Question

Find dy and evaluate when x = −2 and dx = −0.1 for the function y = 4sin(3x).

A) dy/dx = -12cos(6) at x = -2
B) dy/dx = 12cos(6) at x = -2
C) dy/dx = -12cos(6) at x = -0.1
D) dy/dx = 12cos(6) at x = -0.1

Answer 1

The derivative of the function y = 4sin(3x) is dy/dx = 12cos(3x). At x = -2, this becomes dy/dx = 12cos(6), and therefore the differential dy when dx = -0.1 is dy = 12cos(6) × (-0.1).

Step-by-step explanation:

We need to find dy and evaluate it for x = -2 and dx = -0.1 where the function given is y = 4sin(3x). The first step is to find the derivative of y with respect to x, which is dy/dx. By using the chain rule, the derivative of sin(3x) is cos(3x) multiplied by the derivative of the inside function, which is 3. So:

dy/dx = 4 × 3 × cos(3x) = 12cos(3x)

We then need to evaluate dy/dx at x = -2. Plugging in this value, we get:

dy/dx = 12cos(3 × (-2)) = 12cos(-6) = 12cos(6)

Since cosine is an even function, cos(-θ) = cos(θ). Therefore, dy/dx at x = -2 is 12cos(6).

Finally, we calculate dy using the value of dy/dx and the given dx:

dy = (dy/dx) × dx = 12cos(6) × (-0.1)

The correct answer is then:

B) dy/dx = 12cos(6) at x = -2