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# 数学代写|微积分代写Calculus代考|MATH-172 Differentiating Trigonometric Functions

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## 数学代写|微积分代写Calculus代考|Differentiating Trigonometric Functions

Trigonometric functions occur in so many applications that it is useful to know their derivatives. For instance, we would like to know $\frac{d}{d \theta} \sin \theta$. By definition,
$$\frac{d}{d \theta} \sin \theta=\lim _{\Delta \theta \rightarrow 0} \frac{\sin (\theta+\Delta \theta)-\sin \theta}{\Delta \theta} .$$

It is not at all obvious how to evaluate this expression, so let’s take another approach for a minute and try to guess geometrically what the result should be by looking at a plot of $\sin \theta$. Here is a plot of $\sin \theta$ vs. $\theta$ over the interval $0 \leq \theta \leq 2 \pi$. ( $\theta$ is measured in radians. $)$

Draw a sketch of $\frac{d}{d \theta} \sin \theta$ in the space provided. To check your sketch,

Here are drawings of $\sin \theta$ and $\frac{d}{d \theta} \sin \theta$. Note that where the slope of $\sin \theta$ is greatest, at 0 and $2 \pi, \frac{d}{d \theta} \sin \theta$ has its greatest value, and that where the slope is 0 , at $\theta=\pi / 2$ and $\theta=3 \pi / 2$, $\frac{d}{d \theta} \sin \theta$ is 0 .
(If your sketch looked very different from the drawing shown above, you should review frames 160 and 169. This problem is quite similar to problem (c) in frame 168.)

Now, by looking at the graphs, you may be able to guess the correct answer for $\frac{d}{d \theta} \sin \theta$. Can you?
$$\frac{d}{d \theta} \sin \theta=$$

## 数学代写|微积分代写Calculus代考|Higher-Order Derivatives

Suppose a function $f$ depends on $x$, and we have differentiated it to obtain $d f / d x$. If we then differentiate $d f / d x$ with respect to $x$, the result is called the second derivative of $f$ with respect to $x$. This is written $\frac{d^2 f}{d x^2}$. Sometimes this is written as $f^{(2)}$, where the ” $(2)$ ” superscript indicates the second derivative of $f$, not the square of $f$. The variable $x$ is suppressed.
Answer: Frame 238: $\frac{-1}{(\ln x)^2 x}$

Try the following:
$$\text { If } f=2 x^3 \text {, then } f^{(2)}=\frac{d^2 f}{d x^2}=\left[6 x^2|12 x| 0\left|x^2\right| x\right]$$

Here’s how to do the problem in $\mathbf{2 4 2}$.
\begin{aligned} f &=2 x^3, \frac{d f}{d x}=6 x^2, \ f^{(2)} &=\frac{d^2 f}{d x^2}=\frac{d}{d x}\left(\frac{d f}{d x}\right) \frac{d}{d x}\left(6 x^2\right)=12 x . \end{aligned}
Try this:
\begin{aligned} f(x) &=x+\frac{1}{x} \ f^{(2)} &=\frac{d^2 f}{d x^2}=\left[-\frac{1}{x^2}\left|\frac{1}{x}\right|+\frac{2}{x^3} \mid \text { none of these }\right] \end{aligned}

## 数学代写|微积分代写Calculus代考|Differentiating Trigonometric Functions

$$\frac{d}{d \theta} \sin \theta=\lim _{\Delta \theta \rightarrow 0} \frac{\sin (\theta+\Delta \theta)-\sin \theta}{\Delta \theta} .$$

(如果你的草图看起来与上图有很大不同，你应该龺看第 160 帧和第 169 帧。这个问题与第 168 帧中的问题 (c) 非常相似。)

$$\frac{d}{d \theta} \sin \theta=$$

## 数学代写|微积分代写Calculus代考|Higher-Order Derivatives

If $f=2 x^3$, then $f^{(2)}=\frac{d^2 f}{d x^2}=\left[6 x^2|12 x| 0\left|x^2\right| x\right]$

$$f=2 x^3, \frac{d f}{d x}=6 x^2, f^{(2)} \quad=\frac{d^2 f}{d x^2}=\frac{d}{d x}\left(\frac{d f}{d x}\right) \frac{d}{d x}\left(6 x^2\right)=12 x .$$

$$f(x)=x+\frac{1}{x} f^{(2)} \quad=\frac{d^2 f}{d x^2}=\left[-\frac{1}{x^2}\left|\frac{1}{x}\right|+\frac{2}{x^3} \mid \text { none of these }\right]$$

## MATLAB代写

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