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# 数学代写|微积分代写Calculus代考|The Inverse of $\ln x$ and the Number $e$

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## 数学代写|微积分代写Calculus代考|The Inverse of $\ln x$ and the Number $e$

The function $\ln x$, being an increasing function of $x$ with domain $(0, \infty)$ and range $(-\infty, \infty)$, has an inverse $\ln ^{-1} x$ with domain $(-\infty, \infty)$ and range $(0, \infty)$. The graph of $\ln ^{-1} x$ is the graph of $\ln x$ reflected across the line $y=x$. As you can see in Figure 7.10,
$$\lim {x \rightarrow \infty} \ln ^{-1} x=\infty \quad \text { and } \quad \lim {x \rightarrow-\infty} \ln ^{-1} x=0 .$$
The function $\ln ^{-1} x$ is usually denoted as $\exp x$. We now show that $\exp x$ is an exponential function with base $e$.
The number $e$ was defined to satisfy the equation $\ln (e)=1$, so $e=\exp$ (1). We can raise the number $e$ to a rational power $r$ in the usual algebraic way:
$$e^2=e \cdot e, \quad e^{-2}=\frac{1}{e^2}, \quad e^{1 / 2}=\sqrt{e},$$
and so on. Since $e$ is positive, $e^r$ is positive too, so we can take the logarithm of $e^r$. When we do, we find that for $r$ rational
$$\ln e^r=r \ln e=r \cdot 1=r . \quad \text { Theorem 2, Rule } 4$$
Then applying the function $\ln ^{-1}$ to both sides of the equation $\ln e^r=r$, we find that
$$e^r=\exp r \quad \text { for } r \text { rational. } \quad \exp \text { is } \ln ^{-1} \text {. }$$
We have not yet found a way to give an obvious meaning to $e^x$ for $x$ irrational. But $\ln ^{-1} x$ has meaning for any $x$, rational or irrational. So Equation (1) provides a way to extend the definition of $e^x$ to irrational values of $x$. The function exp $x$ is defined for all $x$, so we use it to assign a value to $e^x$ at every point.

## 数学代写|微积分代写Calculus代考|The Derivative and Integral of ex

According to Theorem 1, the natural exponential function is differentiable because it is the inverse of a differentiable function whose derivative is never zero. We calculate its derivative using the inverse relationship and the Chain Rule:
\begin{aligned} \ln \left(e^x\right) & =x & & \text { Inverse relationship } \ \frac{d}{d x} \ln \left(e^x\right) & =1 & & \text { Differentiate both sides. } \ \frac{1}{e^x} \cdot \frac{d}{d x}\left(e^x\right) & =1 & & \text { Eq. (2), Section 7.2, with } u=e^x \ \frac{d}{d x} e^x & =e^x . & & \text { Solve for the derivative. } \end{aligned}
That is, for $y=e^x$, we find that $d y / d x=e^x$ so the natural exponential function $e^x$ is its own derivative. Moreover, if $f(x)=e^x$, then $f^{\prime}(0)=e^0=1$. This means that the natural exponential function $e^x$ has slope 1 as it crosses the $y$-axis at $x=0$.

The Chain Rule extends the derivative result for the natural exponential function to a more general form involving a function $u(x)$ :
If $u$ is any differentiable function of $x$, then
$$\frac{d}{d x} e^u=e^u \frac{d u}{d x}$$

## 数学代写|微积分代写Calculus代考|The Inverse of $\ln x$ and the Number $e$

$$\lim {x \rightarrow \infty} \ln ^{-1} x=\infty \quad \text { and } \quad \lim {x \rightarrow-\infty} \ln ^{-1} x=0 .$$

$$e^2=e \cdot e, \quad e^{-2}=\frac{1}{e^2}, \quad e^{1 / 2}=\sqrt{e},$$

$$\ln e^r=r \ln e=r \cdot 1=r . \quad \text { Theorem 2, Rule } 4$$

$$e^r=\exp r \quad \text { for } r \text { rational. } \quad \exp \text { is } \ln ^{-1} \text {. }$$

## 数学代写|微积分代写Calculus代考|The Derivative and Integral of ex

\begin{aligned} \ln \left(e^x\right) & =x & & \text { Inverse relationship } \ \frac{d}{d x} \ln \left(e^x\right) & =1 & & \text { Differentiate both sides. } \ \frac{1}{e^x} \cdot \frac{d}{d x}\left(e^x\right) & =1 & & \text { Eq. (2), Section 7.2, with } u=e^x \ \frac{d}{d x} e^x & =e^x . & & \text { Solve for the derivative. } \end{aligned}

$$\frac{d}{d x} e^u=e^u \frac{d u}{d x}$$

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