To find the linear approximation of the function f(x) = 4x at the point a = 0, we start by calculating the derivative of the function:
Step 1: Calculate the derivative
The derivative of f(x) is:
f'(x) = 4
Since the function is linear itself, its slope is constant, and it equals 4.
Step 2: Find the value of the function at a = 0
Now, we evaluate f(x) at the point a = 0:
f(0) = 4 * 0 = 0
This gives us the point (0, 0) on the function.
Step 3: Write the linear approximation formula
The formula for the linear approximation is:
L(x) = f(a) + f'(a)(x - a)
Plugging in our values:
L(x) = 0 + 4(x - 0)
Thus, the linear approximation simplifies to:
L(x) = 4x
Step 4: Approximate the values
Now, we will use this linear approximation to find the approximate values:
For 39:
We approximate by substituting x = 39 into the linear function:
L(39) = 4 * 39 = 156
For 399:
Similarly, substituting x = 399 gives us:
L(399) = 4 * 399 = 1596
Conclusion:
Thus, using the linear approximation at a = 0, we find that f(39) ≈ 156 and f(399) ≈ 1596.