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Home < 70 Calculus Starter Lessons < 38 Lessons on Calculating Derivatives << 3 Motivation for Limit Definition - Take 2
[1] [2] [3][4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36]
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Saying how to compute a number directly or via a limiting process defines it. Motivation for Limit Definition or Codification of DerivativesNote sure that is clear enough, a rewrite or elimination in order For straight lines, slopes can be defined by a simple formula
But for curves $y = f(x)$, we can approximate what we think the slope should be at point (x1,y1) on the graph of $y = f(x)$, and see whether or not the approximation get closer and closer to a single number m as the the approximations get better.
Here the slope of the secant is \[ \mbox{slope } m = \frac{\Delta y}{\Delta x} = \frac{y_2-y_1}{x_2-x_1} =\frac{\mbox{rise}}{\mbox{run}} \] So we take the limit of approximations, if it exists, to be the slope or derivative f '(x1) of a function or curve y = f(x) at at x = x1. That is \begin{eqnarray*} f'(x_1) &=& \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x} \\ &=& \lim_{\Delta x \to 0} \frac{f(x_1+\Delta x)-f(x_1)}{\Delta x} \end{eqnarray*} In mathematics, saying how to compute a number or quantity directly or via a convergent process, a limit, defines it. In calculus, first slopes or derivatives and later area of regions are defined using limits. So we need to understand the theory or properties of limits. The question becomes what is a limit and how do with obtain their values. So we are going to cover two simplest view of limits. A more complicated view is left for later. In general there is no direct definition of what is the slope at a point x = a in the domain of a curve or function $y = f(x)$. But we can approximate the slope algebraically by computing the slope of a shorter and shorter secant chord between a pair of point $(x,y) = (a, f(a))$ and $(X,Y) = (a+h, f(a+h)$ with $h = \Delta x$ on the graph. The limiting value of the slope of the secant as h tends to 0 is taken to be the slope of the curve at x = a. So in the first instance, we use a limit-based definition of the slope m at a single point x= a. The process is called differentiation. The value of the slope m is obtained or derived from the formula or function $y =f (x)$. That may justify calling the slope m at x = a the derivative of function $f(x)$ at x = a. Saying how to compute a number directly or via a limiting process defines it. Now the slope m computed gives a value dependent on the location x = a. So the slope $m = g(a)$ for some function $g(x)$. Properties of limits lead to algebraic rules for hiding the limit-based calculation of slope $m = g(a)$ by providing limit-free rules for obtaining or deriving a formula for the slope function $g(x)$ from formulas for the curve height function $y= f(x)$. That may justify calling $g(x)$, the derivative of function $f(x)$ and writing $g(x) =f'(x)$ to indicate that $g(x)$ is derived from $f(x)$.
Theory: Introduce derivatives via limits. This theory to practice pattern of defining or introducing a number via limit-based considerations - limits of approximations - and then seeking algebraically simpler ways to evaluate the limit is repeated often in calculus. Watch for it. This theory to practice pattern of defining or introducing a number via limit-based considerations (approximations) and then seeking algebraically simpler ways to evaluate the limit is repeated in calculus. |
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Home < 70 Calculus Starter Lessons < 38 Lessons on Calculating Derivatives << 3 Motivation for Limit Definition - Take 2
[1] [2] [3][4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36]
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