Find the local linear approximation of f(x) = e^(3x) at
x = 1.
y = e^3
y = e^(3(x − 1))
y = 3e^(3)(x − 1)
y = 3e^(3)x − 2e^3
i need help
To find the local linear approximation of f(x) = e^(3x) at x = 1, we can use the linear approximation formula:
L(x) = f(a) + f'(a)(x - a)
where a is the point of approximation (in this case, a = 1), f(a) is the value of the function at a, and f'(a) is the derivative of the function at a.
Let's calculate the derivative of f(x) = e^(3x) first. The derivative of e^(3x) with respect to x is 3e^(3x).
Now, we can plug in the values into the linear approximation formula:
L(x) = f(1) + f'(1)(x - 1)
L(x) = e^(3 * 1) + 3e^(3 * 1)(x - 1)
L(x) = e^3 + 3e^3(x - 1)
Simplifying the expression, we get:
L(x) = 3e^3x - 2e^3
Therefore, the local linear approximation of f(x) = e^(3x) at x = 1 is given by the equation L(x) = 3e^3x - 2e^3.
To find the local linear approximation of f(x) = e^(3x) at x = 1, we can use the formula for linear approximation:
L(x) = f(a) + f'(a)(x - a)
where a is the value at which we are approximating (in this case, a = 1), f(a) is the value of the function at a, f'(a) is the derivative of the function evaluated at a, and x is the input value.
First, let's find f(a), which is f(1):
f(1) = e^(3*1)
f(1) = e^3
Next, let's find f'(a), which is the derivative of f(x) evaluated at x = 1. The derivative of e^(3x) is found by applying the chain rule:
f'(x) = 3e^(3x)
Now, we can evaluate f'(1):
f'(1) = 3e^(3*1)
f'(1) = 3e^3
Finally, we plug these values into the formula for linear approximation:
L(x) = f(a) + f'(a)(x - a)
L(x) = e^3 + 3e^3(x - 1)
L(x) = 3e^3x - 2e^3
So, the local linear approximation of f(x) = e^(3x) at x = 1 is y = 3e^3x - 2e^3.