Secants and tangents
On rates of change and difference quotients.
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Study Secants and tangents
Many processes are dynamic and cannot be described by static equations or functions.
Scientifically observable quantities depend on the rates of change of other quantities:
The speed is the change in distance per time
The electric current is the change of charges per time
The momentum is the change in kinetic energy per time
The diffusion of a substance depends on the change in its concentration per distance traveled.
The description of such dependencies requires the description of rates of change
The rates of change can be represented by tangents on the curve.
But: A line is only uniquely defined by two points.
Question: How do you determine a tangent that touches the curve at only one point?
Answer: by a sequence of secants \(=\) lines that intersects the curve at two points
Question: What do secants have to do with the tangent? Or: What does it mean that the sequence of secants has the tangent as its limit?
Example 1. Let \(f(x):=\frac{1}{2}x^{2}\). The secants through the points \(A=\left(\frac{1}{2},f\left(\frac{1}{2}\right)\right)\) and \(\left(\frac{1}{2}+h,f\left(\frac{1}{2}+h\right)\right)\) for \(h=1\), \(h=0.5\), \(h=0.18\) are
The tangent at the point \(A=\left(\frac{1}{2},f\left(\frac{1}{2}\right)\right)\) is
Observation: The sequence of secants approaches for decreasing \(h\) the tangent of graph of \(f\) at the point \(A\)
Question: How can the calculate the slope of a secant?
Definition 2. The slope of the secants are called difference quotients: \[m_{S}(z):=\frac{f(z)-f(x)}{z-x}\] The limit of the difference quotient is the slope of the tangent at the point \(x\).
Example 3.
Let \(f(x):= ax+b\). Then it holds \[m_{s}(z)=\frac{\left(az+b\right)-\left(ax+b\right)}{z-x}=\frac{az+b-ax-b}{z-x}=\frac{a\left(z-x\right)}{z-x}=a\] and \(\lim\limits_{z\rightarrow x}m_{S}(z)=a.\)
Let \(f(x):= ax^{2}\). Then it holds \[m_{s}(z)=\frac{az^{2}-ax^{2}}{z-x}=\frac{a\left(z^{2}-x^{2}\right)}{z-x}=\frac{a\left(z+x\right)\left(z-x\right)}{z-x}=a\left(z+x\right)\] and \(\lim\limits_{z\rightarrow x}m_{S}(z)=\lim_{z\rightarrow x}a\left(z+x\right)=2ax.\)
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