37 lines
1.8 KiB
Markdown
37 lines
1.8 KiB
Markdown
A derivative can be used to describe the rate of change at a single point, or the *instantaneous velocity*.
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The formula used to calculate the average rate of change looks like this:
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$$ \dfrac{f(b) - f(a)}{b - a} $$
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Interpreting it, this can be described as the change in $y$ over the change in $x$.
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- Speed is always positive
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- Velocity is directional
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As the distance between the two points $a$ and $b$ grow smaller, we get closer and closer to the instantaneous velocity of a point. Limits are suited to describing the behavior of a function as it approaches a point.
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If we have the coordinate pair $(a, f(a))$, and the value $h$ is the distance between $a$ and another $x$ value, the coordinates of that point can be described as ($(a + h, f(a + h))$. With this info:
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- The slope of the secant line can be described as $\dfrac{f(a + h) - f(a)}{a + h - a}$, which simplifies to $\dfrac{f(a + h) - f(a)}{h}$.
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- The slope of the *tangent line* or the *instantaneous velocity* can be found by taking the limit of the above function as the distance ($h$) approaches zero:
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$$\lim_{h \to 0}\dfrac{f(a + h) - f(a)}{h}$$
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The above formula can be used to find the *derivative*. This may also be referred to as the *instantaneous velocity*, or the *instantaneous rate of change*.
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# Line Types
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## Secant Line
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A **Secant Line** connects two points on a graph.
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A **Tangent Line** represents the rate of change or slope at a single point on the graph.
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# Notation
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Given the equation $y = f(x)$, the following are all notations used to represent the derivative of $f$ at $x$:
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- $f'(x)$
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- $\dfrac{d}{dx}f(x)$
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- $y'$
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- $\dfrac{dy}{dx}$
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- $\dfrac{df}{dx}$
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- "Derivative of $f$ with respect to $x$"
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# Functions that are not differentiable at a given point
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- Where a function is not defined
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- Where a sharp turn takes place
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- If the slope of the tangent line is vertical
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