The Graph of an Equation

Excerpts

The Graph of an Equation
A point (like $M$ or $M'$ in Fig. 14) is called a *bend point* of the graph of % $y=f(x)$ if the tangent to the graph at that point is horizontal and if all of the adjacent points of the graph lie below the tangent or all above the tangent.
The Graph of an Equation
We call $3x^2 + 8x$ the *derivative* of $x^3 + 4x^2 - 11$.
The Graph of an Equation
Thus the derivative of $a_0 x^n$ is $na_0 x^{n-1}$, and hence is obtained by multiplying the given term by its exponent $n$ and then diminishing its exponent by unity.
The Graph of an Equation
The true curve between two points below the $x$-axis may not cross the $x$-axis, or may have a peak and actually cross the $x$-axis twice, or may be an M-shaped curve crossing it four times, etc.
The Graph of an Equation
This formula $(8)$ is known as *Taylor’s theorem* for the present case of % a polynomial $f(x)$ of degree $n$.
The Graph of an Equation
To find geometrically the real roots of a real equation $f(x)=0$, we construct a graph of $y=f(x)$ and measure the distances from the origin $O$ to the intersections of the graph and the $x$-axis, whose equation is $y=0$.
The Graph of an Equation
The purpose of the example was, however, not to point out this obvious fact, but rather to emphasize the chance of serious error in sketching a curve through a number of points, however numerous.
The Graph of an Equation
In this sense the tangent at $O$ is said to meet the curve in three coincident points, their abscissas being the three coinciding roots of $x^3 = 0$.
The Graph of an Equation
Hence *$x^3 - 3lx + q = 0$ has three distinct real roots if and only if $q^2 < 4l^3$, a single real root if and only if $q^2 > 4l^3$, a double root necessarily real if and only if $q^2 = 4l^3$ and $l\neq 0$, and a triple root if $q^2 = 4l^3 = 0$*.
The Graph of an Equation
Between two consecutive real roots $a$ and $b$ of $f(x)=0$, there is an odd number of real roots of $f'(x) = 0$, a root of multiplicity $m$ being counted as $m$ roots.
The Graph of an Equation
For example, $x^4 + 2x^3 =0$ has the triple root $x = 0$ since $0$ is a root, and since the first and second derivatives $4x^3 +6x^2$ and $12x^2 +12x$ are zero for $x = 0$, while the third derivative $24x + 12$ is not zero for $x = 0$.
The Graph of an Equation
The use of the bend points insures greater accuracy to the graph than the use of dozens of points whose abscissas are taken at random.

Equations

The Graph of an Equation
x^2 - 6x - 3 = 0
The quadratic equation whose real roots are found graphically as the example of Use of Graphs.
The Graph of an Equation
y = x^2 - 6x - 3
The equation of the graph (a parabola) whose intersections with the x-axis give the roots of the quadratic equation.
The Graph of an Equation
y = 8x^4 - 14x^3 - 9x^2 + 11x - 2
The quartic whose graph is used to illustrate that a curve sketched through integral points can mislead about the real roots.
The Graph of an Equation
8x^4 - 14x^3 - 9x^2 + 11x - 2 = 0
The quartic equation whose real roots are -1, 2, 1/4 and 1/2 as shown by the full graph.
The Graph of an Equation
y = x^3 + 4x^2 - 11
The cubic whose graph crosses the x-axis only once.
The Graph of an Equation
\frac{Y - y}{h} = \frac{f(x+h) - f(x)}{h}
The slope of the secant PQ of the graph is the difference quotient of f over the increment h.
The Graph of an Equation
f(x) = a_0 x^n + a_1 x^{n-1} + \dotsb + a_{n-1} x + a_n
The general polynomial of degree n with real coefficients a_0 through a_n.
The Graph of an Equation
f'(x) = na_0 x^{n-1} + (n-1)a_1 x^{n-2} + \dotsb + 2a_{n-2} x + a_{n-1}
The derivative of the polynomial f, obtained by multiplying each term by its exponent and lowering the exponent by one.
The Graph of an Equation
f''(x) = n(n-1)a_0 x^{n-2} + (n-1)(n-2)a_1 x^{n-3} + \dotsb + 2a_{n-2}
The second derivative of the polynomial f, the derivative of f'.
The Graph of an Equation
0! = 1
By definition the factorial of zero is one.
The Graph of an Equation
y = f(\alpha) + f'(\alpha)(x-\alpha)
The equation of the tangent to the graph of y = f(x) at the point with abscissa alpha.
The Graph of an Equation
y-\beta = s(x -\alpha)
The equation of the straight line through the point (alpha, beta) with slope s.
The Graph of an Equation
\tan\theta=f'(\alpha)
The angle theta between the tangent and the X-axis has tangent equal to the slope f'(alpha).
The Graph of an Equation
X = x\cos\theta
Relation between the old oblique-coordinate abscissa X and the new coordinate x, with the tangent angle theta.
The Graph of an Equation
Y = f'(\alpha)X + f''(\alpha)\frac{X^2}{2} + \dotsb
The graph near the point of tangency, referred to axes parallel to the old ones through (alpha, beta), expanded in powers of X.
The Graph of an Equation
y = cx^m + dx^{m+1} + \dotsb
The curve near its point of tangency, in oblique axes with the tangent as x-axis, begins with the term c x^m.
The Graph of an Equation
c = \frac{f^{(m)}(\alpha)\cos^m \theta}{m!}
The leading coefficient c of the curve relative to the tangent, nonzero when f^(m)(alpha) is nonzero.
The Graph of an Equation
f(x) = x^3 - 3lx + q
The reduced real cubic whose real roots are classified by the sign of q^2 - 4l^3.
The Graph of an Equation
q^2 = 4l^3
The condition under which a bend point of the reduced cubic lies on the x-axis, so that the cubic has a double root.
The Graph of an Equation
q^2 < 4l^3
The condition under which the reduced cubic has three distinct real roots.
The Graph of an Equation
D = f(a+h) - f(a)
The difference whose smallness as h tends to zero defines continuity of f at a.
The Graph of an Equation
F = a_1 h + a_2 h^2 + \dotsb + a_n h^n
The polynomial in h with no constant term, shown to be numerically small for small h.
The Graph of an Equation
k < \frac{p}{p + g}
The bound on |h| that guarantees the polynomial's increment is less than p.
The Graph of an Equation
f(x) = x^n (a_0 + \phi)
Factoring x^n out of the polynomial shows that for large x its sign is that of a_0 x^n.
The Graph of an Equation
\phi = \frac{a_1}{x} + \frac{a_2}{x^2} + \dotsb + \frac{a_n}{x^n}
The quantity phi collects the lower-order terms divided by powers of x.
The Graph of an Equation
\frac{(x-a)(x-b)f'(x)}{f(x)} \equiv r(x-b) + s(x-a) + (x-a)(x-b) \frac{Q'(x)}{Q(x)}
The logarithmic-derivative identity for f = (x-a)^r (x-b)^s Q(x), used in the proof of Rolle's theorem.
The Graph of an Equation
\tfrac{1}{15}f'(x) = x^4 - 5x^2 + 4 = (x^2 - 1)(x^2 - 4)
Dividing the derivative of 3x^5 - 25x^3 + 60x - 20 by 15 gives a quartic that factors into (x^2 - 1)(x^2 - 4), locating the roots of f'(x) = 0.

Problems

Exercise Page55

The data holds no problems for this exercise yet.

Exercise Page59

Exercise Page59, problem 1, p. 59
Show that the slope of the tangent to $y = 8x^3 - 22x^2 + 13x - 2$ at $(x, y)$ is $24x^2 - 44x + 13$, and that the bend points are $(0.37, 0.203)$, $(1.46, -5.03)$, approximately. Draw the graph.
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page59, problem 10, p. 59
Prove that if $g$ and $k$ are polynomials in $x$, the derivative of $gk$ is $g'k + gk'$. Hint: multiply the members of $g(x+h) = g(x) + g'(x)h + \dotsb$ and $k(x+h) = k(x) + k'(x)h + \dotsb$ and use $(8)$ for $f = gk$.
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page59, problem 2, p. 59
Prove that the bend points of $y = x^3 - 2x - 5$ are $(.82, -6.09)$, $(-.82$, $-3.91)$, % [** PP: Allow line break between coordinates] approximately. Draw the graph and locate the real roots.
Printed answer:
- $2.1$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 2.1
Exercise Page59, problem 3a, p. 59
Find the bend points of $y = x^3 + 6x^2 + 8x + 8$. Locate the real roots.
Printed answer:
- $(-0.845, 4.921)$, $(-3.155, 11.079)$
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page59, problem 3b, p. 59
Find the bend points of $y = x^3 + 6x^2 + 8x + 8$. Locate the real roots.
Printed answer:
- between $-4$ and $-5$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page59, problem 4, p. 59
Locate the real roots of $f(x) = x^4 + x^3 - x - 2 = 0$. Hints: The abscissas of the bend points are the roots of $f'(x) = 4x^3 + 3x^2 - 1 = 0$. The bend points of $y = f'(x)$ are $(0, -1)$ and $(-\frac{1}{2}, -\frac{3}{4})$, so that $f'(x)= 0$ has a single real root (it is just less than $\frac{1}{2}$). The single bend point of $y=f(x)$ is $(\frac{1}{2}, -\frac{37}{16})$, approximately.
Printed answer:
- $1.1$, $-1.3$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [1.1, -1.3]
Exercise Page59, problem 5, p. 59
Locate the real roots of $x^6 - 7x^4 - 3x^2 + 7 = 0$.
Printed answer:
- Between $0$ and $1$, $0$ and $-1$, $2.5$ and $3$, $-2.5$ and $-3$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page59, problem 6, p. 59
Prove that $f''(x)$, given by $(7)$, is equal to the first derivative of $f'(x)$.
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page59, problem 7, p. 59
If $f(x) = f_1(x) + f_2(x)$, prove that the $k$th derivative of $f$ is equal to the sum of the $k$th derivatives of $f_1$ and $f_2$. Use $(8)$.
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page59, problem 8, p. 59
Prove that $f^{(k)}(x)$ is equal to the first derivative of $f^{(k-1)}(x)$. Hint: prove this for $f = ax^m$; then prove that it is true for $f=f_1 + f_2$ if true for $f_1$ and $f_2$.
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page59, problem 9a, p. 59
Find the third derivative of $x^6 + 5x^4$ by forming successive first derivatives; also that of $2x^5 - 7x^3 + x$.
Printed answer:
- $120(x^3 + x)$
unverified: no computed check settled this one (yet)
How it was checked
- differentiate: the printed answer does not match the problem 120*(x**3 + x)
Exercise Page59, problem 9b, p. 59
Find the third derivative of $x^6 + 5x^4$ by forming successive first derivatives; also that of $2x^5 - 7x^3 + x$.
Printed answer:
- $120x^2 - 42$.
unverified: no computed check settled this one (yet)
How it was checked
- differentiate: the printed answer does not match the problem 120*x**2 - 42

Exercise Page62

Exercise Page62, problem 1, p. 62
Prove that $x^3 - 7x^2 + 15x - 9 = 0$ has a double root.
Printed answer:
- $3$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles 3
Exercise Page62, problem 2, p. 62
Show that $x^4 - 8x^2 + 16 = 0$ has two double roots.
Printed answer:
- $2$, $-2$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles [2, -2]
Exercise Page62, problem 3, p. 62
Prove that $x^4 - 6x^2 - 8x - 3 = 0$ has a triple root.
Printed answer:
- $-1$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles -1
Exercise Page62, problem 4, p. 62
Test $x^4 - 8x^3 + 22x^2 - 24x + 9 = 0$ for multiple roots.
Printed answer:
- Double roots, $1$, $3$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles [1, 3]
Exercise Page62, problem 5, p. 62
Test $x^3 - 6x^2 + 11x - 6 = 0$ for multiple roots.
Printed answer:
- None.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles []
Exercise Page62, problem 6, p. 62
Test $x^4 - 9x^3 + 9x^2 + 81x - 162 = 0$ for multiple roots.
Printed answer:
- $3$, $3$, $-3$, $6$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles [3, 3, -3, 6]

Exercise Page64

Exercise Page64, problem 1, p. 64
If $f(x) = 3x^5 + 5x^3 + 4$, the only real root of $f'(x)=0$ is $x = 0$. Show that $(0, 4)$ inflexion point, and thus that there is no bend point and hence that $f(x)=0$ has a single real root.
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: the record may be misread
Exercise Page64, problem 2, p. 64
Prove that $x^3 - 3x^2 + 3x + c = 0$ has an inflexion point, but no bend point.
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page64, problem 3, p. 64
Show that $x^5 - 10x^3 - 20x^2 - 15x + c = 0$ has two bend points and no horizontal inflexion tangents.
Printed answer:
- Use Ex. 3, p. 62, abscissas $-1$, $3$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page64, problem 4, p. 64
Prove that $3x^5 - 40x^3 + 240x + c = 0$ has no bend point, but has two horizontal inflexion tangents.
Printed answer:
- Use Ex. 2, p. 62.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page64, problem 5, p. 64
Prove that any function $x^3 - 3\alpha x^2 + \dotsb$ of the third degree can be written in % [** PP: Not using page range] the form $f(x) = (x-\alpha)^3 + ax + b$. The straight line having the equation $y = ax+b$ meets the graph of $y=f(x)$ in three coincident points with the abscissa $\alpha$ and hence is an inflexion tangent. If we take new axes of coordinates parallel to the old and intersecting at the new origin $(\alpha, 0)$, i.e., if we make the transformation $x = X+\alpha$, $y = Y$, %% -----File: 071.png---Folio 65------- of coordinates, we see that the equation $f(x)=0$ becomes a reduced cubic equation $X^3 + pX + q = 0$ (§42).
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page64, problem 6, p. 64
Find the inflexion tangent to $y = x^3 + 6x^2 - 3x + 1$ and transform $x^3 + 6x^2 - 3x + 1 = 0$ into a reduced cubic equation.
Printed answer:
- $y = -15x - 7$, $X^3 - 15X + 23 = 0$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles

Exercise Page66

Exercise Page66, problem 1, p. 66
$x^3 + 2x - 4 = 0$.
Printed answer:
- One real.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page66, problem 10, p. 66
Prove that no straight line crosses the graph of $y = f(x)$ in more than $n$ points if the degree $n$ of the real polynomial $f(x)$ exceeds unity. [Apply §16.] This fact serves as a check on the accuracy of a graph.
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page66, problem 2, p. 66
$x^3 - 7x + 7 = 0$.
Printed answer:
- $(±\sqrt{\frac{7}{3}}, 7\mp\frac{14}{3}\sqrt{\frac{7}{3}})$, three real.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page66, problem 3, p. 66
$x^3 - 2x - 1 = 0$.
Printed answer:
- $(±\sqrt\frac{2}{3}, -1\mp\frac{4}{3}\sqrt{\frac{2}{3}})$, three.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page66, problem 4, p. 66
$x^3 + 6x^2 - 3x + 1 = 0$.
Printed answer:
- $(-2±\sqrt{5}, 23\mp10\sqrt{5})$, one.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page66, problem 5, p. 66
Prove that the inflexion point of $y = x^3 - 3lx + q$ is $(0, q)$.
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page66, problem 6, p. 66
Show that the theorem in the text is equivalent to that in §45.
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page66, problem 7, p. 66
Prove that, if $m$ and $n$ are positive odd integers and $m>n$, $x^m + px^n + q = 0$ has no bend point and hence has a single real root if $p>0$; but, if $p<0$, it has just two bend points which are on the same side or opposite sides of the $x$-axis according as (npm)^m + (nqm-n)^m-n is positive or negative, so that the number of real roots is $1$ or $3$ in the respective cases.
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page66, problem 8, p. 66
Draw the graph of $y = x^4 - x^2$. By finding its intersections with the line $y = mx + b$, solve $x^4 - x^2 - mx - b= 0$.
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- solve: no printed answer to check
Exercise Page66, problem 9, p. 66
Prove that, if $p$ and $q$ are positive, $x^{2m} - px^{2n} + q = 0$ has four distinct real roots, two pairs of equal roots, or no real root, according as (npm)^m - (nqm-n)^m-n > 0, ${} = 0$, or ${} < 0$.
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles

Exercise Page68

The data holds no problems for this exercise yet.

Exercise Page69

The data holds no problems for this exercise yet.

Exercise Page70

The data holds no problems for this exercise yet.