The Slope of Parametric Curves, calculus lesson

A parametric curve may loop and cross itself, so it is often not the graph of one function $y = F(x)$ . Near most of its points, though, it does behave like one, and then it has a slope. The chain rule finds that slope from $x(t)$ and $y(t)$ , without ever solving for $y$ in terms of $x$ .

Key terms

The slope formula

Definition Let $x = f(t)$ and $y = g(t)$ be differentiable on an open interval containing $t_0$ , with $f'$ continuous at $t_0$ and $f'(t_0) \ne 0$ . Then the part of the curve traced for $t$ near $t_0$ is the graph of a differentiable function $y = F(x)$ , and its slope at the point $(f(t_0), g(t_0))$ is

\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{g'(t_0)}{f'(t_0)}.

The reason is that $f'$ keeps one sign near $t_0$ , so $x = f(t)$ can be solved for $t$ there. Writing $y = F(x)$ , the chain rule gives $g'(t) = F'(f(t)) \, f'(t)$ , and dividing by $f'(t) \ne 0$ gives the formula.

In plain words The slope is how fast $y$ changes divided by how fast $x$ changes. It works whenever $x$ is really moving, that is when $dx/dt \ne 0$ .

Example For $x = t^2 + 1$ and $y = t^3$ the slope is

\frac{dy}{dx} = \frac{3t^2}{2t} = \frac{3t}{2}, \qquad t \ne 0.

At $t = 2$ the point is $(5, 8)$ and the slope is $3$ .

Tangent and normal lines

Definition Under the hypotheses of the slope formula, with $m = \frac{g'(t_0)}{f'(t_0)}$ and $(x_0, y_0) = (f(t_0), g(t_0))$ , the tangent line is

y - y_0 = m (x - x_0).

If $m \ne 0$ the normal line is

y - y_0 = -\frac{1}{m}(x - x_0),

and if $m = 0$ it is the vertical line $x = x_0$ . More generally, whenever $(f'(t_0), g'(t_0)) \ne (0, 0)$ , the tangent line is the line through $(x_0, y_0)$ in the direction of the vector $(f'(t_0), g'(t_0))$ .

In plain words Once you have the slope and the point, the tangent line comes from the point slope form as usual. The normal is the perpendicular line through the same point.

Example For $x = t^2 + 1$ , $y = t^3$ at $t = 2$ , the tangent is $y - 8 = 3(x - 5)$ , or $y = 3x - 7$ . The normal is

y - 8 = -\frac{1}{3}(x - 5).

Horizontal and vertical tangents

Definition Let $f$ and $g$ be differentiable near $t_0$ , with $f'$ and $g'$ continuous at $t_0$ . The curve has a horizontal tangent at $t_0$ if

g'(t_0) = 0 \quad \text{and} \quad f'(t_0) \ne 0,

and a vertical tangent at $t_0$ if $f'(t_0) = 0$ and $g'(t_0) \ne 0$ .

In plain words Horizontal means $y$ stops changing while $x$ keeps moving. Vertical means $x$ stops while $y$ keeps moving. If both stop, these tests say nothing.

Example Take $x = t^2$ , $y = t^3 - 3t$ . Here $x' = 2t$ and $y' = 3t^2 - 3$ .

$t$	$x'$	$y'$	Point	Tangent
$-1$	$-2$	$0$	$(1, 2)$	horizontal
$0$	$0$	$-3$	$(0, 0)$	vertical
$1$	$2$	$0$	$(1, -2)$	horizontal

This curve passes through $(3, 0)$ twice, at $t = \pm\sqrt{3}$ , with slopes $\frac{6}{2\sqrt{3}} = \sqrt{3}$ and $\frac{6}{-2\sqrt{3}} = -\sqrt{3}$ . So it has two different tangent lines at the same point.

Stationary parametrizations

Definition Let $f$ and $g$ be differentiable on an open interval containing $t_0$ . The parameter value $t_0$ is stationary, or singular, for the parametrization if $f'(t_0) = g'(t_0) = 0$ . At such a value the slope formula does not apply. Suppose $f'(t) \ne 0$ for all $t \ne t_0$ near $t_0$ and the finite limit

L = \lim_{t \to t_0} \frac{g'(t)}{f'(t)}

exists. Then the curve has a tangent line of slope $L$ at that point. The point can also be a cusp or a corner, and a cusp can occur even when $L$ exists. For example $x = t^2$ , $y = t^3$ has a cusp at the origin with $L = 0$ .

In plain words The moving point comes to a stop for an instant. What the curve looks like there depends on how it slows down and starts again, so you have to look more closely.

Example For $x = t^3$ , $y = t^2$ , both derivatives vanish at $t = 0$ . Since $y = x^{2/3}$ , the curve has a cusp at the origin, and the quotient

\frac{g'(t)}{f'(t)} = \frac{2t}{3t^2} = \frac{2}{3t}

is unbounded near $0$ . For $x = t^3$ , $y = t^6$ , again both derivatives vanish at $t = 0$ , but the curve is the smooth parabola $y = x^2$ , and

\frac{g'(t)}{f'(t)} = \frac{6t^5}{3t^2} = 2t^3 \to 0,

giving the horizontal tangent $y = 0$ .

Second derivative and concavity

Definition Let $f$ and $g$ be twice differentiable on an open interval where $f'(t) \ne 0$ . Then $\frac{dy}{dx}$ is a differentiable function of $t$ , and

\frac{d^2 y}{dx^2} = \frac{\dfrac{d}{dt}\left( \dfrac{dy}{dx} \right)}{dx/dt}.

The curve is concave up on parts where this is positive and concave down where it is negative. Note that $\frac{d^2y}{dx^2}$ is not $\frac{g''(t)}{f''(t)}$ .

In plain words To differentiate anything with respect to $x$ , differentiate it with respect to $t$ and divide by $dx/dt$ . Apply that rule to the slope itself.

Example For $x = t^2 + 1$ , $y = t^3$ the slope is $\frac{3t}{2}$ , so

\frac{d^2 y}{dx^2} = \frac{3/2}{2t} = \frac{3}{4t}, \qquad t \ne 0.

At $t = 2$ this is $\frac{3}{8} > 0$ , so the curve is concave up there. For $t < 0$ it is concave down.

Check yourself

Find $\frac{dy}{dx}$ for $x = \cos t$ , $y = \sin t$ at $t = \frac{\pi}{3}$ .

Show the answer

The slope is

\frac{dy}{dx} = \frac{\cos t}{-\sin t}, \qquad \sin t \ne 0,

so at $t = \frac{\pi}{3}$ it is

\frac{1/2}{-\sqrt{3}/2} = -\frac{1}{\sqrt{3}}.

Where does the cycloid $x = \theta - \sin\theta$ , $y = 1 - \cos\theta$ have horizontal tangents?

Show the answer

The conditions $y' = \sin\theta = 0$ and $x' = 1 - \cos\theta \ne 0$ require $\theta = (2k + 1)\pi$ for an integer $k$ . These are the tops of the arches, the points $((2k + 1)\pi, 2)$ .

For $x = t^2$ , $y = t^3 - 3t$ , find $\frac{d^2y}{dx^2}$ and decide the concavity at $t = 1$ .

Show the answer

For $t \ne 0$ the slope is

\frac{dy}{dx} = \frac{3t^2 - 3}{2t} = \frac{3}{2}\left(t - \frac{1}{t}\right),

whose $t$ derivative is $\frac{3}{2}\left(1 + \frac{1}{t^2}\right)$ . Dividing by $2t$ gives

\frac{d^2y}{dx^2} = \frac{3(t^2 + 1)}{4t^3}.

At $t = 1$ it is $\frac{3}{2} > 0$ , so the curve is concave up there.

The Slope of Parametric Curves

Key terms

The slope formula

Tangent and normal lines

Horizontal and vertical tangents

Stationary parametrizations

Second derivative and concavity

Check yourself

How this lesson connects

Builds on

Used later in