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The problem in the image has two parts involving a function \( g(t) = (0.4)^t \). Let's go through both parts step by step.

### Part (a)
You are asked to sketch a graph of \( g(t) = (0.4)^t \) and determine whether \( g'(2) \) is positive, negative, or zero. 

#### Solution:
The function \( g(t) = (0.4)^t \) is an exponential function with base less than 1, which means the graph is decreasing. The derivative of a function gives us the slope of the tangent line at any point, so to determine whether \( g'(2) \) is positive, negative, or zero, we need to understand the behavior of the graph at \( t = 2 \).

Since \( g(t) = (0.4)^t \) is a decreasing function, the slope at \( t = 2 \) is negative. Thus, we can conclude that:

- **Answer: Negative**

### Part (b)
You are asked to use a small interval to estimate \( g'(2) \) and round the answer to three decimal places.

#### Solution:
The derivative \( g'(2) \) can be approximated using the definition of the derivative:
\[
g'(2) \approx \frac{g(2 + h) - g(2)}{h}
\]
for a small value of \( h \). Let's choose \( h = 0.001 \) and compute an estimate for \( g'(2) \).

We know \( g(t) = (0.4)^t \), so:

\[
g(2) = (0.4)^2 = 0.16
\]
\[
g(2 + h) = (0.4)^{2.001} \approx 0.1596 \quad \text{(using calculator approximation)}
\]

Now, using the values for \( g(2) \) and \( g(2 + h) \):

\[
g'(2) \approx \frac{0.1596 - 0.16}{0.001} = \frac{-0.0004}{0.001} = -0.4
\]

So, the estimated value of \( g'(2) \) is approximately **-0.400**.

### Summary of answers:
- (a) The derivative \( g'(2) \) is **Negative**.
- (b) The estimate for \( g'(2) \) is approximately **-0.400**.

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Would you like more details on any of the steps? 

Here are some related questions you could explore:
1. How is the derivative of exponential functions with base less than 1 generally computed?
2. Why do functions with bases less than 1 always have negative slopes?
3. How would the graph of \( g(t) = (0.4)^t \) look for larger values of \( t \)?
4. What is the geometric interpretation of the derivative at a point?
5. How does changing the base in an exponential function affect the behavior of its graph?

**Tip:** For small changes in \( t \), using the difference quotient formula is a great way to approximate the derivative if calculating it directly is complicated.

Let g(t) = (0.4)^t. Sketch a graph to determine whether g'(2) is positive, negative, or zero. Estimate g'(2) using a small interval and round the answer to three decimal places.

Learn how to sketch a graph for the function g(t) = (0.4)^t and determine whether its derivative g'(2) is positive, negative, or zero. Use a small interval to estimate the derivative g'(2) and round your answer to three decimal places. This calculus problem covers key concepts such as exponential functions and derivatives using the difference quotient.

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