Chapter I

CHAPTER I

FUNCTIONS

1. Dependence.

There are countless instances in which onequantity depends upon another. The speed of a body fallingfrom rest depends upon the time it has fallen. One’s incomefrom a given investment depends upon the amount investedand the rate of interest realized. The crops depend upon rainfall,soil fertility and proper cultivation.In mathematics we usually deal with quantities that aredefinitely and completely determined by certain others. Thusthe area $A$ of a square is determined precisely when the length$s$ of its side is given: $A=s^2$; the volume of a sphere is $\pi r^3/3$;the force of attraction between two bodies is $k\cdot m\cdot m^{\prime }/d^2$, where$m$ and $m^{\prime }$ are their masses, $d$. the distance between them, and $k$a certain number given by experiment. The Calculus is thestudy of the relations between such interdependent quantities,with special reference to their rates of change.

2. Variables. Constants. Functions. A quantity whichmay change is called a variable. The quantities mentioned in§1, except $k$ and $\pi$, are examples of variables.

A quantity which has a fixed value is called a constant. Examplesof constants are ordinary numbers: 1, $\sqrt{2},-7,2/3,\pi$,${30}^{\mathrm{o}},\log 6$, and the number $k$ in §1.

If one variable $y$ depends on another variable $x$, so that $y$ isdetermined when $x$ is known, $y$ is said to be a function of $x$.The variable $x$, thus thought of as determining the other, iscalled the independent variable; the other variable $y$ is calledthe dependent variable. Thus, in §1, thearea $A$ of a square is a function, $A=s^s$, ofthe side $s$.

In Algebra we learn how to express suchrelations by means of equations.

In Analytic Geometry such relations arerepresented graphically. For example, ifthe principal at simple interest is a fixedsum $p$ and if the interest rate $r$ also isfixed, then the amount $a$, of principal andinterest, varies solely with (is a function of)the time $t$ that the principal has been at interest. In fact, if $p$$=100$ and $r=6\%$,

$$a=p+ptr=100+6t.$$

This is represented graphically in Fig. 1. In practice fractionalparts of a day are neglected.

The relation $A=s^2$ of §1 is represented in Fig. 2.

EXERCISES I.–FUNCTIONS AND GRAPHS

Represent graphically the following: –1. $a=100+3t,a=300+4t,a=160+7t$.

2. The number of feet $f$ in terms of thenumber of yards $y$ in a given length is givenby the equation $f=3y$.

3. The temperature in degrees Fahrenheit, F, is 32 more than 9/5 thetemperature in degrees Centigrade, C.

4. The distance $s$ that a body falls from rest in a time $t$ Is given by$s=16t^2$. (Measure $t$ horizontally and $s$ vertically downward.)

5. (a)$y=x^2+3x+1$. $(b)y=2x^2-6x$.

(c) $y=x+2$. (d)$y=\frac{1}{x+1}\cdot$

(e)$y=\frac{x-1}{x+2}$. (f)$y=\frac{x^2+2x+3}{x-6}$.

6. The volume $v$ of a fixed quantity of gas at a constant temperaturevaries inversely as the pressure $p$ upon the gas.

7. The amount of $ 1.00 at compound interest at 10% per annum fort years i8 $a={(1+1/10)}^t$.

8. The area $A$ of an equilateral triangle is a function of its side $s$.Determine this function, and represent the relation graphioally. Expressthe side in terms of the area.

9. Determine the area $a$ of a circle in terms of its radius $r$. Determinethe radius in terms of the area.

10. The radius, surface, and volume of a sphere are functionallyrelated. Find the equations connecting each pair. Also express each ofthe three as a function of the circumference of a great circle of the sphere.

11. The area $A$ bounded by the straight line $y=ax+b$, the ordinate$y$, and the axes, is a function of $x$. Determine it; and also express $y$ asa function of the area.

3. The Function Notation.A very useful abbreviation forfunctions consists in writing $f(x)$ (read $f$ of $x$) in place of thegiven expression.

Thus if $f(x)=x^2+3x+1$, we may write $f(2)=2^2+3\cdot 2+1=11$, that is,the value of $x^2+3x+1$ when $x=2$ is 11.Likewise $f(3)=19,f(-1)=-1,f(0)=1$, and so on. $f(a)=a^2+3a+1$.$f(u+v)={(u+v)}^2+3(u+v)+1$.

Other letters than $f$ are often used, to avoid confusion, but$f$ is used most often, because it is the initial of the word function.Other letters than $x$ are often used for the variable. Inany case, given $f(x)$, to find $f(a)$, simply substitute $a$ for $x$ inthe given expression.

EXERCISES II. SUBSTITUTION  FUNCTION NOTATION

1. If $f(x)=x^2-6x+2$ find $f(1),f(2),f(3),f(4),f(0),f(-1)$,$f(-2)$. From these values (and others, if needed) draw the graph ofthe curve $y=f(x)$. Mark its lowest point, and estimate the values of $x$and $y$ there.

2. Proceed as in Ex. 1 for each of the following functions using thefunction notation in calculating values; mark the highest and lowestpoints if any exist, and estimate the values of $x$ and $y$ at these points.

(a)$d-2x+4$. (b)$3x^2-2x+1$. (c) ${\displaystyle \frac{x+1}{2x-3}}$. (d)$\frac{1}{x+1}+\frac{2}{x-1}$.

$(e)y=\sin x$, taking $x=\pi /6,\pi /4,\pi /2,3\pi /4,\pi ,0,-\pi /2$.

$(f)y={\log }_{10}x$, taking $x=1,2,10,1/10,1/100$.

3. If $f(x)=x^4-6x^3+3x^2-2x+3$, calculate $f(1),f(4),f(6)$.Hence show that one solution of the equation $f(x)=0$ is $x=1$; andthat another solution lies between 4 and 5.

(This work is simplified by using the theorem that $f(a)$ is equal to theremainder obtained by dividing $f(x)$ by $(x-a)$; and by using syntheticdivision.)

4. If $f(x)=2x^2-3x+5$, show that $f(a)=2a^2-3a+5$,$f(m+n)=2{(m+n)}^2-3(m+n)+5$; find $f(a-b),f(a+2b),f(a/b)$.

5. If $f(x)=x^2+3$ and $\varphi (x)=3x+1$, show that $f(1)=\varphi (1)$ and$f(2)=\varphi (2)$. Show that $f(3)\mathrm{\&}gt;\varphi (3)$. Draw $y=f(x)$ and $y=\varphi (x)$.

6. In Ex. 5, draw the curve $y=f(x)-\varphi (x)$. Mark the pointswhere $f(x)-\varphi (x)=0$. Mark the lowest point.

7. If $f(x)=-2x^2+1$ and $\varphi (x)=x^2+2x+4$, find the value forwhich $f(x)=\varphi (x)$ by use of $f(x)-\varphi (x)$. Sketch all of the curves$y=f(x),y=\varphi (x),y=f(x)-\varphi (x)$.

8. If $f(x)=\sin x$ and $\varphi (x)=\cos x$, show that ${[f(x)]}^2+{[\varphi (x)]}^2=1$ ;$f(x)÷\varphi (x)=\tan x;f(x+y)=f(x)\varphi (y)+f(y)\varphi (x);\varphi (x+y)=\mathrm{?};f(x)=\varphi (\pi /2-x);\varphi (x)=f(\pi /2-x)=-\varphi (\pi -x);f(-x)=-f(+x);\varphi (-x)=\varphi (x)$.

9. If $f(x)={\log }_{10}x$, show that

$$f(x)+f(y)=f(x\cdot y);f(x^2)=2f(x);$$

$f(m/n)-f(n/m)=2f(m)-2f(n)$; $f(m/n)+f(n/m)=0$.

10. If $f(x)=\tan x,\varphi (x)=\cos x$, draw the curves $y=f(x),y=\varphi (x)$,$y=f(x)-\varphi (x)$. Mark the points where $f(x)=\varphi (x)$ and estimate thevalues of $x$ and $y$ there.

11. Taking $f(x)=x^2$, compare the graph of $y=f(x)$ with that of$y=f(x)+1$, and with that of $y=f(x+1)$.

12. Taking any two curves $y=f(x),y=\varphi (x)$, how can you mosteasily draw $y=f(x)-\varphi (x)\mathrm{?}y=f(x)+\varphi (x)$ ? Draw $y=x^2+1/x$.

13. How can you most easily draw $y=f(x)+5$ ? $y=f(x+5)$ ?assuming that $y=f(x)$ is drawn.

l4. Draw $y=x^2$ and show how to deduce from it the graph of$y=2x^2$; the graph of $y=-x^2$.

Assuming that $y=f(x)$ is drawn, show how to draw the graph of$y=2f(x)$; that of $y=-f(x)$.

15. From the graph of $y=x^2$, show how to draw the graph of$y={(2x)}^2$; that of $y=x^2+2$; that of $y={(x+2)}^2$;that of $y={(2x-3)}^2$.

16. What change is made in a curve if $x$, in the equation, is replacedby $-x$ ? if $y$ by $-y$ ? if both things are done ? Compare the graphs of$y=f(x),y=f(-x),-y=f(x);y=2f(x);y=f(x)+2$.

17. What change is made in a curve if $x$ is replaced by $2x,3x,x/2$?Compare the graphs of $y=f(x),y=f(2x),y=f(3x),y=f(x/2)$;$y=f(x+2)$.

18. What Is the effect upon a curve if, In the equation, $x$ and $y$ areinterchanged ? Compare the graphs of $y=f(x),x=f(y)$.

19. Plot the following curves: $(a)y+2=\sin (3x+2),(b)y=x+\sin x$,(c) $y=2^l-\sin x,(d)y=2^x\cos x,(e)3x+4y=4\sin (4x-3y)$,(f) $y=(\cos x)/(2x+3)$, (g)$\sin y=\cos 2x$, (h)$y={\log }_2(x^2+1)$.

20. In polar coordinates $(r,\theta )$, what change is made in a curve if, inthe equation, $\theta$ is replaced by 2 $\theta$, if $r$ is replaced by $2r$?

21. What change in $\theta$ is equivalent to a change in the sense of $r$.

22. From the graph of $r=f(\theta )$ derive those of (a) $f=f(2\theta )$,(b)$r=2f(\theta )$, (c)$r=f(-\theta )$, (d)$r=-f(\theta )$, (e)$f+1=f(\theta )$,(f)$r=f(\theta +1)$, (g)$r+1=f(\theta +2)$.

Take, for example, $f(\theta )=1,f(\theta )=\theta ,f(\theta )=\sin \theta ,f(\theta )=2\theta ,f(\theta )=\arctan \theta$, and draw the variations from the original graph.

23. Plot the following: (a)$r=2+3\cos \theta$, (b)$r=3+2\cos \theta$,(c)$r=2+2\cos \theta$, (d)$r=2^{\theta }$, (e)$r^2=a\theta$, (f)$\theta =2^r$, (g)${\theta }^2=ar$,(h)$\theta =\sin r$, (i)$\theta =\cos r$, (j)$\theta =\tan r$, (k)$r=\sec (\theta -a)$, (l)$\theta =\sec r$.

24. Show how to obtain the graph of $y=A\sin (at+b)$ by suitablemodification of the simple sine curve $y=\sin t$.

25. Draw the graphs from the following equations: $(a)2s=e^t+e^{-t}$,(b)$2s=e^t-e^{-t}$, (c) $s=(e^t+e^{-l})/(e^l-e^{-l})$, (d)$s=\sin t+\sin 2t$,(e)$s=\sin t+e^{-t}\sin 2t$. Take $e=2.7$, and use logarithms in computations.