To find the linear approximation of the function g(x) = 51x at the point a = 0, we start by calculating the value of the function and its derivative at that point.
1. **Calculate g(0):**
First, substitute 0 into the function:
g(0) = 51(0) = 0.
2. **Calculate g'(x):**
Next, we need to find the derivative of g(x). Since the function is a simple linear function, the derivative is:
g'(x) = 51.
3. **Calculate g'(0):**
Now, substitute 0 into the derivative:
g'(0) = 51.
4. **Use the linear approximation formula:**
The linear approximation (or tangent line) at the point (a, g(a)) is given by the formula:
g(x) ≈ g(a) + g'(a)(x – a).
Substituting our values (where a = 0):
g(x) ≈ 0 + 51(x – 0) = 51x.
Thus, the linear approximation of g(x) = 51x at a = 0 is simply g(x) ≈ 51x, which coincidentally is the same as the original function since it’s a linear equation.