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?
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.
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.
Out of the following, which ones are true? 🤔 Justify your answer with the domain of the proper function.
Find a function that models the simple harmonic motion with the following properties. Assume the displacement is zero at $t = 0$.
Amplitude 4 centimeters, period 3 seconds
Amplitude 8 meters, frequency $\frac{1}{2}$ Hz (cycles per second)
Find a function that models the simple harmonic motion with the following properties. Assume the displacement is at the maximum at $t = 0$.
Amplitude 3 inches, period 2 minutes
Amplitude 1.2 meters, frequency $5$ Hz
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.
What is the maximum population?
Find the length of time between successive periods of maximum population.
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.
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.