[Notes & HW Answers] 3.5 The Trigonometric Functions

[Notes & HW Answers] 3.5 The Trigonometric Functions.pdf

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[Prepwork 3.5]

Q1. The Bay of Monterey in California is known for extreme tides. The depth of the water, y, in meters can be modeled as a function of time, t, in half-hours after midnight, by y=12+6cos(t). How quickly is the depth of the water rising or falling at 3 a.m.? (Make sure you compute this in radians and give your answers to three decimal places.)

Rising at (in m/half-hours)

A: 1.676

Q2. For how many values of its domain is the derivative of the cosine function equal to zero?

A: Infinitely many values

 

[HW 3.5]

Q1. Find the derivative of 𝑠(𝑞)=18 cos𝑞 sin𝑞

A: s'(q) =  18(cos^2(q)-sin^2(q)

Q2. Find the derivative of R(x) = 16 - 6cos(𝜋𝑥)

A: R'(x) =  6𝜋sin(𝜋𝑥)

Q3. Find the derivative of f(x) = x^6*cos(x)

A: f'(x) =  6x^5*cos(x) - sin(x)*x^6

Q4. Find the derivative of f(x) = 4.15^(cos(x))

A: f'(x) =  (ln (4.15)(4.15^(cos(x))(-sin(x))

Q5. Find the derivative of f(x) = (1-sin(x))^(1/7)

A: f'(x) =  (1/7)(1-sin(x))^(-6/7)*(-cos(x))

Q6. Find the derivative of z = tan(e^(-8w))

A: dz/dw =  (1/(cos^2*(e^(-8w))*(-8e^(-8w))

Q7. Find the derivative of y = cos^2(w) + cos(w^2)

A: dy/dw =  2*cos(w)*(-sin(w))+(-sin(w^2))*2w

Q8. A boat at anchor is bobbing up and down in the sea. The vertical distance, y, in feet, between the sea floor and the boat is given as a function of time, t, in minutes, by y=29+cos⁡(2πt) ft.

Find the vertical velocity, v, of the boat at time t. 

A: v =  -sin(2πt)*2π ft/min