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Calculus I: Activities

Mark A. Fitch

Section 2.3 Derivative as a Function

We developed the derivative as the limit on the slope (of secant lines). This defined a sort of slope at one point. Here we consider what that slope tells us, practice using it, and realize what that implies about how we treat derivatives.
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Subsection 2.3.1 Derivatives and Tangent Slopes

Consider Figure 2.2.2. The line through points A and B intersects the curve in at least those two points. But as A approaches B those intersections are close together. In the limit, points A and B overlap (become one point). That means the line intersects this curve at only one point. This is a common description of a tangent line. They are sometimes known as osculating curves. Osculate is Latin for barely touch (also used to mean “kiss”). In context the line barely touches the curve.
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Another perspective on this tangent line comes from considering the direction a person travelling along the curve is facing. Suppose a person is at point A looking toward point B. As they approach B they would not be looking across the curve but just along the curve. This defines the direction of the curve. It is also the direction the person would go if they slipped (direction they are headed at that moment).
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Example 2.3.1.

Consider the function \(f(x)=x^2\text{.}\) We will write an equation for the line tangent to \(f(x)\) at \(x=3\text{.}\) To write the equation of a line we need a point and the slope at that point.
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Because of the meaning of a derivative (limit on the slopes is the tangent slope), the slope of the tangent line will be the value of derivative.
\begin{align*} \lim_{x \to 3} \frac{x^2-3^2}{x-3} = \amp \\ \lim_{x \to 3} \frac{(x-3)(x+3)}{(x-3)} = \amp \\ \lim_{x \to 3} (x+3) = \amp 6. \end{align*}
The point is \((3,3^2)=(3,9)\text{.}\) Now we can write an equation for this line:
\begin{equation*} \frac{y-9}{x-3}=6\text{.} \end{equation*}
Graph this line and the parabola to confirm that they are tangential (osculate at the point \((3,9)\)).
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Subsection 2.3.2 The derivative as a function

Finally in this section consider that we can determine if the derivative is defined at every point on a curve. For many of the curves we know, they are differentiable at all points where they are defined (a list is below). This means that while a derivative is defined as a property of a point on a curve, we can expand the concept to refer to the derivative as a property of the whole function.
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We define the derivative of a function \(f(x)\) to be the function, denoted \(f'(x)\text{,}\) to be the function such that \(f'(a)=\lim_{x \to a} \frac{f(x)-f(a)}{x-a}\) for all \(a\) where it is defined. For a function \(y=f(x)\) this is also denoted \(\frac{dy}{dx}=f'(x)\text{.}\)
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Example 2.3.2.

Consider the function \(f(x)=sin(x)\text{.}\) We will write an equation for the line tangent to \(f(x)\) at \(x=\pi/4\text{.}\) To write the equation of a line we need a point and the slope at that point.
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The point is \(\left( \frac{\pi}{4}, \sin\left( \frac{\pi}{4} \right) \right) = \left( \frac{\pi}{4}, 1 \right)\text{.}\) The sloe is given by the derivative.
\begin{align*} f'(\pi/4) = \amp \\ \cos(\pi/4) = \amp 1\text{.} \end{align*}
Thus an equation of the tangent line to sine at \(\pi/4\) is
\begin{equation*} \frac{y-1}{x-\pi/4} = 1\text{.} \end{equation*}
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Subsection 2.3.3 Known Derivatives

For ease of calculation it is useful for us to memorize the following functions and their derivatives.
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Table 2.3.3. Derivative Forms
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Function
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Derivative
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\begin{equation*} x^r \end{equation*}
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\begin{equation*} rx^{r-1} \end{equation*}
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\begin{equation*} \sin(x) \end{equation*}
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\begin{equation*} \cos(x) \end{equation*}
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\begin{equation*} \cos(x) \end{equation*}
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\begin{equation*} -\sin(x) \end{equation*}
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\begin{equation*} \tan(x) \end{equation*}
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\begin{equation*} \sec^2(x) \end{equation*}
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\begin{equation*} \sec(x) \end{equation*}
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\begin{equation*} \sec(x)\tan(x) \end{equation*}
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\begin{equation*} \csc(x) \end{equation*}
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\begin{equation*} -\csc(x)\cot(x) \end{equation*}
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\begin{equation*} \cot(x) \end{equation*}
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\begin{equation*} -\csc^2(x) \end{equation*}
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\begin{equation*} e^x \end{equation*}
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\begin{equation*} e^x \end{equation*}
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\begin{equation*} \ln(x) \end{equation*}
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\begin{equation*} \frac{1}{x} \end{equation*}
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