# 数学代写|微积分代写Calculus代写|МАTH141

## 数学代写|微积分代写Calculus代写|Pricing a New Product

You have just invented a new soccer helmet to better protect players from head injuries. You need to decide how many helmets you need to sell in order to maximize your total profit $P$. Your revenue $r$ will be the product of the price $p$ and the quantity $q$ you sell,
$$r=p q .$$
The number of helmets sold naturally depends on the price: as the price goes up, sales go down. The costs also depend on the quantity of helmets sold: the more helmets sold, the less the price per helmet but the more your total cost. Your profit will be the difference between your revenue and your costs
$$P=r-c \text {. }$$
If the price is low, you will sell many helmets, but the income may not be very large. If the price is high, you will sell very few. Somewhere between the extremes your profit will be greatest. For purposes of planning, you assume that the price of helmets sold depends on the number sold:
$$p=p_0\left[1-\left(q / q_0\right)\right],$$
where $p_0$ is the maximum price that you can charge per helmet (above that price you will not sell any helmets), and $q_0$ is the maximum amount you can sell if you lower the price to zero.

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

Definition of a Limit:
Let $f(x)$ be defined for all $x$ in an interval about $x=a$, but not necessarily at $x=a$. If there is a number $L$ such that to each positive number $\varepsilon$ there corresponds a positive number $\delta$ such that
$$|f(x)-L|<\varepsilon \text { provided } 0<|x-a|<\delta,$$
we say that $L$ is the limit of $f(x)$ as $x$ approaches $a$ and write
$$\lim {x \rightarrow a} f(x)=L .$$ The ordinary algebraic manipulations can be performed with limits as shown in Appendix A2; thus, $$\lim {x \rightarrow a}[F(x)+G(x)]=\lim {x \rightarrow a} f(x)+\lim {x \rightarrow a} G(x) .$$
Two trigonometric limits are of particular interest (Appendix A3):
$$\lim {\theta \rightarrow 0} \frac{\sin \theta}{\theta}=1 \text { and } \lim {\theta \rightarrow 0} \frac{1-\cos \theta}{\theta}=0 .$$
The following limit is of such great interest in calculus that it is given the special name $e$, as discussed in frame 109 and Appendix A3:
$$e=\lim _{x \rightarrow 0}(1+x)^{1 / x}=2.71828 \ldots$$

# 微积分代考

## 数学代写|微积分代写Calculus代写|为新产品定价

$$r=p q .$$

$$P=r-c \text {. }$$

$$p=p_0\left[1-\left(q / q_0\right)\right],$$

## 数学代写|微积分代写Calculus代写|函数的极限

.函数的极限

$f(x)$ 为所有人定义 $x$ 在一段时间内 $x=a$，但不一定在 $x=a$。如果有数字的话 $L$ 对于每一个正数 $\varepsilon$ 对应一个正数 $\delta$ 这样
$$|f(x)-L|<\varepsilon \text { provided } 0<|x-a|<\delta,$$

$$\lim {x \rightarrow a} f(x)=L .$$ 一般的代数运算可以在附录A2所示的极限下进行;因此， $$\lim {x \rightarrow a}[F(x)+G(x)]=\lim {x \rightarrow a} f(x)+\lim {x \rightarrow a} G(x) .$$两个三角极限特别有趣(附录A3):
$$\lim {\theta \rightarrow 0} \frac{\sin \theta}{\theta}=1 \text { and } \lim {\theta \rightarrow 0} \frac{1-\cos \theta}{\theta}=0 .$$下面的极限在微积分中非常有趣，因此给了它一个特殊的名字 $e$，如第109帧和附录A3所述:
$$e=\lim _{x \rightarrow 0}(1+x)^{1 / x}=2.71828 \ldots$$

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