# Enrichment, Extension, and Applications

1 point

## Exercises

1. And the wheel goes 'round...

A sawmill is powered by a water wheel of diameter $$10$$ m. A flowing river turns the wheel at a rate of one revolution per minute. At what speed is a paddle on the wheel rising when it is $$2$$ m vertically from the top of the wheel?

2. And one flew off...

Two particles travel together counter-clockwise at a speed of $$1$$ m/s around a unit circle with equation $$x^2 + y^2 = 1$$.

Let one particle be named $$P$$ and the other be named $$Q$$. At the point $$(1, 0)$$, after $$t = 2\pi$$ seconds have passed, particle $$Q$$ flies tangentially (i.e., vertically) off the circle, heading upwards at the same speed (assume the particles are not affected by gravity). Particle $$P$$ continues along its circular path. How fast are the particles separating at $$t = 3\pi$$ s?

3. A Bounded Graph

Consider the function $$f(x) = x - 2\cos(x)$$ on the interval $$\left[ -2\pi, 2\pi \right]$$.

1. Sketch the graphs of $$y = x$$ and $$y = 2\cos(x)$$ on $$\left[ -2\pi, 2\pi \right]$$. How many points of intersection (i.e., zeros of $$f$$) occur on this interval?
2. Determine any relative extremes and points of inflection of $$y = f(x)$$ on the interval $$\left[ -2\pi, 2\pi \right]$$.
3. Sketch the graph of $$f(x)$$ on $$\left[ -2\pi, 2\pi \right]$$.

Hint: First note where $$f(x) = 0$$ (using part a) and where $$f(x) = x$$.

4. Explain why the sketch from part c) lies between the lines $$y = x - 2$$ and $$y = x + 2$$. Add these lines to the sketch, showing clearly where $$f(x)$$ meets these lines.
5. Challenge: Make a sketch, similar to the one in part d), for the general case $$y = x - c \cos(x)$$ for each of the cases
• $$0 \lt c \lt 1$$,
• $$c = 1$$, and
• $$c \gt 1$$.
4. Implicit Curves

Sometimes implicit equations define curves in the plane. For example, $$x^2 + y^2 = 4$$ defines a circle centred at the origin of radius $$2$$.

There is also the possibility that more than one curve is defined. For example, the equation $$x^2 = y^2$$ implies $$x^2 - y^2 = 0$$, or $$(x - y)(x + y) = 0$$, which means that both $$y = x$$ and $$y = -x$$ satisfy the equation, defining two straight lines.

For each of the following equations, determine any curves defined. Illustrate your solutions with suitable sketches.

1. $$\cot(x) = \tan(y)$$, where $$x$$ and $$y$$ are acute angles $$\left( 0 \lt x, y \lt \frac{\pi}{2} \right)$$.
2. $$\sin{(2x)} = 2y\sin(x)$$
3. $$e^{8x^2y} = e^{y^4}$$
4. $$\ln( y^2 ) = \ln(4) + 2\sin(\pi x)$$
5. Challenge: A computer-generated plot of points satisfying the equation $$x \ln(y) = y \ln(x)$$ (or $$y^x = x^y$$) on the region $$x \gt 0, y \gt 0$$ is shown.

1. State the equation of the linear solution.
2. Explain the obvious symmetry of the other solution.
3. Given that you know $$\dfrac{\ln(x)}{x} = k$$ has
• one solution if $$k \leq 0$$ or $$k = \frac{1}{e}$$, or
• two solutions if $$0 \lt k \lt \frac{1}{e}$$
explain why $$P$$ has coordinates $$(e, e)$$ and why the second, non-linear solution cannot meet the lines $$x = 1$$ or $$y = 1$$.
5. A Maxed-Out Triangle

A triangle with vertices $$O~(0, 0)$$, $$P~(x, y)$$ (on the curve $$y = x^2e^{-ax}$$, where $$a \gt 0$$), and $$Q~(x, 0)$$ has area $$A(x)$$.

Suppose that $$x = x_m$$ maximizes $$A(x)$$ and that $$x = x^{*}$$ maximizes the height of the curve.

1. Determine the value of $$x^{*}$$ in terms of $$a$$.
2. Determine the value of $$x_m$$ in terms of $$a$$.
3. Which of the two values, $$x^{*}$$ or $$x_m$$, is greater? Think about why this is so, given the geometry of the curve $$y = x^2e^{-ax}$$ (a sample curve is shown for some value of $$a \gt 0$$).
6. Sliding along...

Note: This question assumes familiarity with Newton's laws of motion. Expositions can be found in most high school physics courses/textbooks.

A child on a toboggan is pulled along a horizontal surface. The force $$F$$ exerted on the rope is given by $F( \theta ) = \dfrac{\mu W}{\mu \sin(\theta) + \cos(\theta)} ~ \text{for} ~ 0 \leq \theta \leq \tfrac{\pi}{2}$ where $$\mu$$ is a positive constant satisfying $$0 \lt \mu \lt 1$$, $$W$$ is the combined weight of the child and toboggan, and $$\theta$$ is the angle of the rope to the horizontal, as shown in the diagram.

1. Evaluate $$F(0)$$ and $$F\left( \frac{\pi}{2} \right)$$. Explain why these values make sense, physically.
2. Find the absolute extremes of $$F(\theta)$$ on the interval $$0 \leq \theta \leq \frac{\pi}{2}$$.
3. What value of $$F$$ is easiest on the person pulling the toboggan?
4. Assuming the toboggan moves horizontally, what value of $$F$$ gives the child the fastest ride?
7. How square is this?

Two smaller squares, $$S_1$$ and $$S_2$$, lie completely within a larger square, $$S$$.

$$S$$ has side length $$1$$.

The smaller of the two inner squares, $$S_1$$, lies flush with the bottom left corner of the largest square, and has side length $$x$$.

The larger of the two inner squares, $$S_2$$, has side length $$y$$. $$S_2$$ lies on an angle such that each of its vertices lies on exactly one side of $$S$$. The vertex of $$S_2$$ which lies on the bottom side of $$S$$, side $$AB$$, is labelled as point $$P$$; the vertex of $$S_2$$ which lies on the rightmost side of $$S$$ is labelled $$Q$$. $$S_2$$ also has one of its sides resting on the top-right corner of $$S_1$$.

Let $$\theta = \angle{QPB}$$. Then, as $$P$$ moves along $$AB$$, the values of $$\theta$$, $$x$$, and $$y$$ will vary.

1. Show that, in terms of angle $$\theta$$, $y = \dfrac{1}{\sin(\theta) + \cos(\theta)}, \quad x = \dfrac{1}{2} \left( \dfrac{\sin{(2\theta)}}{\sin{(2\theta)} + 1} \right)$ for $$0 \leq \theta \leq \frac{\pi}{2}$$.
2. Determine the values of $$x$$ and $$y$$ for which $$x^2 + y^2$$ is a minimum, i.e., the values that minimize the area of the shaded region.

Hint: Show, after some simplification, that $$x^2 + y^2$$ has the form $$\dfrac{1}{4} \left( \dfrac{u + 2}{u + 1} \right)^2$$, where $$u$$ is a simple expression in $$\theta$$. Then write $$\dfrac{u + 2}{u + 1}$$ in a simpler form.

8. To Row or To Jog — that is the question...

Standing at $$A$$, on the edge of a circular pond as shown in the diagram, you observe a toddler straight across at $$B$$, dangerously close to the edge of the water.

Thinking quickly, you realize there are three possibilities:

• Using your rowboat at point $$A$$, you can row to some point $$P$$ on the edge of the pond, located at an angle $$\theta$$ to $$A$$ as shown (with $$\theta$$ to be determined by you), and then jog the rest of the way to $$B$$, or
• Jog the entire way around the edge of the pond from $$A$$ to $$B$$, or
• Row directly from $$A$$ to $$B$$ across the pond

If you can jog $$1.5$$ times faster than you can row, which of the three options will get you from $$A$$ to $$B$$ in the least time? If it is the first option, find the angle $$\theta$$ which optimizes your route.

9. In the Limit...

1. Evaluate each limit, or show that it does not exist.
1. $$\displaystyle \lim_{x \to 0} e^{x \ln{\left\lvert x \right\rvert}}$$
2. $$\displaystyle \lim_{x \to 0} \dfrac{e^{x \ln{\left\lvert x \right\rvert}} - 1}{x}$$
Hence, comment on the differentiability at $$x = 0$$ of the function $f(x) = \begin{cases} \left\lvert x \right\rvert^x & x \neq 0 \\ 1 & x = 0 \end{cases}$
2. Given that $$\displaystyle \lim_{x \to 0} \dfrac{x^2 + \sin(bx) + \sin(x)}{ax^2 + 3x^4 + 2x^3} = 5$$, determine the values of $$a$$ and $$b$$.
3. For what values of the constant $$a$$ is $$\displaystyle \lim_{x \to \infty} \left( \dfrac{x + a}{x - a} \right)^x = e^{a^2}$$?
10. Falling Down, Down, Down...

A plausible model for the forces acting on a falling body near the Earth's surface is to describe them as a net force $F_{\text{net}} = F_{gr} - F_{r}$ where $$F_{gr} = mg$$ is the mass of the body times the gravitational acceleration constant $$g$$, and $$F_r$$ is the resistance force created by atmospheric drag.

If we assume that $$F_r$$ is proportional to the velocity of the object, then Newton's law of motion $(\text{mass}) \times (\text{acceleration}) = (\text{net force})$ implies that the velocity, $$v(t)$$, obeys the differential equation $m \dfrac{dv}{dt} = F_{gr} - F_r = mg - kv$ for some positive constant $$k$$.

1. Show that, if the body starts from rest, with $$v(0) = 0$$, then $$v(t) = \dfrac{mg}{k} \left( 1 - e^{-\frac{kt}{m}} \right)$$ satisfies the given differential equation.
2. Find $$\displaystyle \lim_{t \to \infty} v(t)$$, and hence give a physical interpretation for the constant $$\dfrac{mg}{k}$$.
3. A value of $$k = 0$$ implies that the motion occurs in a vacuum (no atmospheric drag). Thinking of $$t$$ as a constant (i.e., fixing a moment in time), use l'Hospital's rule to determine $$\displaystyle \lim_{k \to 0^{+}} v(t)$$. Hence, determine the (well-known) velocity for a body falling from rest in a vacuum.
11. Keeping Order

Suppose that $$\displaystyle \lim_{x \to \infty} f(x) = \infty$$ and $$\displaystyle \lim_{x \to \infty} g(x) = \infty$$. We say that "$$g$$ grows faster than $$f$$" if $\lim_{x \to \infty} \dfrac{f(x)}{g(x)} = 0$ and "$$g$$ and $$f$$ grow at the same rate" if $\lim_{x \to \infty} \dfrac{f(x)}{g(x)} = c \neq 0$ That is, $$f(x) \approx cg(x)$$ as $$x \to \infty$$.

1. Use l'Hospital's rule to show that
1. $$\displaystyle \lim_{x \to \infty} \dfrac{\ln(x)}{x^p} = 0$$ for all $$p \gt 0$$, and
2. $$\displaystyle \lim_{x \to \infty} \dfrac{x^p}{e^{ax}} = 0$$ for all $$a \gt 0$$, $$p \gt 0$$
2. Explain why, if $$g$$ grows faster than $$f$$, then $$e^g$$ grows faster than $$e^f$$ as $$x \to \infty$$.
3. Order the following functions from slowest to fastest growing, as $$x \to \infty$$:

$x^{\ln(x)}, x^{e}, x^{x}, 2^{x}, e^{x^2}$

Hint: the results from parts a) and b) may be useful.

12. A Little Geometry...

In the diagram, $$P$$ and $$Q$$ are points on the circumference of a circle with center $$C$$ and radius $$r$$, and $$O$$ is a point which lies outside the circle. The lines $$OP$$ and $$OQ$$ are tangent to the circle, and the line $$OR$$ is perpendicular to the line $$PQ$$, where $$R$$ lies on $$PQ$$.

Let $$x$$ denote the portion of $$OR$$ that lies within the circle, and let $$y$$ denote the portion of $$OR$$ which lies outside the circle. If $$\theta = \angle{PCQ}$$, evaluate $$\displaystyle \lim_{\theta \to 0} ~ \dfrac{x}{y}$$.