Curvature of Curves

Excerpts

Curvature of Curves
Clearly it means the rate (per unit of length $x$) at which the slope is changing---in brief, it is *a measure of the curvature of the slope*.
Curvature of Curves
If $\dfrac{d^2y}{dx^2}$ comes out *positive*, then you know that the value of $y$ which you got was a *minimum*; but if $\dfrac{d^2y}{dx^2}$ comes out *negative*, then the value of $y$ which you got must be a *maximum*. That’s the rule.
Curvature of Curves
To the left of $M$ the slope is downward, that is, negative, and is getting less negative.
Curvature of Curves
But what can $\dfrac{d^2 y}{dx^2}$ mean in this case? Clearly it means the rate (per unit of length $x$) at which the slope is changing---in brief, it is *a measure of the curvature of the slope*.
Curvature of Curves
It is now time to initiate you into another secret---how to tell whether the result that you get by “equating to zero” is a maximum or a minimum. The trick is this: After you have differentiated (so as to get the expression which you equate to zero), you then differentiate a second time, and look whether the result of the second differentiation is *positive* or *negative*. If $\dfrac{d^2y}{dx^2}$ comes out *positive*, then you know that the value of $y$ which you got was a *minimum*; but if $\dfrac{d^2y}{dx^2}$ comes out *negative*, then the value of $y$ which you got must be a *maximum*. That’s the rule.
Curvature of Curves
Clearly the change of slope as the curve passes through $M$ is such that $\dfrac{d^2y}{dx^2}$ is *positive*, for its operation, as $x$ increases toward the right, is to convert a downward slope into an upward one.
Curvature of Curves
In this case, as the curve passes through $M$ from left to right, its upward slope is converted into a downward or negative slope, so that in this case the “slope of the slope” $\dfrac{d^2y}{dx^2}$ is *negative*.
Curvature of Curves
The denominator is always positive, so it is sufficient to ascertain the sign of the numerator.
Curvature of Curves
The expense $C$ of handling the products of a certain factory varies with the weekly output $P$ according to the relation $C = aP + \dfrac{b}{c+P} + d$, where $a$, $b$, $c$, $d$ are positive constants. For what output will the expense be least?

Equations

Curvature of Curves
C = aP + \dfrac{b}{c+P} + d
The expense C of handling the factory's products is modelled as a function of the weekly output P, plus a term b/(c+P) and a constant d.
Curvature of Curves
\dfrac{dC}{dP} = a - \frac{b}{(c+P)^2} = 0
Setting the derivative of the expense with respect to the output to zero gives the candidate outputs for a maximum or minimum.
Curvature of Curves
P = ±\sqrt{\dfrac{b}{a}} - c
The stationary output is P equal to the square root of b/a minus c; the book keeps only the positive sign, since output cannot be negative.
Curvature of Curves
C = N\left(\frac{C_l}{t} + \frac{EPC_e}{1000}\right)
The total cost per hour of lighting a building with N lamps is the renewal cost per lamp plus the energy cost per lamp.
Curvature of Curves
t = mE^n
The average life of a lamp is approximately a constant power of its commercial efficiency.
Curvature of Curves
\frac{PC_e}{1000} - \frac{nC_l}{m} E^{-(n+1)} = 0
Setting the derivative of the total cost with respect to the commercial efficiency to zero gives the candidate efficiency for a maximum or minimum.
Curvature of Curves
E = \sqrt[n+1]{\frac{1000 × nC_l}{mPC_e}}
The commercial efficiency that makes the total cost of lighting least is the (n+1)th root of 1000 n C_l divided by m P C_e.
Curvature of Curves
\frac{d^2C}{dE^2} = (n + 1) \frac{nC_l}{m} E^{-(n+2)}
The second derivative of the total cost with respect to efficiency is positive for positive E, so the stationary value found is a minimum.

Problems

Exercise X

Exercise X, problem 101, p. 120
Suppose it to be known that consumption of coal by a certain steamer may be represented by the formula $y = 0.3 + 0.001v^3$; where $y$ is the number of tons of coal burned per hour and $v$ is the speed expressed in nautical miles per hour. The cost of wages, interest on capital, and depreciation of that ship are together equal, per hour, to the cost of $1$ ton of coal. What speed will make the total cost of a voyage of $1000$ nautical miles a minimum? And, if coal costs $10$ shillings per ton, what will that minimum cost of the voyage amount to?
Printed answer:
- Speed $8.66$ nautical miles per hour. Time taken $115.47$ hours. Minimum cost £$112$. $12$*s*.
verified: the printed answer passed a computed check
How it was checked
- extremum: passes, with the problem read into an equation 650**(Rational(1,3))
- evaluate: passes 650**(Rational(1,3))
Exercise X, problem 102, p. 118
Suppose it to be known that consumption of coal by a certain steamer may be represented by the formula $y = 0.3 + 0.001v^3$; where $y$ is the number of tons of coal burned per hour and $v$ is the speed expressed in nautical miles per hour. The cost of wages, interest on capital, and depreciation of that ship are together equal, per hour, to the cost of $1$ ton of coal. What speed will make the total cost of a voyage of $1000$ nautical miles a minimum? And, if coal costs $10$ shillings per ton, what will that minimum cost of the voyage amount to?
Printed answer:
- Speed $8.66$ nautical miles per hour. Time taken $115.47$ hours. Minimum cost £$112$. $12$*s*.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 650**(Rational(1,3))
Exercise X, problem 103, p. 120
Suppose it to be known that consumption of coal by a certain steamer may be represented by the formula $y = 0.3 + 0.001v^3$; where $y$ is the number of tons of coal burned per hour and $v$ is the speed expressed in nautical miles per hour. The cost of wages, interest on capital, and depreciation of that ship are together equal, per hour, to the cost of $1$ ton of coal. What speed will make the total cost of a voyage of $1000$ nautical miles a minimum? And, if coal costs $10$ shillings per ton, what will that minimum cost of the voyage amount to?
Printed answer:
- Speed $8.66$ nautical miles per hour. Time taken $115.47$ hours. Minimum cost £$112$. $12$*s*.
unverified: no computed check settled this one (yet)
How it was checked
- evaluate: no printed answer to check
Exercise X, problem 11, p. 118
Find the maxima and minima of y = x^3 + x^2 - 10x + 8.
Printed answer:
- Max.: $x = -2.19$, $y = 24.19$; min.:, $x = 1.52$, $y = -1.38$.
unverified: no computed check settled this one (yet)
How it was checked
- extremum: the printed answer does not match the problem (-1 - sqrt(31))/3
- evaluate: the printed answer does not match the problem (-1 - sqrt(31))/3
- evaluate: passes (-1 - sqrt(31))/3
Exercise X, problem 111, p. 120
Find the maxima and minima ofEx:X11% y = ±x6x(10-x).
Printed answer:
- Max. and min. for $x = 7.5$, $y = ±5.414$. (See example no. 10, Example10.)
unverified: no computed check settled this one (yet)
How it was checked
- extremum: passes Rational(15,2)
- evaluate: passes Rational(15,2)
- evaluate: the printed answer does not match the problem Rational(15,2)
Exercise X, problem 112, p. 120
Find the maxima and minima ofEx:X11% y = ±x6x(10-x).
Printed answer:
- Max. and min. for $x = 7.5$, $y = ±5.414$. (See example no. 10, Example10.)
unverified: no computed check settled this one (yet)
How it was checked
- extremum: passes Rational(15,2)
- evaluate: passes Rational(15,2)
- evaluate: the printed answer does not match the problem Rational(15,2)
Exercise X, problem 12, p. 118
Find the maxima and minima of y = x^3 + x^2 - 10x + 8.
Printed answer:
- Max.: $x = -2.19$, $y = 24.19$; min.:, $x = 1.52$, $y = -1.38$.
verified: the printed answer passed a computed check
How it was checked
- extremum: passes (-1 + sqrt(31))/3
- evaluate: passes (-1 + sqrt(31))/3
- evaluate: passes (-1 + sqrt(31))/3
Exercise X, problem 121, p. 120
Find the maxima and minima of y= 4x^3 - x^2 - 2x + 1.
Printed answer:
- Min.: $x = \frac{1}{2}$, $y= 0.25$; max.: $x = - \frac{1}{3}$, $y= 1.408$.
verified: the printed answer passed a computed check
How it was checked
- extremum: passes Rational(1,2)
- evaluate: passes Rational(1,2)
- evaluate: passes Rational(1,2)
Exercise X, problem 122, p. 120
Find the maxima and minima of y= 4x^3 - x^2 - 2x + 1.
Printed answer:
- Min.: $x = \frac{1}{2}$, $y= 0.25$; max.: $x = - \frac{1}{3}$, $y= 1.408$.
verified: the printed answer passed a computed check
How it was checked
- extremum: passes -Rational(1,3)
- evaluate: passes -Rational(1,3)
- evaluate: PASS-LOOSE -Rational(1,3)
Exercise X, problem 21, p. 118
Given $y = \dfrac{b}{a}x - cx^2$, find expressions for $\dfrac{dy}{dx}$, and for $\dfrac{d^2y}{dx^2}$, also find the value of $x$ which makes $y$ a maximum or a minimum, and show whether it is maximum or minimum.
Printed answer:
- $\dfrac{dy}{dx} = \dfrac{b}{a} - 2cx$; $\dfrac{d^2 y}{dx^2} = -2c$; $x = \dfrac{b}{2ac}$ (*a maximum*).
verified: the printed answer passed a computed check
How it was checked
- differentiate: passes b/a - 2*c*x
Exercise X, problem 22, p. 118
Given $y = \dfrac{b}{a}x - cx^2$, find expressions for $\dfrac{dy}{dx}$, and for $\dfrac{d^2y}{dx^2}$, also find the value of $x$ which makes $y$ a maximum or a minimum, and show whether it is maximum or minimum.
Printed answer:
- $\dfrac{dy}{dx} = \dfrac{b}{a} - 2cx$; $\dfrac{d^2 y}{dx^2} = -2c$; $x = \dfrac{b}{2ac}$ (*a maximum*).
verified: the printed answer passed a computed check
How it was checked
- differentiate2: passes -2*c
Exercise X, problem 23, p. 118
Given $y = \dfrac{b}{a}x - cx^2$, find expressions for $\dfrac{dy}{dx}$, and for $\dfrac{d^2y}{dx^2}$, also find the value of $x$ which makes $y$ a maximum or a minimum, and show whether it is maximum or minimum.
Printed answer:
- $\dfrac{dy}{dx} = \dfrac{b}{a} - 2cx$; $\dfrac{d^2 y}{dx^2} = -2c$; $x = \dfrac{b}{2ac}$ (*a maximum*).
verified: the printed answer passed a computed check
How it was checked
- extremum: passes b/(2*a*c)
Exercise X, problem 3a, p. 118
Find how many maxima and how many minima there are in the curve, the equation to which is y = 1 - x^22 + x^424; and how many in that of which the equation is y = 1 - x^22 + x^424 - x^6720.
Printed answer:
- (*a*) One maximum and two minima. (*b*) One maximum. ($x = 0$; other points unreal.)
unverified: no computed check settled this one (yet)
How it was checked
- other: no printed answer to check
Exercise X, problem 3b, p. 118
Find how many maxima and how many minima there are in the curve, the equation to which is y = 1 - x^22 + x^424; and how many in that of which the equation is y = 1 - x^22 + x^424 - x^6720.
Printed answer:
- (*a*) One maximum and two minima. (*b*) One maximum. ($x = 0$; other points unreal.)
unverified: no computed check settled this one (yet)
How it was checked
- other: no printed answer to check
Exercise X, problem 4, p. 119
Find the maxima and minima of y=2x+1+5x^2.
Printed answer:
- Min.: $x = 1.71$, $y = 6.14$.
unverified: no computed check settled this one (yet)
How it was checked
- extremum: passes 5**(Rational(1,3))
- evaluate: passes 5**(Rational(1,3))
- evaluate: the printed answer does not match the problem 5**(Rational(1,3))
Exercise X, problem 5, p. 119
Find the maxima and minima of y=3x^2+x+1.
Printed answer:
- Max: $x = -.5$, $y = 4$.
verified: the printed answer passed a computed check
How it was checked
- extremum: passes -Rational(1,2)
- evaluate: passes -Rational(1,2)
- evaluate: passes -Rational(1,2)
Exercise X, problem 61, p. 119
Find the maxima and minima of y=5x2+x^2.
Printed answer:
- Max.: $x = 1.414$, $y = 1.7675$. Min.: $x = -1.414$, $y = 1.7675$.
unverified: no computed check settled this one (yet)
How it was checked
- extremum: passes sqrt(2)
- evaluate: passes sqrt(2)
- evaluate: the printed answer does not match the problem sqrt(2)
Exercise X, problem 62, p. 119
Find the maxima and minima of y=5x2+x^2.
Printed answer:
- Max.: $x = 1.414$, $y = 1.7675$. Min.: $x = -1.414$, $y = 1.7675$.
unverified: no computed check settled this one (yet)
How it was checked
- extremum: passes -sqrt(2)
- evaluate: passes -sqrt(2)
- evaluate: the printed answer does not match the problem -sqrt(2)
Exercise X, problem 71, p. 119
Find the maxima and minima of y=3xx^2-3 + x2 + 5.
Printed answer:
- Max.: $x = -3.565$, $y = 2.12$. Min.: $x = +3.565$, $y = 7.88$.
verified: the printed answer passed a computed check
How it was checked
- extremum: passes -sqrt(6 + 3*sqrt(5))
- evaluate: passes -sqrt(6 + 3*sqrt(5))
- evaluate: passes -sqrt(6 + 3*sqrt(5))
Exercise X, problem 72, p. 119
Find the maxima and minima of y=3xx^2-3 + x2 + 5.
Printed answer:
- Max.: $x = -3.565$, $y = 2.12$. Min.: $x = +3.565$, $y = 7.88$.
verified: the printed answer passed a computed check
How it was checked
- extremum: passes sqrt(6 + 3*sqrt(5))
- evaluate: passes sqrt(6 + 3*sqrt(5))
- evaluate: passes sqrt(6 + 3*sqrt(5))
Exercise X, problem 8, p. 119
Divide a number $N$ into two parts in such a way that three times the square of one part plus twice the square of the other part shall be a minimum.
Printed answer:
- $0.4N$, $0.6N$.
verified: the printed answer passed a computed check
How it was checked
- extremum: passes, with the problem read into an equation 2*N/5
Exercise X, problem 9, p. 119
The efficiency $u$ of an electric generator at different values of output $x$ is expressed by the general equation: u=xa+bx+cx^2; where $a$ is a constant depending chiefly on the energy losses in the iron and $c$ a constant depending chiefly on the resistance of the copper parts. Find an expression for that value of the output at which the efficiency will be a maximum.
Printed answer:
- $x = \sqrt{\dfrac{a}{c}}$.
verified: the printed answer passed a computed check
How it was checked
- extremum: passes sqrt(a/c)