answer
Q = B + A(P - 0)
steps
To linearize the equation Q = AP + B + P^2, we can use the following steps:
1. Write the equation in the form y = f(x). In this case, y = Q and x = P.
2. Find the first derivative of f(x). In this case, f'(x) = A + 2P.
3. Substitute the value of x = 0 into f'(x). In this case, f'(0) = A.
4. The linearized equation is y = f(0) + f'(0)(x - 0). In this case, the linearized equation is Q = B + A(P - 0).
Therefore, the linearized equation of Q = AP + B + P^2 is Q = B + A(P - 0).
Here is a graph of the original equation and the linearized equation:
[Image of a graph of the original equation and the linearized equation]
As you can see, the linearized equation is a good approximation of the original equation near P = 0. However, as P moves away from 0, the linearized equation becomes less accurate.
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ANOTHER WAY :
To linearize the expression `q=ap+b+p²`, we can use the first two terms of the Taylor series expansion of `q` around a point `p0`. The first two terms of the Taylor series expansion are:
```
q(p) = q(p0) + (dq/dp)|p=p0 * (p - p0)
```
where `dq/dp` is the derivative of `q` with respect to `p`.
We can find the derivative of `q` with respect to `p` as follows:
```
dq/dp = a + 2p
```
Substituting this into the Taylor series expansion, we get:
```
q(p) = q(p0) + (a + 2p0) * (p - p0)
```
This is a linear equation in `q` and `p`. Therefore, we have linearized the expression.
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