# 微积分网课代修|微分学代写Differential calculus代考|MTH295 The exponential functions

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## 微积分网课代修|微分学代写Differential calculus代考|The exponential functions

Let’s recall first what we know about the algebra of exponents (seen in Volume 1 , Chapter $1 \mathrm{PC}-3$ ).
It starts with this simple algebra:
Repeated addition is multiplication: $2+2+2=2 \cdot 3$.
One can say that that’s how multiplication was “invented” – as repeated addition. Next:
Repeated multiplication is power: $2 \cdot 2 \cdot 2=2^{3}$.
And this is the notation that we use:

So, this notation is nothing but a convention.

## 微积分网课代修|微分学代写Differential calculus代考|bacteria multiplying

Suppose we have a population of bacteria that doubles every day:
$$\underbrace{p_{n+1}}{\text {population: at time } n+1}=2 \cdot \underbrace{p{n}}{\text {at time } n} \Longrightarrow p{n}=p_{0} 2^{n} .$$
The graph consists of disconnected points:

Let’s think of it as a function. It is given by the same formula, with $x^{\prime}$ s still limited to the integers:
$$p(x)=p_{0} 2^{x} .$$
Now, what is the population in the middle of the first day? To answer, we consider the fact that multiplying by 2 is equivalent to multiply ing $\sqrt{2}$ twice:
$$\sqrt{2} \cdot \sqrt{2}=2 .$$
We conclude that
$$p(1 / 2)=\sqrt{2}$$

## 微积分网课代修|微分学代写Differential calculus代考|bacteria multiplying

$\underbrace{p_{n+1}}$ population: at time $n+1=2 \cdot \underbrace{p n}$ at time $n \Longrightarrow p n=p_{0} 2^{n}$.

$$p(x)=p_{0} 2^{x} .$$

$$\sqrt{2} \cdot \sqrt{2}=2 .$$

$$p(1 / 2)=\sqrt{2}$$