Essential Calculus Early Transcendentals

A crystal growth furnace is used in research to determine

how best to manufacture crystals used in electronic com-

ponents for the space shuttle. For proper growth of the

crystal, the temperature must be controlled accurately by

adjusting the input power. Suppose the relationship is

given by

$T(w)=0.1 w^{2}+2.155 w+20$

where $T$ is the temperature in degrees Celsius and $w$ is the

power input in watts.

(a) How much power is needed to maintain the temperature

at $200^{\circ} \mathrm{C} ?$

(b) If the temperature is allowed to vary from $200^{\circ} \mathrm{C}$ by up

to $\pm 1^{\circ} \mathrm{C},$ what range of wattage is allowed for the input

power?

(c) In terms of the $\varepsilon, \delta$ definition of $\lim _{x \rightarrow a} f(x)=L,$ what

is $x ?$ What is $f(x) ?$ What is $a ?$ What is $L$ ? What value

of $\varepsilon$ is given? What is the corresponding value of $\delta$ ?

Essential Calculus Early Transcendentals

(a) Evaluate $h(x)=(\tan x-x) / x^{3}$ for $x=1,0.5,0.1$

$0.05,0.01,$ and 0.005

(b) Guess the value of $\lim _{x \rightarrow 0} \frac{\tan x-x}{x^{3}}$

(c) Evaluate $h(x)$ for successively smaller values of $x$ until

you finally reach 0 values for $h(x)$ . Are you still con-

fident that your guess in part (b) is correct? Explain

why you eventually obtained 0 values. (In Section 3.7

a method for evaluating the limit will be explained.)

(d) Graph the function $h$ in the viewing rectangle $[-1,1]$

by $[0,1] .$ Then zoom in toward the point where the

graph crosses the $y$ -axis to estimate the limit of $h(x)$ as

$x$ approaches $0 .$ Continue to zoom in until you observe

distortions in the graph of $h .$ Compare with the results

of part (c).

Essential Calculus Early Transcendentals

(a) Evaluate the function $f(x)=x^{2}-\left(2^{x} / 1000\right)$ for

$x=1,0.8,0.6,0.4,0.2,0.1,$ and $0.05,$ and guess the

value of

$\lim _{x \rightarrow 0}\left(x^{2}-\frac{2^{x}}{1000}\right)$

(b) Evaluate $f(x)$ for $x=0.04,0.02,0.01,0.005,0.003$

and $0.001 .$ Guess again.