- In chapter 4, we have learnt how get information about a graph \(C:y=f(x)\) via the derivatives of \(f(x)\):
- For a point \(P(a,f(x=a))\) on \(C\), the derivative \(f'(x=a)\) gives the slope of the tangent line of \(C\). Correspondingly, an equation of the tangent line is \[
y-f(x=a)=f'(x=a)(x-a).
\]
- This also tells us that the function \(f(x)\) is increasing when \(f'(x)>0\), and \(f(x)\) is decreasing, when \(f'(x)<0\).

- The second derivative \(f''(x)\) gives information about concavity.
- Recall that a curve is
*concave upward*if all of its tangent lines are below the curve. A curve is*concave downward*if all of its tangent lines are above the curve. - The graph \(C:y=f(x)\) is concave upward where \(f''(x)>0\), and it's concave downward where \(f''(x)<0\).

- Recall that a curve is

- For a point \(P(a,f(x=a))\) on \(C\), the derivative \(f'(x=a)\) gives the slope of the tangent line of \(C\). Correspondingly, an equation of the tangent line is \[
y-f(x=a)=f'(x=a)(x-a).
\]
- We can directly translate these statements to the case of a parametric curve \[
C:x=f(t),\quad y=g(t)
\] using the chain rule.
- For the first derivative, we get \[ \frac{\mathrm dy}{\mathrm dx}=\frac{\mathrm dy/\mathrm dt}{\mathrm dx/\mathrm dt}=\frac{g'(t)}{f'(t)}. \]
- Therefore, for a point \(P(f(t=a),g(t=a))\) on \(C\), an equation for the tangent line at \(P\) is \[ y-g(t=a)=\frac{g'(t=a)}{f'(t=a)}(x-f(t=a)). \]

- Note that the slope \(\frac{g'(t)}{f'(t)}\) is calculated with respect to the \((x,y)\) coordinate system, and you need to interpret your results accordingly.
- For example, in the case of the standard parametrization of the unit circle: \[ x=\cos t,\quad y=\sin t,\quad0\le t\le2\pi \] we have \(\frac{g'(t)}{f'(t)}=-\cot t\). This function is positive when \(\frac{\pi}{2}<t<\pi\) or \(\frac{3\pi}{2}<t<2\pi\). Note that those are the areas where increasing \(x\) increases \(y\).

- We can get a formula for the second derivative by using the chain rule again: \[
\frac{\mathrm d}{\mathrm dx}\left(\frac{\mathrm dy}{\mathrm dx}\right)=\frac{\frac{\mathrm d}{\mathrm dt}\left(\frac{\mathrm dy}{\mathrm dx}\right)}{\frac{\mathrm dx}{\mathrm dt}}.
\]
- In the case of \(f(t)=\cos t,\,g(t)=\sin t\), we get \[ \frac{\mathrm d^2y}{\mathrm dx^2}=\frac{(-\cot t)'}{(\cos t)'}=-\frac{\csc^2t}{\sin t}=-\frac{1}{\sin^3t}. \] This is positive precisely when \(\pi<t<2\pi\). Note that that's indeed where the circle is concave upward.

- Exercises. 10.2: 3, 5, 11, 17, 19, 27, 29

- Consider a graph \(C:y=F(x),\,a\le x\le b\). Suppose that \(C\) can also be described with the parametric equations \(x=f(t),\,y=g(t),\,\alpha\le t\le\beta\), where \(f'(t)>0\). Then we can use the substitution formula to rewrite the arc length formula: \[
L=\int_a^b\sqrt{1+\left(\frac{\mathrm dy}{\mathrm dx}\right)^2}\,\mathrm dx=\int_\alpha^\beta\sqrt{1+\left(\frac{\mathrm dy/\mathrm dt}{\mathrm dx/\mathrm dt}\right)^2}\frac{\mathrm dx}{\mathrm dt}\mathrm dt=\int_\alpha^\beta\sqrt{\left(\frac{\mathrm dx}{\mathrm dt}\right)^2+\left(\frac{\mathrm dy}{\mathrm dt}\right)^2}\,\mathrm dt.
\]
*Theorem.*The same formula works for any parametric curve \(x=f(t),\,y=g(t),\,\alpha\le t\le\beta\), even when it is not a graph.

- Example. Let's calculate the arc length function of a circle of radius \(r\): \[
x=r\cos\theta,\,y=r\sin\theta,\,\theta\ge0.
\]
- We have \(f'(\theta)=-r\sin\theta\) and \(g'(\theta)=r\cos\theta\), therefore we get \[ s(\theta)=\int_0^\theta\sqrt{(r\sin t)^2+(r\cos t)^2}\,\mathrm dt=\int_0^\theta r\,\mathrm dt=r\theta. \]

- Example. The curve traced out by a point \(P\) on the circumference of a circle of radius \(r\) as the circle rolls along a straight line is called a
*cycloid*. We can parametrize it as follows.- Let \(\theta\) be the angle of \(P\) in the circle with respect to the rightmost point.
- We have seen in the previous example that as \(P\) travels an angle of \(\theta\) on the circle, it covers a path length of \(r\theta\). Therefore, at angle \(\theta\), the centre of the circle is at \[ x_1=r\theta,\quad y_1=r. \]
- On the circle, the point is moving in the clockwise direction. For some reason, it's customary to make it start at the bottom. This can be parametrized by \[ x_2=-r\sin\theta,\quad y_2=-r\cos\theta. \]
- To get the parametrization of the cycloid, we can add the two functions: \[ x=x_1+x_2=r(\theta-\sin\theta),\quad y=r(1-\cos\theta) \]
- To get at the arc length, we first compute the derivatives: \[ f'(\theta)=r(1-\cos\theta),\quad g'(\theta)=r\sin\theta \]
- Let's see the path length over a full revolution: \[ L=\int_0^{2\pi}\sqrt{r^2(1-\cos\theta)^2+r^2\sin^2\theta)}\,\mathrm d\theta=r\int_0^{2\pi}\sqrt{2-2\cos\theta}\,\mathrm d\theta=r\int_0^{2\pi}\sqrt{4\sin^2(\theta/2)}\,\mathrm d\theta\stackrel{\sin(\theta/2)\ge0\text{ for }0\le\theta\le2\pi}{=}2r\int_0^{2\pi}\sin(\theta/2)\,\mathrm d\theta=8r. \]

- Exercises. 10.2: 41, 43