Geometrical Meaning of Differentiation

Excerpts

Geometrical Meaning of Differentiation
This tangent to the curve has evidently the same slope as $QT$, so that $\dfrac{dy}{dx}$ is the slope of the tangent to the curve at the point $Q$ for which the value of $\dfrac{dy}{dx}$ is found.
Geometrical Meaning of Differentiation
For a horizontal line, or a horizontal place in a curve, $dy=0$, and therefore $\dfrac{dy}{dx}=0$.
Geometrical Meaning of Differentiation
We have seen that the short expression “the slope of a curve” has no precise meaning, because a curve has so many slopes---in fact, every small portion of a curve has a different slope.
Geometrical Meaning of Differentiation
“The slope of a curve *at a point*” is, however, a perfectly defined thing; it is the slope of a very small portion of the curve situated just at that point; and we have seen that this is the same as “the slope of the tangent to the curve at that point.”
Geometrical Meaning of Differentiation
Observe that $dx$ is a short step to the right, and $dy$ the corresponding short step upwards. These steps must be considered as short as possible---in fact indefinitely short,---though in diagrams we have to represent them by bits that are not infinitesimally small, otherwise they could not be seen.
Geometrical Meaning of Differentiation
If a curve is sloping up at $45°$ at a particular point, as in [fig:8]Fig. 8, $dy$ and $dx$ will be equal, and the value of $\dfrac{dy}{dx} = 1$.
Geometrical Meaning of Differentiation
If a curve slopes *downward*, as in [fig:11]Fig. 11, $dy$ will be a step down, and must therefore be reckoned of negative value; hence $\dfrac{dy}{dx}$ will have negative sign also.
Geometrical Meaning of Differentiation
The characteristic of a minimum is that $y$ must increase *on either side* of it.
Geometrical Meaning of Differentiation
The slope of the tangent is the slope of the curve at the point where they touch one another (see slope); that is, it is the $\dfrac{dy}{dx}$ of the curve for that point.
Geometrical Meaning of Differentiation
In all exercises dealing with curves, students will find it extremely instructive to verify the deductions obtained by actually plotting the curves.

Equations

Geometrical Meaning of Differentiation
y=x+b
A straight line of slope 1 that crosses the y-axis at height b and rises at 45 degrees.
Geometrical Meaning of Differentiation
\dfrac{dy}{dx} = 1
Differentiating y = x + b gives a slope of 1 at every point, so the line has constant slope.
Geometrical Meaning of Differentiation
y = ax+b
A straight line with constant slope a that crosses the y-axis at height b.
Geometrical Meaning of Differentiation
\dfrac{dy}{dx} = a
Differentiating y = ax + b gives a constant slope equal to a.
Geometrical Meaning of Differentiation
y = ax^2 + b
A parabola with its vertex at height b on the y-axis, whose steepness changes with x.
Geometrical Meaning of Differentiation
\frac{dy}{dx} = 2ax
Differentiating y = ax^2 + b gives a slope that is proportional to x, so the steepness increases with x.
Geometrical Meaning of Differentiation
\dfrac{dy}{dx}=0
At a horizontal place on a curve the slope dy/dx is zero.
Geometrical Meaning of Differentiation
\dfrac{dy}{dx}= 0
At the x where y is a minimum or a maximum, the slope dy/dx is zero.

Problems

Exercise VIII

Exercise VIII, problem 1, p. 91
Plot the curve $y = \tfrac{3}{4} x^2 - 5$, using a scale of millimetres. Measure at points corresponding to different values of $x$, the angle of its slope. Find, by differentiating the equation, the expression for slope; and see, from a Table of Natural Tangents, whether this agrees with the measured angle.
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: no printed answer to check
Exercise VIII, problem 10a, p. 92
A straight line $y = 2x - b$ touches a curve $y = 3x^2 + 2$ at one point. What are the coordinates of the point of contact, and what is the value of $b$?
Printed answer:
- $x = \frac{1}{3}$, $y = 2 \frac{1}{3}$, $b = -\frac{5}{3}$.
unverified: no computed check settled this one (yet)
How it was checked
- other: the check does not fit this problem 1/3
Exercise VIII, problem 10b, p. 92
A straight line $y = 2x - b$ touches a curve $y = 3x^2 + 2$ at one point. What are the coordinates of the point of contact, and what is the value of $b$?
Printed answer:
- $x = \frac{1}{3}$, $y = 2 \frac{1}{3}$, $b = -\frac{5}{3}$.
unverified: no computed check settled this one (yet)
How it was checked
- other: the check does not fit this problem 7/3
Exercise VIII, problem 10c, p. 92
A straight line $y = 2x - b$ touches a curve $y = 3x^2 + 2$ at one point. What are the coordinates of the point of contact, and what is the value of $b$?
Printed answer:
- $x = \frac{1}{3}$, $y = 2 \frac{1}{3}$, $b = -\frac{5}{3}$.
unverified: no computed check settled this one (yet)
How it was checked
- other: the check does not fit this problem -5/3
Exercise VIII, problem 2, p. 91
Find what will be the slope of the curve y = 0.12x^3 - 2, at the particular point that has as abscissa $x = 2$.
Printed answer:
- $1.44$.
verified: the printed answer passed a computed check
How it was checked
- differentiate: passes 0.36*x**2
- evaluate: passes 0.36*x**2
Exercise VIII, problem 3, p. 91
If $y = (x - a)(x - b)$, show that at the particular point of the curve where $\dfrac{dy}{dx} = 0$, $x$ will have the value $\tfrac{1}{2} (a + b)$.
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: no printed answer to check
Exercise VIII, problem 4, p. 91
Find the $\dfrac{dy}{dx}$ of the equation $y = x^3 + 3x$; and calculate the numerical values of $\dfrac{dy}{dx}$ for the points corresponding to $x = 0$, $x = \tfrac{1}{2}$, $x = 1$, $x = 2$.
Printed answer:
- $\dfrac{dy}{dx} = 3x^2 + 3$; and the numerical values are: $3$, $3 \frac{3}{4}$, $6$, and $15$.
verified: the printed answer passed a computed check
How it was checked
- differentiate: passes 3*x**2 + 3
- evaluate: passes 3*x**2 + 3
- evaluate: passes 3*x**2 + 3
- evaluate: passes 3*x**2 + 3
- evaluate: passes 3*x**2 + 3
Exercise VIII, problem 5, p. 91
In the curve to which the equation is $x^2 + y^2 = 4$, find the values of $x$ at those points where the slope ${} = 1$.
Printed answer:
- $ ± \sqrt{2}$.
unverified: no computed check settled this one (yet)
How it was checked
- other: no printed answer to check sqrt(2)
Exercise VIII, problem 6, p. 91
Find the slope, at any point, of the curve whose equation is $\dfrac{x^2 }{3^2} + \dfrac{y^2}{2^2} = 1$; and give the numerical value of the slope at the place where $x = 0$, and at that where $x = 1$.
Printed answer:
- $ \dfrac{dy}{dx} = - \dfrac{4}{9} \dfrac{x}{y}$. Slope is zero where $x = 0$; and is $\mp \dfrac{1}{3 \sqrt{2}}$ where $x = 1$.
unverified: no computed check settled this one (yet)
How it was checked
- other: no printed answer to check -4*x/(9*y)
Exercise VIII, problem 7a, p. 91
The equation of a tangent to the curve $y = 5 - 2x + 0.5x^3$, being of the form $y = mx + n$, where $m$ and $n$ are constants, find the value of $m$ and $n$ if 104.png92% the point where the tangent touches the curve has $x=2$ for abscissa.
Printed answer:
- $m = 4$, $n = -3$.
verified: the printed answer passed a computed check
How it was checked
- differentiate: passes -2 + 1.5*x**2
- evaluate: passes -2 + 1.5*x**2
Exercise VIII, problem 7b, p. 91
The equation of a tangent to the curve $y = 5 - 2x + 0.5x^3$, being of the form $y = mx + n$, where $m$ and $n$ are constants, find the value of $m$ and $n$ if 104.png92% the point where the tangent touches the curve has $x=2$ for abscissa.
Printed answer:
- $m = 4$, $n = -3$.
unverified: no computed check settled this one (yet)
How it was checked
- other: the check does not fit this problem -3
Exercise VIII, problem 8, p. 92
At what angle do the two curves y = 3.5x^2 + 2 and y = x^2 - 5x + 9.5 cut one another?
Printed answer:
- Intersections at $x = 1$, $x = -3$. Angles $153°\;26'$, $2°\;28'$.
unverified: no computed check settled this one (yet)
How it was checked
- other: no printed answer to check
Exercise VIII, problem 9a, p. 91
Tangents to the curve $y = ± \sqrt{25-x^2}$ are drawn at points for which $x = 3$ and $x = 4$. Find the coordinates of the point of intersection of the tangents and their mutual inclination.
Printed answer:
- Intersection at $x = 3.57$, $y = 3.50$. Angle $16°\;16'$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 25/7
Exercise VIII, problem 9b, p. 92
Tangents to the curve $y = ± \sqrt{25-x^2}$ are drawn at points for which $x = 3$ and $x = 4$. Find the coordinates of the point of intersection of the tangents and their mutual inclination.
Printed answer:
- Intersection at $x = 3.57$, $y = 3.50$. Angle $16°\;16'$.
unverified: no computed check settled this one (yet)
How it was checked
- other: no printed answer to check 3.5
Exercise VIII, problem 9c, p. 92
Tangents to the curve $y = ± \sqrt{25-x^2}$ are drawn at points for which $x = 3$ and $x = 4$. Find the coordinates of the point of intersection of the tangents and their mutual inclination.
Printed answer:
- Intersection at $x = 3.57$, $y = 3.50$. Angle $16°\;16'$.
unverified: no computed check settled this one (yet)
How it was checked
- other: no printed answer to check