课程： 15.12 摘要:微积分预览

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选择节摘要:微积分预览

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摘要:微积分预览
Chapter Summary
::章次摘要

In this chapter, we learned about:
::在本章中,我们了解到:

Limits:
::限制 :

To solve limits graphically, graph the function and determine if the limit exists.

The limit exists only when the left and right one-sided limits are equal .
$\begin{aligned} lim_{x \to a^{-}} f (x) = lim_{x \to a^{+}} f (x) = lim_{x \to a} f (x) = L \end{aligned}$

::限制只在左和右的单向限制相等时才存在。 limxa- f( x) =limxa+f( x) =limxaf( x) =L

If the left and right one-sided limits are not equal, then the limit does not exist, or DNE.
::如果左侧和右侧的界限不相等,则该界限不存在,或 DNE。

::要以图形方式解析限制, 请绘制函数图, 并确定限制是否存在。限制只有在左右单向限制相等时才存在。 limxa- f( x) =limxa+f( x) =limxaf( x) =L 如果左右单向限制不相等, 那么限制不存在, 或者 DNE 。

To solve limits numerically, use a table with $\begin{aligned} x \end{aligned}$ -values to the left and to the right on the number line to the number in the limit. Then plug these values into the function to obtain the function value at this point.
::要从数字上解析限制, 请使用一个表格, 其左侧和右侧的数字行的 X 值与限制的编号。然后将这些值插入函数中, 以便在此点获得函数值。

To solve limits analytically, substitute the value that $\begin{aligned} x \end{aligned}$ approaches into the function.

I f the function is a rational expression with a hole, then algebraically factor the numerator and denominator. Next, cancel any common factors in the numerator and denominator. Finally, substitute the value that $\begin{aligned} x \end{aligned}$ approaches into the resulting expression.
::如果函数是带有洞的合理表达式,则代数因素为分子和分母。接下来,取消分子和分母中的任何共同系数。最后,将x对结果表达式的数值替换为X。

Rationalization may also be used to help simply the rational expression.
::合理化也可以用来帮助简单的理性表达。

::要在分析中解析限制, 请替换 x 向函数中切换的值。如果函数是用空洞表示的合理表达式, 则代数乘数和分母。下一步, 取消分子和分母中的任何共同系数。最后, 替换 x 向结果表达式中切换的值。合理化也可以用来帮助简单的理性表达式。

Continuity:
::连续性:

A function is continuous at $\begin{aligned} x = a \end{aligned}$ if $\begin{aligned} lim_{x \to a} f (x) = f (a), \end{aligned}$ where both $\begin{aligned} f (a) \end{aligned}$ and $\begin{aligned} lim_{x \to a} f (x) \end{aligned}$ exist.
::在 f(a) 和 limx* *f(x) 存在的情况下, x=a 如果 limx=f(a) 则在 x=a 和 limx=af(x) 存在时,一个函数是连续的。

::在 f(a) 和 limx* *f(x) 存在的情况下, x=a 如果 limx=f(a) 则在 x=a 和 limx=af(x) 存在时,一个函数是连续的。

Intermediate and Extreme Value Theorem:
::中间和极端价值理论:

The Intermediate Value Theorem states: If a function is continuous on a closed interval $\begin{aligned} [a, b] \end{aligned}$ and $\begin{aligned} u \end{aligned}$ is a value between $\begin{aligned} f (a) \end{aligned}$ and $\begin{aligned} f (b) \end{aligned}$ , then there exists a $\begin{aligned} c \in [a, b] \end{aligned}$ such that $\begin{aligned} f (c) = u \end{aligned}$ .
::中间值理论指出:如果一个函数在封闭间隔[a、b]和u之间的一个数值为f(a)和f(b)之间,则存在f(c)=u的 c[a、b]。

The Extrem e Value Theorem states: If a function is continuous on a closed interval $\begin{aligned} [a, b] \end{aligned}$ , then there is at least one maximum and one minimum value.
::极端值定理表示:如果一个函数在封闭间隔[a,b] 连续,则至少有一个最大值和一个最低值。

Derivative :
::衍生因素:

A secant line is a line that passes through two distinct points on a function.
::秒线是指通过函数上两个不同点的行。

The average rate of change of a function is the slope of the secant line through two points.
::函数的平均变化速率是通过两个点的分隔线坡度。

A tangent line is a line that passes through one point on a function.
::相切线是函数通过一个点的线条。

The instantaneous rate of change is the slope of the tangent line at a point.
::瞬时变化率是某一点相切线的斜坡。

A derivative function is a function of the slopes of the original function.
::衍生函数是原函数的斜坡函数。

Area Under a Curve:
::曲线下的区域 :

The area under a curve is calculated by $\begin{aligned} Area = \sum_{i = 1}^{n} f (c_{i}) \cdot Δ x, \end{aligned}$ where $\begin{aligned} c_{i} \end{aligned}$ is the point of the box that hits the curve, $\begin{aligned} f (c_{i}) \end{aligned}$ is the function value at that point, and $\begin{aligned} Δ x \end{aligned}$ is the width of the base of each rectangle.
::曲线下的区域由“区域”=%i=1nf(ci)%x计算,其中“c”是单击曲线的框点,“f(ci)”是该点的函数值,“x”是每个矩形基底的宽度。

For a curve defined on the interval $\begin{aligned} [a, b] \end{aligned}$ , $\begin{aligned} Δ x = \frac{b - a}{n}, \end{aligned}$ where $\begin{aligned} n \end{aligned}$ is the number of rectangles or boxes.
::对于在间距[a,b], x=b-an, 其中 n 是矩形或框数的曲线。

Review
::回顾

Try the following cumulative review problems to practice the concepts in this chapter:
::尝试下列累积审查问题来实践本章中的概念:

章节大纲

常规

摘要:微积分预览

Chapter Summary ::章次摘要

Review ::回顾

Chapter Summary
::章次摘要

Review
::回顾