# Calculus_微积分_Applications of the Derivative

## Calculus_微积分_The approximation principle still applies

Any time we approximate with hyperreals, in any setting, the approximation principle still applies.
Example 9 Calculate $\lim {x \rightarrow \infty}\left(x-\frac{3 x^2-6}{3 x+4}\right)$. Solution We begin by evaluating the expression at $x=\Omega$ : $$\lim {x \rightarrow \infty}\left(x-\frac{3 x^2-6}{3 x+4}\right)=\Omega-\frac{3 \Omega^2-6}{3 \Omega+4} .$$
Next it seems reasonable to approximate the fraction by approximating the numerator and the denominator, which gives
$$\approx \Omega-\frac{3 \Omega^2}{3 \Omega}=\Omega-\Omega=0,$$
which violates the approximation principle. We approximated (terms were thrown away) and the result is zero.

We must try something different. Some sort of exact arithmetic must be used to change the form of the expression before we approximate. Let’s get a common denominator and perform the subtraction:
\begin{aligned} \Omega-\frac{3 \Omega^2-6}{3 \Omega+4} &=\Omega \cdot \frac{3 \Omega+4}{3 \Omega+4}-\frac{3 \Omega^2-6}{3 \Omega+4} \ &=\frac{3 \Omega^2+4 \Omega}{3 \Omega+4}-\frac{3 \Omega^2-6}{3 \Omega+4}=\frac{4 \Omega+6}{3 \Omega+4} . \end{aligned}
Now we can try approximation again:
$$\approx \frac{4 \Omega}{3 \Omega}=\frac{4}{3} .$$
The limit is $\frac{4}{3}$.
Suppose that a continuous function $f$ has the horizontal asymptote $y=k$ on the right and $k \neq 0$. Then, for any positive infinite hyperreal $\Omega, f(\Omega) \doteq k$. Now suppose that $A$ and $B$ are two different positive infinite hyperreals with $A \approx B$. We then have
\begin{aligned} &f(A) \doteq k_3 \ &f(B) \doteq k . \end{aligned}

## Calculus_微积分_Turning information into a sketch

The information we can collect about a function using calculus has now grown to a rather long list:

• discontinuities, including vertical asymptotes, removable discontinuities, and jump discontinuities
• corners or vertical tangents (continuous but not differentiable)
• intervals of increase/decrease
• extreme points
• intervals of concavity
• inflection points
• horizontal asymptotes
If desired, additional information such as $x$ – or $y$-intercepts or other individual points can be added. The issue becomes how to take this information and turn it into a sketch. To keep the process organized without having to memorize a too-detailed 20 -step procedure, we use the following steps:
(1) Plot extreme points, points of inflection, and other points such as intercepts (when expedient).
2) Produce graph snippets for any discontinuities found (based on limit information), as practiced in sections $1.4$ and 1.5. Draw the dotted lines for any horizontal asymptotes as well (also based on limit information).
• (3) Identify and plot points where the function is continuous but not differentiable.
• Identify the pieces of the curve to be drawn. Pieces begin and end at discontinuities, points from step (3), and inflection points.
5) Draw the pieces of the curve one at a time, using the correct concavity, passing through relevant points appropriately (i.e., ensuring extrema are represented correctly), and connecting graph snippets. For the left- and right-most pieces, which should be drawn first, any horizontal asymptotes must also be approached.
• For the examples and exercises we examine, step (3) is needed only occasionally.

# 微积分代考

## Calculus_微积分_The approximation principle still applies

$$\lim _{x \rightarrow \infty}\left(x-\frac{3 x^2-6}{3 x+4}\right)=\Omega-\frac{3 \Omega^2-6}{3 \Omega+4} .$$

$$\approx \Omega-\frac{3 \Omega^2}{3 \Omega}=\Omega-\Omega=0,$$

$$\Omega-\frac{3 \Omega^2-6}{3 \Omega+4}=\Omega \cdot \frac{3 \Omega+4}{3 \Omega+4}-\frac{3 \Omega^2-6}{3 \Omega+4} \quad=\frac{3 \Omega^2+4 \Omega}{3 \Omega+4}-\frac{3 \Omega^2-6}{3 \Omega+4}=\frac{4 \Omega+6}{3 \Omega+4} .$$

$$\approx \frac{4 \Omega}{3 \Omega}=\frac{4}{3} .$$

$$f(A) \doteq k_3 \quad f(B) \doteq k .$$

## Calculus_微积分_Turning information into a sketch

• 间断点，包括垂直渐近线、可移动间断点和跳跃间断点
• 角点或垂直切线（连续但不可微）
• 增加/减少的间隔
• 极值点
• 凹度区间
• 拐点
• 水平渐近线
如果需要，附加信息，例如X- 或者是-可以添加截距或其他单独的点。问题变成了如何获取这些信息并将其转化为草图。为了使过程井井有条，而不必记住过于详细的 20 步程序，我们使用以下步骤：
(1) 绘制极值点、拐点和其他点，例如截距（权宜之计）。
2) 为发现的任何不连续性生成图形片段（基于限制信息），如章节中所实践1.4和 1.5。也为任何水平渐近线绘制虚线（也基于极限信息）。
• (3) 识别并绘制函数连续但不可微的点。
• 确定要绘制的曲线段。片段在不连续点、步骤 (3) 的点和拐点处开始和结束。
5) 一次绘制一条曲线，使用正确的凹度，适当地通过相关点（即确保正确表示极值），并连接图形片段。对于应该首先绘制的最左边和最右边的部分，还必须接近任何水平渐近线。
• 对于我们检查的示例和练习，步骤 (3) 只是偶尔需要。

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