Questions: Find Δy and dy if y=x^3+2x^2-1 x=-1 Δx=dx=0.01 Round your answers to 6 decimal places. (a) Δy (a) (b) dy (b)

Solution Steps

To find \(\Delta y\) and \(d y\), we need to evaluate the change in the function \(y = x^3 + 2x^2 - 1\) as \(x\) changes from \(-1\) to \(-1 + \Delta x\). For \(\Delta y\), calculate the difference in \(y\) values at these two points. For \(d y\), use the derivative of the function to estimate the change, which is \(dy = f'(x) \cdot dx\).

We start by evaluating the function \(y = x^3 + 2x^2 - 1\) at \(x = -1\): \[ y(-1) = (-1)^3 + 2(-1)^2 - 1 = -1 + 2 - 1 = 0 \]

Next, we calculate \(y\) at \(x = -1 + 0.01\): \[ y(-1 + 0.01) = (-0.99)^3 + 2(-0.99)^2 - 1 \] Calculating this gives: \[ y(-0.99) \approx -0.010099 \]

Now, we find \(\Delta y\) as the difference between \(y(-1 + 0.01)\) and \(y(-1)\): \[ \Delta y = y(-1 + 0.01) - y(-1) = -0.010099 - 0 = -0.010099 \]

To find \(dy\), we first compute the derivative \(f'(x)\): \[ f'(x) = 3x^2 + 4x \] Evaluating the derivative at \(x = -1\): \[ f'(-1) = 3(-1)^2 + 4(-1) = 3 - 4 = -1 \] Now, we calculate \(dy\): \[ dy = f'(-1) \cdot dx = -1 \cdot 0.01 = -0.01 \]

Final Answer