# 建築系微積分 Exercise 2 Solution
###### tags: `Calculus` `微積分` `Arch` `建築系`

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### Section 1.4

* Evaluate the difference and simplify the result.

**30.** $h(x)=x^2+x+3$, $\frac{h(x+\Delta x)-h(x)}{\Delta x}$
***Solution:***
We compute
$$\frac{h(x+\Delta x)-h(x)}{\Delta x}=\frac{[(x+\Delta x)^2+(x+\Delta x)+3]-[x^2+x+3]}{\Delta x}$$
$$=\frac{(x^2+2\Delta x\cdot x+\Delta x^2+x+\Delta x+3)-(x^2+x+3)}{\Delta x}$$
$$=\frac{2\Delta x\cdot x+\Delta x^2+\Delta x}{\Delta x}=2x+\Delta x+1,\ \Delta x\neq0.$$

**70.** From 2008 to 2012, the total numbers of births $B$ (in thousands) and deaths $D$ (in thousands) in the United States can be approximated by the models
$$B(t)=4.917t^3-124.71t^2+925.9t+2308$$
and
$$D(t)=-7.083t^3+222.64t^2-2281.8t+10104,$$
where $t$ represents the year, with $t=8$ corresponding to 2008. Find $B(t)-D(t)$ and interpret this function.
***Solution:***
$B(t)-D(t)=12t^3-347.35t^2+3207.7t-7796$, this function represent the approximated increase or decrease in people from 2008 to 2012.

**75.** A company invests $98,000 for equipment to produce a new product. Each unit of the product costs $12.30 and is sold for $17.98. Let $x$ be the number of units produced and sold.
a. Write the total cost $C$ as a function of $x$.
b. Write the revenue $R$ as a function of $x$.
c. Write the profit $P$ as a function of $x$.
***Solution:***
a. $C(x)=98000+12.3x$.
b. $R(x)=17.98x$.
c. $P(x)=R(x)-C(x)=5.68x-98000$.

### Section 1.5
**4.** Use the graph to find the limit.
a. $\lim_{x\rightarrow-2}h(x)$
b. $\lim_{x\rightarrow0}h(x)$
***Solution:***
a. $\lim_{x\rightarrow-2}h(x)=-5$
b. $\lim_{x\rightarrow0}h(x)=-3$

**6.** Complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result.
$\lim_{x\rightarrow1}x^2-4x-1$

| $x$ | 0.9 | 0.99 | 0.999 | 1 | 1.001 | 1.01 | 1.1 |
| -------- | --- | ---- | ----- | - | ----- | ---- | --- |
| $f(x)$ | | | | ? | | | |

***Solution:***

| $x$ | 0.9 | 0.99 | 0.999 | 1 | 1.001 | 1.01 | 1.1 |
| -------- | --- | ---- | ----- | - | ----- | ---- | --- |
| $f(x)$ |-3.79|-3.9799|-3.998 | -4 |-4.002 |-4.0199 | -4.19|

* Find the limit (if it exists).

**42.** $\lim_{x\rightarrow-3}\frac{x^3+27}{x+3}$
***Solution:***
$$\lim_{x\rightarrow-3}\frac{x^3+27}{x+3}=\lim_{x\rightarrow-3}\frac{(x+3)(x^2-3x+9)}{x+3}=^{x\neq-3}\lim_{x\rightarrow-3}x^2-3x+9=27.$$

**51.** $\lim_{x\rightarrow2}f(x)$, where
$$f(x)=\left\{\begin{array}{ll} 4-x, & x\neq 2 \\
0, & x=2\end{array}\right..$$
***Solution:***
We can find that $\lim_{x\rightarrow2^+}f(x)=2$ and that$\lim_{x\rightarrow2^-}f(x)=2$, so $\lim_{x\rightarrow2}f(x)=2$.

* Use the graph to find the limit (if it exists).
a. $\lim_{x\rightarrow c^+}f(x)$
b. $\lim_{x\rightarrow c^-}f(x)$
c. $\lim_{x\rightarrow c}f(x)$

**60.**
***Solution:***
a. $\lim_{x\rightarrow c^+}f(x)=-2$
b. $\lim_{x\rightarrow c^-}f(x)=-2$
c. $\lim_{x\rightarrow c}f(x)=-2$

**64.**
***Solution:***
a. $\lim_{x\rightarrow c^+}f(x)=0$
b. $\lim_{x\rightarrow c^-}f(x)=2$
c. $\lim_{x\rightarrow c}f(x)$ does not exist since $\lim_{x\rightarrow c^+}f(x)\neq\lim_{x\rightarrow c^-}f(x)$.

### Section 1.6
* Determine whether the function is continuous on the entire real number line. Explain your reasoning.

**4.** $f(x)=\frac{1}{9-x^2}$

***Solution:***
Since $f$ is not defined at $x=\pm2$, that is, the denominator(分母) equals zero for $x=\pm2$, $f$ is not continuous on the entire real number line.

**5.** $f(x)=\frac{1}{4+x^2}$
***Solution:***
Since the deniminator is nonzero for every real $x$ and is polynomial, the function $f$ is continuous on the entire real number line.

**61.** The cost $C$ (in millions of dollars) of removing $x$ percent of the pollutants emitted from the smokestack of a factory can be modeled by $$C=\frac{2x}{100-x}.$$
a. What is the implied domain of $C$? Explain your reasoning.
b. Use a graphing utility to graph the cost function. Is the function continuous on its domain? Explain your reasoning.
c. Find the cost of removing 75% of the pollutants from the smokestack.
***Solution:***
a. Since the denominator of $C$ is nonzero for $x\neq100$, and the implementation of percentage implies the bound $0\leq x\leq100$, the implied domain of $C$ is $\{x\in\mathbb{R}:0\leq x<100\}$.
b.
c. Substitute $x$ with 75, we have $$C=\frac{2\cdot75}{100-75}=6.$$