Question

A point is moving along the graph of a given function such that dx/dt is 9 centimeters per second. Find dy/dt for the given values of x. (Round your answers to three decimal places.)

y = 9 cos(x)
a) x = /6
dy/dt =
b) x = /3
dy/dt =
c) x = /4
dy/dt =

Answer 1

To find dy/dt, we apply the chain rule to the function y = 9 cos(x) and multiply the derivative dy/dx by dx/dt = 9 cm/s. We then calculate -9 sin(x) × 9 cm/s for each given value of x and round to three decimal places.

Step-by-step explanation:

The student is asking how to find dy/dt for the function y = 9 cos(x), given that dx/dt = 9 cm/s, for different values of x. Using the chain rule, we can find dy/dt as follows: since y = 9 cos(x), by taking the derivative with respect to x we get dy/dx = -9 sin(x). Then, using the chain rule, we find dy/dt by multiplying dy/dx by dx/dt, which gives dy/dt = dy/dx × dx/dt = -9 sin(x) × 9 cm/s.

• a) x = π/6, dy/dt = -9 sin(π/6) × 9 cm/s = -9 × 0.5 × 9 cm/s = -40.500 cm/s.
• b) x = π/3, dy/dt = -9 sin(π/3) × 9 cm/s = -9 × √3/2 × 9 cm/s = -70.200 cm/s.
• c) x = π/4, dy/dt = -9 sin(π/4) × 9 cm/s = -9 × √2/2 × 9 cm/s = -56.727 cm/s.

Note that the values are rounded to three decimal places as per the student's instruction.