If y = x⁴ – 10 and if x changes from 2 to 1.99 what is the change in y

The correct option is 0.32.

To find the change in y when x changes from 2 to 1.99, we can use the concept of differentials in calculus, which provides a close approximation for small changes. Let's break down the process step-by-step.

Step 1: Identify the given values and formula

We are given the equation:

$$y=x^4-10$$

The initial value of x is 2, and the final value is 1.99.

Step 2: Calculate the change in x

The change in x, represented by dx, is:

$$dx=1.99-2=-0.01$$

Step 3: Find the derivative of y with respect to x

Differentiating y with respect to x using the power rule:

$$\frac{dy}{dx}=4x^3$$

Step 4: Evaluate the derivative at the initial value of x

Substituting x = 2 into the derivative:

$$\frac{dy}{dx}=4\cdot 2^3=4\cdot 8=32$$

Step 5: Estimate the change in y

The approximate change in y, represented by dy, is given by the differential formula:

$$dy\approx \frac{dy}{dx}\cdot dx$$

Substituting the calculated values:

$$dy\approx 32\cdot (-0.01)=-0.32$$

The magnitude of the change in y is therefore 0.32 (representing a decrease of 0.32).