Question

Written Assignment unit 7

Complete the following questions utilizing the concepts introduced in this unit.

1. Find the length of an arc in a circle of radius 10 centimetres subtended by the central angle of 50°. Show your work.

2. Graph f(x) = x Sin x on [-4π, 4π] and verbalize how the graph varies from the graphs of f(x) + or - x.

Graph f(x) =Sin x / x on the window [−5π, 5π] and describe freely what the graph shows. You can use www.desmos.com/calculator to obtain the graphs.

3. A 23-ft ladder leans against a building so that the angle between the ground and the ladder is 80°. How high does the ladder reach up the side of the building? Show the steps of your reasoning.

Answer 1

1. The circumference of the circle is $L=2\pi r=2\pi(10)=20\pi(cm)$

The length $l$ of yhe arc in a circle subtended by the central angle of 50° is

$$l=L(\dfrac{50\degree}{360\degree})=\dfrac{20\pi(5)}{36}=\dfrac{25\pi}{9}(cm)$$

2.

First draw the graph of $f(x)=±x.$

The function $f(x)=x\sin x$ is even. The graph is symmetric with respect to the $y$ -axis.

Points $(-\dfrac{\pi}{2}+2\pi n,\dfrac{\pi}{2}+2\pi n),n\in \Z$ lie on the graph of $f(x)=-x$

Points $(\dfrac{\pi}{2}+2\pi n,\dfrac{\pi}{2}+2\pi n),n\in \Z$ lie on the graph of $f(x)=x$

Points $(-\dfrac{3\pi}{2}+2\pi n,-\dfrac{3\pi}{2}+2\pi n),n\in \Z$ lie on the graph of $f(x)=x$

Points $(\dfrac{3\pi}{2}+2\pi n,-\dfrac{3\pi}{2}+2\pi n),n\in \Z$ lie on the graph of $f(x)=-x$

(ii)

The function $f(x)=\dfrac{\sin x}{x}$ is not defined at $x=0.$

$$\lim\limits_{x\to 0}f(x)=\lim\limits_{x\to 0}\dfrac{\sin x}{x}=1$$

The function $f(x)=\dfrac{\sin x}{x}$ has a removable discontinuity at $x=0.$

The function $f(x)=\dfrac{\sin x}{x}$ is even. The graph is symmetric with respect to the $y$ -axis.

$y\to0$ as $x\to\pm \infin.$

The graph of $f(x)=\dfrac{\sin x}{x}$ is decaying oscillations (oscillations of continuously decreasing amplitude). The oscillations never stop, but go on decreasing in strength. The amplitude of oscillations at any point $x\not=0$ is $1/x.$

3.

From right triangle

$$h=L\sin \theta$$ $$h=23\sin 80\degree \ ft\approx22.65\ ft$$