s say 10 ft.
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Just like we did with implicit differentiation, we will take the derivative of both sides of this relationship, but this time with respect to \(t\) (instead of with respect to \(x\)).
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We get \(2x\frac{dx}{dt} + 2y \frac{dy}{dt} = 0\text{.}\)
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We can now answer lots of questions, provided we know two or three of these four quantities/rates. For example, how fast is the top of the ladder moving if when the base of the ladder is 6 feet from the wall, it is moving at a rate of 3 feet per second? Note that we can determine the value of \(y\) from knowing \(x\text{,}\) using the original (static) relationship (we get \(y = 8\)). Since we also know \(\frac{dx}{dt}\text{,}\) we can substitute three values and get the fourth.
Return to the handout from last class with the other related rates scenarios. Try using implicit differentiation to find the relationship between their rates. Do as many as there is time for (start from the last question, since we didn’t ever write down the original static relationship).
Subsection Wednesday 10/22
Today in class we will practice with related rates questions by completing the Related Rates Practice activity.
Handout Friday 10/24
Today we will finish the related rates activity from last time and then start exploring section 3.2 on L’Hôpital’s Rule.
Our next application of derivatives takes us back to evaluating limits. Suppose we want to evaluate the limit
\begin{equation*}
\lim_{x \to 2}\frac{x^3 - x^2 + 4x - 12}{x^2 - 4}\text{.}
\end{equation*}
Note that if we try to plug in \(x = 2\text{,}\) we get \(\frac{0}{0}\text{.}\) This is called an indeterminate form.
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Try graphing this function on Desmos to make a guess at the limit value.
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Now just for fun, let’s take the derivative, not of the entire function, but of the numerator and denominator separately.
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What is \(\lim_{x \to 2}\frac{3x^2 - 2x + 4}{2x}\text{?}\) Now we can just plug in \(x = 2\) and get \(\frac{12}{4} = 3\text{.}\) That is exactly what the original limit it appeared to be.
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Why does this work?
Remember that a differentiable function can be approximated by its tangent line, at least very close to the point at which it is tangent. This suggests that if we replace both the numerator and denominator with their tangent lines, the ratios will be the same.
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What is the equation of the tangent line for the numerator and denominator in this example?
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Notice that the \((x-2)\) terms cancel.
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This means we are only left with the slopes. That means that the ratio of the functions is well-approximated by the ratio of their slopes.
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This doesn’t just happen for this example though. What do we get for arbitrary \(\frac{f(x)}{g(x)}\text{?}\) Write down the general tangent line approximation for each. Notice that the \(f(a)\) and \(g(a)\) will both be zero, which is exactly why this only works when you start with an indeterminate form.
