$$  $$$$ f\left(t\right)=150\left(1.126^{10}\right) $$  Find the rate of change, the the linearization of F 

To address your request, let's first find the **rate of change** of the function $ f(t) = 150 \cdot 1.126^{10} $, and then proceed to find the **linearization** of $ f(t) $.

### 1. Finding the rate of change:
The function given is a constant function, as $ 1.126^{10} $ is a number independent of $ t $. Let's calculate it:

\[
f(t) = 150 \cdot 1.126^{10} \approx 150 \cdot 3.239 = 485.85
\]

Since there is no variable $ t $ involved, the rate of change (which refers to the derivative) is 0. Therefore, the derivative is:

\[
f'(t) = 0
\]

### 2. Linearization of the function:
For the **linearization**, we usually find the tangent line to the function at a point $ t = a $, but since $ f(t) $ is constant (no dependence on $ t $), its linearization is simply the constant value of the function:

\[
L(t) = f(a) + f'(a)(t - a) = 485.85 + 0 \cdot (t - a) = 485.85
\]

Thus, the **linearization** of $ f(t) $ is $ L(t) = 485.85 $, which is just a horizontal line.

Would you like more details or further clarification on this? Here are five related questions:

1. What is the interpretation of the linearization for a constant function?
2. How would you linearize a function that actually depends on $ t $?
3. How does the rate of change (derivative) behave for functions of different types (e.g., polynomials, exponentials)?
4. What is the general formula for the linear approximation of a differentiable function?
5. How do second derivatives provide insights into the curvature or concavity of a function?

**Tip**: For a constant function, the derivative is always zero, and its linear approximation remains the constant itself, since there is no variation in the slope.

Find the rate of change, the linearization of f(t) = 150(1.126^{10})

Explore how to find the rate of change and linearization of the constant function f(t) = 150(1.126^{10}). This problem involves concepts from differentiation, including calculating the derivative of constant functions and applying the linearization formula. Suitable for college calculus students.

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