Question

A point moves along the curve y=2x^2+1 in such a way that the y value is decreasing at the rate of 2 units per second. What rate is x changing when x=3/2. Also, please show your work.

a) -1
b) 1
c) -2
d) 2

Answer 1

The rate at which x is changing when y=2x^2+1 is decreasing at the rate of 2 units per second and x=3/2 is -1/3 units per second. This is found by using implicit differentiation and substituting the given values into the derived equation. None of the given options match this result, indicating a potential typo in the question.

Step-by-step explanation:

We need to find the rate at which x is changing when the y value of the curve y=2x^2+1 is decreasing at the rate of 2 units per second, given that x=3/2. We will use implicit differentiation with respect to time t to find dx/dt when dy/dt = -2.

First, take the derivative with respect to t of both sides of the equation:

y = 2x^2 + 1

Applying the chain rule we get:

dy/dt = 2 · 2x · dx/dt

Since we know that dy/dt = -2 and x = 3/2, we can substitute these values in:

-2 = 4 · (3/2) · dx/dt

We solve for dx/dt:

dx/dt = -2 / (4 · (3/2))
dx/dt = -2 / 6
dx/dt = -1/3

This means the rate at which x is changing is -1/3 units per second, which is not one of the options provided. Thus, there may have been a typo in the options, but based on the calculations, none of the options given (a) -1, (b) 1, (c) -2, (d) 2 are correct.