Topic being the linearization (finding the equation of the tangent lines), how do you determine if it's overestimate or underestimate?
thanks!
When linearizing a function and finding the equation of the tangent line, we usually use it to approximate the behavior of the function around a specific point. Whether the linearization overestimates or underestimates the function depends on the curvature of the function's graph at that point.
To determine whether the linearization overestimates or underestimates, you need to compare the value of the function to the corresponding value of the linearization at a specific input. Follow these steps:
1. Find the equation of the linearization by using the point-slope form of a line. Start with the formula:
y - y₁ = m(x - x₁)
Here, (x₁, y₁) is the point where you want to find the tangent line, and m is the slope of the function at that point.
2. Calculate the value of the function at the specific input. Let's call it f(x).
3. Calculate the value of the linearization at the same input. Substitute the x-value into the equation of the linearization and solve for y. Let's call it L(x).
4. Compare the values of f(x) and L(x):
- If f(x) > L(x), it means the function is above the linearization, suggesting that the linearization underestimates the function.
- If f(x) < L(x), it means the function is below the linearization, indicating that the linearization overestimates the function.
- If f(x) = L(x), it means the function and the linearization have the same value at that point.
Note that this comparison is only accurate locally. As you move away from the specific point, the linearization may become less accurate, and the overestimation or underestimation may change.
It's important to remember that linearization is an approximation technique, and its accuracy depends on the behavior of the function at that particular point.