以下是完整 markdown：
---
title: Calculus hw
author: Jewel
date: April 21, 2023
---
> Question 1
$$
f(x) =
\begin{cases}
Asin(x)+Bcos(x) & \text{if } x < 0 \\
x^2+1 & \text{if } x \geqslant 0
\end{cases}
$$
For $f(x)$ to be continuous at $x=0$ ,
$RHL = LHL = f(0)$
$RHL = \lim_{x \to 0} (x^2+1)\ = 1$
$LHL = \lim_{x \to 0} (Asin(x)+Bcos(x))\ = B$
$f(0)=1 \implies B =1$
for $f$ to be differentiable at 0 :
RHD = LHD
##### RHD at $x=0$
$\lim_{h \to 0} {\dfrac{(0+h)^2 + 1 - 0^2-1}{h}} = 0$
<!-- ##### LHD at $x=0$ : -->
$= \lim_{h \to 0} { \dfrac{Asin(0+h)+Bcos(0+h)-Asin(0)-Bcos(0)}{h}}$
$= \lim_{h \to 0} { \dfrac{Asin(h)+Bcos(h)-B}{h}}$
$= \lim_{h \to 0} { \dfrac{Asin(0+h)+Bcos(0+h)-Asin(0)-Bcos(0)}{h}}$
$= A+ B\lim_{h \to 0} { \dfrac{cos(h)-1}{h}}$
Applying L'Hopital as it is a $\dfrac{0}{0}$ case
$ = A+ \lim_{h \to 0}{-sin(h)} = A+0 = A$
so for RHD = LHD , $A=0$
$(A,B)=(0,1)$
---
> Question 2
Let $f(t)$ denote distance (in km) travelled in time $t$ ( in minutes) .
$ f(0)=0, f(5) = 5$ *( in question )*
$f$ is continuous , differentiable in $[0,5]$ assuming smooth motion of car
So by Mean-Value Thereom , $ \exists$ $c$ $\in [0,5] $ such that :
$$ f'(c) = \dfrac{f(5)-f(0)}{5-0} = 1 km/min = 60km/hr$$
so at some $c \in (0,5)$ , Alice's speed $> 58km/hr$ . So she will be fined.
---
> Question 3
$f : (0,\frac{\pi}{2}) \to \mathbb{R} $.
$f(x) = sin(x)^\pi-\pi sin(x) +\pi $
I = $(0,\frac{\pi}{2})$
$f(x)$ is continuous , diffrentiable in I by algebra of continuous and differentiable functions
$f'(c) = \pi sin(c)^ {(\pi-1)} cos(c) - \pi cos(c) \leqslant 0$
Take $m,n \in I , n>m$
$ f$ is diffrentiable , continuous in [m,n] :
By Mean-Value thereom , there $ \exists c \in [m,n]$ st :
$f'(c) = \frac{f(n)-f(m)}{n-m} \leq 0 \implies f(n)< f(m)$
so $f$ is decreasing in I as we chose **any** two points in I
$f(0) = \pi , f(\frac{\pi}{2}) = 1 $
as $f$ is decreasing , its minimum value in I is 1 .
So $f > 0 $ in I
这是错误信息
PS C:\Users\jewel> cd .\OneDrive\Desktop\
PS C:\Users\jewel\OneDrive\Desktop> pandoc nmims.md -o xyz.pdf
Error producing PDF.
! Missing $ inserted.
<inserted text>
$
l.95 \$ = A+ \lim
这是我第一次使用 markdown。我尝试在线研究该错误，但没有找到任何有用的解决方案。
答案1
最有可能的问题是某些分隔符周围有空格$。Pandoc 手册的Pandoc 的 Markdown > 数学部分，解释道：
两个
$字符之间的任何内容都将被视为 TeX 数学。开头$必须紧靠其右侧有一个非空格字符，而结尾$必须紧靠其左侧有一个非空格字符，并且后面不能紧跟数字。因此，$20,000 and $30,000不会解析为数学。
所以$ f(0)=0, f(5) = 5$应该是$f(0)=0, f(5) = 5$和$f > 0 $应该成为$f > 0$，等等。