# 数学代写|微积分代写Calculus代写|MATH1111

## 数学代写|微积分代写Calculus代写|Continuity of a Function

We noticed in Section $2.3$ that the limit of a function as $x$ approaches $a$ can often be found simply by calculating the value of the function at $a$. Functions having this property are called continuous at $a$. We will see that the mathematical definition of continuity corresponds closely with the meaning of the word continuity in everyday language. (A continuous process is one that takes place without interruption.

Notice that Definition 1 implicitly requires three things if $f$ is continuous at $a$ :

1. $f(a)$ is defined (that is, $a$ is in the domain of $f$ )
2. $\lim _{x \rightarrow a} f(x)$ exists
3. $\lim _{x \rightarrow a} f(x)=f(a)$
The definition says that $f$ is continuous at $a$ if $f(x)$ approaches $f(a)$ as $x$ approaches $a$. Thus a continuous function $f$ has the property that a small change in $x$ produces only a small change in $f(x)$. In fact, the change in $f(x)$ can be kept as small as we please by keeping the change in $x$ sufficiently small.

If $f$ is defined near $a$ (in other words, $f$ is defined on an open interval containing $a$, except perhaps at $a$ ), we say that $f$ is discontinuous at $a$ (or $f$ has a discontinuity at $a$ ) if $f$ is not continuous at $a$.

Physical phenomena are usually continuous. For instance, the displacement or velocity of a moving vehicle varies continuously with time, as does a person’s height. But discontinuities do occur in such situations as electric currents. [The Heaviside function, introduced in Section 2.2, is discontinuous at 0 because $\lim _{t \rightarrow 0} H(t)$ does not exist.]
Geometrically, you can think of a function that is continuous at every number in an interval as a function whose graph has no break in it: the graph can be drawn without removing your pen from the paper.

## 数学代写|微积分代写Calculus代写|Derivatives and Rates of Change

Now that we have defined limits and have learned techniques for computing them, we revisit the problems of finding tangent lines and velocities from Section 2.1. The special type of limit that occurs in both of these problems is called a derivative and we will see that it can be interpreted as a rate of change in any of the natural or social sciences or engineering.

Then we let $Q$ approach $P$ along the curve $C$ by letting $x$ approach $a$. If $m_{P Q}$ approaches a number $m$, then we define the tangent line $\ell$ to be the line through $P$ with slope $m$. (This amounts to saying that the tangent line is the limiting position of the secant line $P Q$ as $Q$ approaches $P$. See Figure 1.)
1 Definition The tangent line to the curve $y=f(x)$ at the point $P(a, f(a))$ is the line through $P$ with slope
$$m=\lim {x \rightarrow a} \frac{f(x)-f(a)}{x-a}$$ provided that this limit exists. In our first example we confirm the guess we made in Example 2.1.1. EXAMPLE 1 Find an equation of the tangent line to the parabola $y=x^2$ at the point $P(1,1)$ SOLUTION Here we have $a=1$ and $f(x)=x^2$, so the slope is \begin{aligned} m &=\lim {x \rightarrow 1} \frac{f(x)-f(1)}{x-1}=\lim {x \rightarrow 1} \frac{x^2-1}{x-1} \ &=\lim {x \rightarrow 1} \frac{(x-1)(x+1)}{x-1} \ &=\lim _{x \rightarrow 1}(x+1)=1+1=2 \end{aligned}
Using the point-slope form of the equation of a line, we find that an equation of the tangent line at $(1,1)$ is
$$y-1=2(x-1) \quad \text { or } \quad y=2 x-1$$
We sometimes refer to the slope of the tangent line to a curve at a point as the slope of the curve at the point. The idea is that if we zoom in far enough toward the point, the curve looks almost like a straight line. Figure 2 illustrates this procedure for the curve $y=x^2$ in Example 1. The more we zoom in, the more the parabola looks like a line. In other words, the curve becomes almost indistinguishable from its tangent line.

# 微积分代考

## 数学代写|微积分代写Calculus代写|Continuity of a Function

1. $f(a)$ 被定义 (即 $a$ 属于 $f$ )
2. $\lim _{x \rightarrow a} f(x)$ 存在
3. $\lim {x \rightarrow a} f(x)=f(a)$ 定义说 $f$ 是连续的 $a$ 如果 $f(x)$ 方法 $f(a)$ 作为 $x$ 方法 $a$. 因此一个连续函数 $f$ 具有微小 变化的特性 $x$ 只产生很小的变化 $f(x)$. 事实上，改变 $f(x)$ 可以通过保持变化保持尽 可能小 $x$ 足够小。 如果 $f$ 定义在附近 $a$ (换句话说， $f$ 定义在一个包含的开区间 $a$ ，除了也许在 $a$ )，我们说 $f$ 在不连续 $a$ （或者 $f$ 有一个不连续点 $a$ ) 如果 $f$ 不连续于 $a$. 物理现象通常是连续的。例如，移动车辆的位移或速度随时间连续变化，人的身高也 是如此。但是在电流等情况下确实会发生不连续性。[在 $2.2$ 节中介绍的 Heaviside 函 数在 0 处不连续，因为 $\lim {t \rightarrow 0} H(t)$ 不存在。] 在
几何上，您可以将一个在区间内的每个数字处连续的函数视为其图形中没有中断的函 数: 无需将笔从纸上移开即可绘制图形。

## 数学代写|微积分代写Calculus代写|Derivatives and Rates of Change

1 定义 曲线的切线 $y=f(x)$ 在这一点上 $P(a, f(a))$ 是线通过 $P$ 带坡度
$$m=\lim x \rightarrow a \frac{f(x)-f(a)}{x-a}$$

$$m=\lim x \rightarrow 1 \frac{f(x)-f(1)}{x-1}=\lim x \rightarrow 1 \frac{x^2-1}{x-1} \quad=\lim x \rightarrow 1 \frac{(x-1)(x+1)}{x-1}$$

$$y-1=2(x-1) \quad \text { or } \quad y=2 x-1$$

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