Some of the highest tides in the world occur in the Bay of...

Some of the highest tides in the world occur in the Bay of Fundy on the Atlantic Coast of Canada. At Hopewell Cape the water depth at low tide is about $2.0 \mathrm{~m}$ and at high tide it is about $12.0 \mathrm{~m}$. The natural period of oscillation is a little more than 12 hours and on a day in June, high tide occurred at $6: 45 \mathrm{AM}$. This helps explain the following model for the water depth $D$ (in meters) as a function of the time $t$ (in hours after midnight) on that day: $$ D(t)=7+5 \cos [0.503(t-6.75)] $$ How fast was the tide rising (or falling) at the following times? (a) $3: 00 \mathrm{AM}$ (b) $6: 00 \mathrm{AM}$ (c) $9: 00 \mathrm{AM}$ (d) Noon

Some of the highest tides in the world occur in the Bay of Fundy on the Atlantic Coast of Canada. At Hopewell Cape the water depth at low tide is about $2.0 \mathrm{~m}$ and at high tide it is about $12.0 \mathrm{~m}$. The natural period of oscillation is a little more than 12 hours and on a day in June, high tide occurred at $6: 45 \mathrm{AM}$. This helps explain the following model for the water depth $D$ (in meters) as a function of the time $t$ (in hours after midnight) on that day:
$$
D(t)=7+5 \cos [0.503(t-6.75)]
$$
How fast was the tide rising (or falling) at the following times?
(a) $3: 00 \mathrm{AM}$
(b) $6: 00 \mathrm{AM}$
(c) $9: 00 \mathrm{AM}$
(d) Noon

Key Concepts

Differentiation of Trigonometric Functions

This concept involves finding the derivative of trigonometric functions such as cosine and sine. In particular, the derivative of cos(g(t)) is -sin(g(t)) multiplied by the derivative of its argument g(t). This procedure is fundamental when dealing with time-dependent periodic functions to determine instantaneous rates of change.

Chain Rule in Differentiation

The chain rule is a key concept in calculus used for differentiating composite functions. When a function is composed of an outer function and an inner function, the derivative is found by differentiating the outer function and multiplying it by the derivative of the inner function. This process is essential in problems that involve a cosine or sine function with a non-linear argument.

Harmonic Motion and Sinusoidal Functions

Harmonic motion refers to motions that can be modeled with sinusoidal functions like sine and cosine due to their periodic nature. Such models are commonly used to describe naturally occurring oscillations, including tides. The sinusoidal function provides a simple and effective way to capture the cyclic behavior observed in many physical phenomena.

Angular Frequency and Period

Angular frequency is a parameter that describes how quickly a periodic function completes its cycles, and it is related to the period of the motion. Knowing either the angular frequency or the period allows one to fully characterize the timing of a cyclic process. These concepts are critical for understanding the detailed timing and scaling of periodic models.

Interpretation of Derivatives in Periodic Contexts

In the context of periodic or harmonic functions, the derivative represents the rate at which the function's value is changing with respect to time. For physical phenomena modeled by sinusoidal functions, this rate of change can indicate the speed at which a quantity, such as tidal depth, increases or decreases. This interpretation is vital for linking the mathematical model to real-world behavior.