Homework 5

Directions:

  1. Show each step of your work and fully simplify each expression.
  2. Turn in your answers in class on a physical piece of paper.
  3. Staple multiple sheets together.
  4. Feel free to use Desmos for graphing.


Answer the following:

  1. Graph one period using transformations:
    1. $y = 1 + \tan\left(\pi x\right)$
    2. $y = 4 \tan (3x - 2\pi)$
    3. $y = -\sec(x + \pi)$
    4. $y = \cot\left(2x - \dfrac{\pi}{4}\right)$
    5. $y = \csc\left(2\left(x + \dfrac{\pi}{2}\right)\right)$
  2. Because $f(x) = \sin(x)$ is not one to one, what is the interval of values we restrict the domain of $f(x)$ to in order for $f(x)$ to have an inverse?
  3. Inverse function property in 2.8 tells us that if $f(x) = \sin(x)$ and $g(x) = \sin^{-1}(x)$, then $(f\circ g)(x) = x$ is valid. For what values of $x$ is $(f \circ g)(x) = x$ true? Moreover, explain the exact reason why these specific values of $x$ are valid.
  4. Inverse function property in 2.8 tells us that $\cos^{-1}(\cos(x)) = x$. For what values of $x$ is this true? Moreover, explain the exact reason why these specific values of $x$ are valid.
  5. Out of the following, which ones are true? 🤔 Justify your answer with the domain of the proper function.
    1. $\sin^{-1}\left(\sin\left(\dfrac{\pi}{3}\right)\right) = \dfrac{\pi}{3}$
    2. $\sin^{-1}\left(\sin\left(-\dfrac{\pi}{4}\right)\right) = -\dfrac{\pi}{4}$
    3. $\sin^{-1}\left(\sin\left(-\dfrac{2\pi}{3}\right)\right) = -\dfrac{2\pi}{3}$
    4. $\cos^{-1}\left(\cos\left(-\dfrac{2\pi}{3}\right)\right) = -\dfrac{2\pi}{3}$
    5. $\cos\left(\cos^{-1}\left(\dfrac{1}{2}\right)\right) = \dfrac{1}{2}$
    6. $\tan^{-1}\left(\tan\left(-\dfrac{2\pi}{3}\right)\right) = -\dfrac{2\pi}{3}$
    7. $\tan\left(\tan^{-1}100000000000\right) = 100000000000$
  6. Evaluate the following:
    1. $\sin^{-1}\left(\dfrac{\sqrt{3}}{2}\right)$
    2. $\sin^{-1}\left(\dfrac{-1}{2}\right)$
    3. $\cos^{-1}\left(-\dfrac{\sqrt{2}}{2}\right)$
    4. $\cos^{-1}(2)$
    5. $\tan^{-1}(0)$
    6. $\tan^{-1}(\sqrt{3})$
    7. $\tan\left(\sin^{-1}\frac{\sqrt{2}}{2}\right)$
  7. Find a function that models the simple harmonic motion with the following properties. Assume the displacement is zero at $t = 0$.
    1. Amplitude 4 centimeters, period 3 seconds
    2. Amplitude 8 meters, frequency $\frac{1}{2}$ Hz (cycles per second)
  8. Find a function that models the simple harmonic motion with the following properties. Assume the displacement is at the maximum at $t = 0$.
    1. Amplitude 3 inches, period 2 minutes
    2. Amplitude 1.2 meters, frequency $5$ Hz
  9. In a predator/prey model, the predator population is modeled by the function \[y = 1100\cos\left(3t\right) + 500\] where $t$ is measured in years.
    1. What is the maximum population?
    2. Find the length of time between successive periods of maximum population.
  10. A mass suspended from a spring is at rest. Force is introduced at time $t = 0$, causing the mass to displace a maximum of four inches. If the mass completes 4 cycles in one second, find an equation that describes its motion.
  11. A mass suspended from a spring is at rest. It is pulled down three centimeters and released at time $t = 0$. If the mass returns to the location it was released at after one second, find an equation that describes its motion. Hint: It starts at the lowest position.