The linearization at a=0 to sqrt(6+6x) is A+Bx where A is:_____ and where B is:_____

y = √(6+6x)

y(0) = √6

y' = 1/2 * 1/√(6+6x) * 6 = 3/√(6+6x)
y'(0) = 3/√6 = √6/2

so, y = √6/2 x + √6

A = √6
B = √6/2

Well, aren't you getting a little too mathematical for this clown? But hey, I'll give it a shot!

For the linearization at a = 0 of √(6 + 6x), let's first find the derivative. We have:

f(x) = √(6 + 6x)

f'(x) = (1/2)(6 + 6x)^(-1/2)(6)

Simplifying that, we get:

f'(x) = 3(6 + 6x)^(-1/2)

Now, plugging in a = 0:

A = f(0) = √(6 + 6(0)) = √6

B = f'(0) = 3(6 + 6(0))^(-1/2) = 3(6)^(-1/2) = 3/√6

So, the clown's answer is A = √6 and B = 3/√6. But always remember, my answers come with a side of humor!

To find the linearization at a=0 to sqrt(6+6x), we need to use the first two terms of the Taylor series expansion. The general form of the Taylor series expansion for a function f(x) at a=0 is:

f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + f'''(a)(x-a)^3/3! + ...

We want to find the linear approximation, so we consider only the first two terms.

First, let's find the first derivative of the function.

f(x) = sqrt(6+6x)

Using the chain rule, we have:

f'(x) = (1/2)(6+6x)^(-1/2) * 6

Simplifying this expression, we get:

f'(x) = 3(6+6x)^(-1/2)

Now, we can find the value of f(0) and f'(0) to determine the linearization.

f(0) = sqrt(6+6(0))
= sqrt(6+0)
= sqrt(6)
= √6

f'(0) = 3(6+6(0))^(-1/2)
= 3(6)^(-1/2)
= 3/√6

So, the linearization at a=0 to sqrt(6+6x) is A + Bx, where A = √6 and B = 3/√6.

To find the linearization of a function at a given point, we can use the formula:

L(x) = f(a) + f'(a)(x - a)

Given that the function is f(x) = sqrt(6 + 6x) and we want to find the linearization at a = 0, we need to evaluate f(0), f'(0), and substitute them into the linearization formula.

1. Evaluate f(0):
Plug in x = 0 into the function f(x) = sqrt(6 + 6x):
f(0) = sqrt(6 + 6(0))
f(0) = √6

2. Evaluate f'(0):
Calculate the derivative of f(x) = sqrt(6 + 6x) with respect to x:
f'(x) = (1/2)(6 + 6x)^(-1/2) *6
f'(0) = (1/2)(6 + 6(0))^(-1/2) *6
f'(0) = (1/2)(6)^(-1/2) *6
f'(0) = (1/2)(√6) *6
f'(0) = 3√6

3. Substitute the values into the linearization formula:
L(x) = f(0) + f'(0)(x - 0)
L(x) = √6 + 3√6 *x
L(x) = (√6)(1 + 3x)

Therefore, the linearization at a=0 to √(6+6x) is A + Bx, where A = √6 and B = 3√6.