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From page Chapter 1 of Calculus: Single and Multiple Variable, 3rd Edition

Greetings,

I'm trying to teach myself calculus from an old textbook and I've been banging my head against this for a few days now. Any help would be appreciated. The chapter doesn't give any overt clues on how to solve this and I'm stumped.

The question is from Section 1.1 of Calculus: Single and Multivariable (3rd Edition, 2002).

Happy New Year!

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    What are the constants? What are the variables? Which variable is the independent variable? Try comparing $l-l_0 = al_0(t-t_0)$ with $y-y_0 = m(x-x_0)$. Try graphing it. Find the slope and vertical intercept. – 2017-01-05

2 Answers 2

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A linear function function $f$ on the variable $t$ is a function $f(t)$ of the form

$$f(t) = mx + b$$ where $m$ and $b$ are constants.

All you have to do is rewrite your expression:

$$l - l_0 = al_0(t - t_0) \iff l = al_0(t - t_0) + l_0$$

Then, expanding the expression inside the parenthesis, you get

$$l = al_0(t - t_0) + l_0 = al_0t - al_0t_0 + l_0 = al_0t + (1 - at_0)l_0$$

Now pay close attention to that expression and try to notice it resembles the first one,

$$l(t) = mt + b$$

If you set

$$\begin{cases}m = al_0\\ b = (1 - at_0)l_0\end{cases}$$

Can you do the slope and the vertical interception?

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    So now it's in slope intercept form and $al_0$ is the slope and $(0, (1-at_0)l_0)$ is the y-intercept. And the question asked us to express in terms of $l_0, t_0,$, and a which we did. So part (a) is actually solved? But how could you know this intuitively if you didn't know the forms in advance? – 2017-01-05
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    Forgot to tag @RSerrao – 2017-01-05
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    @user5319165 sorry for the late reply. To the part of "express $l $ as a function of such-and-such", one can simply isolate $l $ one side of the equation by algebraic manipulation. When they ask for it to be a *linear* function, it pretty much assumes you know what a linear function is and how it looks like ( $mx + b$) – 2017-01-07
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It appears all you need to do is add $l_0$ to both sides:

$$l=al_0(t-t_0)-l_0$$

and since $a,l_0,t_0$ are all constants, we have a linear equation of the form $y=mx+b$. That should help you easily find the slope. Vertical intercept most like refers to the y-intercept, which, as a hint, occurs at $t=0$.

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    In this case then $m = al_0$ ? – 2017-01-04
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    @user5319165 Yes, it is the case. – 2017-01-04
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    hmm would m also equal $$m = (t_0-t)/(l_0-l)$$ ? – 2017-01-04
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    @user5319165 Intuitively, if you flipped those fractions upside down, that would almost be a derivative. And yes, if you worked it out, it still comes out to $al_0$. – 2017-01-04