Question

The approximate value of the function \[ f(x) = x^3 - 3x + 5 \quad \text{at} \quad x = 1.99 \] is

• A. 6.91
• B. 6.94
• C. 7.94
• D. 7.91

Hint:

To approximate a function near a point, use linear approximation with the first derivative. This is especially useful for values near known points.

Answer 1

The Correct Option is A

Step 1: Approximate the function using the first derivative.
We are given the function \( f(x) = x^3 - 3x + 5 \). We approximate \( f(1.99) \) by using a linear approximation: \[ f(x) \approx f(a) + f'(a)(x - a) \] where \( a = 2 \). First, calculate \( f'(x) \): \[ f'(x) = 3x^2 - 3. \] Then, evaluate \( f(2) \) and \( f'(2) \): \[ f(2) = 2^3 - 3(2) + 5 = 8 - 6 + 5 = 7, \] \[ f'(2) = 3(2^2) - 3 = 12 - 3 = 9. \] Now, use the linear approximation to estimate \( f(1.99) \): \[ f(1.99) \approx 7 + 9(1.99 - 2) = 7 + 9(-0.01) = 7 - 0.09 = 6.91. \]
Step 2: Conclusion.
Thus, the approximate value of \( f(1.99) \) is 6.91, which makes option (A) the correct answer.