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# 数学代写|微积分代写Calculus代考|MATH1051 CALCULUS PROPERTIES OF THE EXPONENTIAL

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## 数学代写|微积分代写Calculus代考|CALCULUS PROPERTIES OF THE EXPONENTIAL

Now we want to learn some “calculus properties” of our new function $\exp (x)$. These are derived from the standard formula for the derivative of an inverse, as in Section 2.5.1.
For all $x$ we have
$$\frac{d}{d x}(\exp (x))=\exp (x)$$
In other words,
$$\int \exp (x) d x=\exp (x)$$
More generally,
$$\frac{d}{d x} \exp (u)=\exp (u) \frac{d u}{d x}$$

and
$$\int \exp (u) \frac{d u}{d x} d x=\exp (u)+C .$$
We note for the record that the exponential function is the only function (up to constant multiples) that is its own derivative. This fact will come up later in our applications of the exponential
EXAMPLE 6.9
Compute the derivatives:
$$\frac{d}{d x} \exp (4 x), \quad \frac{d}{d x}(\exp (\cos x)), \quad \frac{d}{d x}([\exp (x)] \cdot[\cot x]) .$$
SOLUTION
For the first problem, notice that $u=4 x$ hence $d u / d x=4$. Therefore we have
$$\frac{d}{d x} \exp (4 x)=[\exp (4 x)] \cdot \frac{d}{d x}(4 x)=4 \cdot \exp (4 x) .$$
Similarly,
\begin{aligned} \frac{d}{d x}(\exp (\cos x)) &=[\exp (\cos x)] \cdot\left(\frac{d}{d x} \cos x\right)=[\exp (\cos x)] \cdot(-\sin x) \ \frac{d}{d x}([\exp (x)] \cdot[\cot x]) &=\left[\frac{d}{d x} \exp (x)\right] \cdot(\cot x)+[\exp (x)] \cdot\left(\frac{d}{d x} \cot x\right) \ &=[\exp (x)] \cdot(\cot x)+[\exp (x)] \cdot\left(-\csc ^2 x\right) \end{aligned}

## 数学代写|微积分代写Calculus代考|THE NUMBER e

The number $\exp (1)$ is a special constant which arises in many mathematical and physical contexts. It is denoted by the symbol $e$ in honor of the Swiss mathematician Leonhard Euler (1707-1783) who first studied this constant. We next see how to calculate the decimal expansion for the number $e$.

In fact, as can be proved in a more advanced course, Euler’s constant $e$ satisfies the identity
$$\lim _{n \rightarrow+\infty}\left(1+\frac{1}{n}\right)^n=e$$

[Refer to the “You Try It” following Example $5.9$ in Subsection 5.2.3 for a consideration of this limit.]
This formula tells us that, for large values of $n$, the expression
$$\left(1+\frac{1}{n}\right)^n$$
gives a good approximation to the value of $e$. Use your calculator or computer to check that the following calculations are correct:
$$\begin{array}{ll} n=10 & \left(1+\frac{1}{n}\right)^n=2.5937424601 \ n=50 & \left(1+\frac{1}{n}\right)^n=2.69158802907 \ n=100 & \left(1+\frac{1}{n}\right)^n=2.70481382942 \ n=1000 & \left(1+\frac{1}{n}\right)^n=2.71692393224 \ n=10000000 & \left(1+\frac{1}{n}\right)^n=2.71828169254 \end{array}$$

## 数学代写|微积分代写Calculus代考|CALCULUS PROPERTIES OF THE EXPONENTIAL

$$\frac{d}{d x}(\exp (x))=\exp (x)$$

$$\int \exp (x) d x=\exp (x)$$

$$\frac{d}{d x} \exp (u)=\exp (u) \frac{d u}{d x}$$

$$\int \exp (u) \frac{d u}{d x} d x=\exp (u)+C$$

$$\frac{d}{d x} \exp (4 x), \quad \frac{d}{d x}(\exp (\cos x)), \quad \frac{d}{d x}([\exp (x)] \cdot[\cot x])$$

$$\frac{d}{d x} \exp (4 x)=[\exp (4 x)] \cdot \frac{d}{d x}(4 x)=4 \cdot \exp (4 x) .$$

$$\frac{d}{d x}(\exp (\cos x))=[\exp (\cos x)] \cdot\left(\frac{d}{d x} \cos x\right)=[\exp (\cos x)] \cdot(-\sin x) \frac{d}{d x}([\exp (x)] \cdot[\cot x]) \quad=\left[\frac{d}{d x} \exp (x)\right] \cdot(\cot x)+[\exp (x)] \cdot\left(\frac{d}{d x} \cot x\right)$$

## 数学代写|微积分代写Calculus代考|THE NUMBER e

$$\lim _{n \rightarrow+\infty}\left(1+\frac{1}{n}\right)^n=e$$
[参考“你试一试”下面的例子 $5.9$ 在 5.2.3 小节中考虑这个限制。]

$$\left(1+\frac{1}{n}\right)^n$$

$$n=10 \quad\left(1+\frac{1}{n}\right)^n=2.5937424601 n=50 \quad\left(1+\frac{1}{n}\right)^n=2.69158802907 n=100 \quad\left(1+\frac{1}{n}\right)^n=2.70481382942 n=1000 \quad\left(1+\frac{1}{n}\right)^n=2.71692393$$

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