Estimate the slope of the tangent line at the point 𝑥=−4,x=−4, for 𝑓(𝑥)=𝑒𝑥.f(x)=ex.

(Use decimal notation. Give your answer to three decimal places.)

To find the slope of the tangent line to the function \( f(x) = e^x \) at the point \( x = -4 \), we need to evaluate the derivative of the function at that point.

The derivative of \( f(x) = e^x \) is:

\[
f'(x) = e^x
\]

Now, we substitute \( x = -4 \) into the derivative:

\[
f'(-4) = e^{-4}
\]

To calculate this, we evaluate \( e^{-4} \) and round the result to three decimal places:

\[
f'(-4) \approx 0.018
\]

So, the slope of the tangent line at \( x = -4 \) is approximately \( \boxed{0.018} \).

Would you like more details on the process, or do you have any other questions?

Here are some related questions to consider:

1. What is the slope of the tangent line at \( x = 0 \) for \( f(x) = e^x \)?
2. How would the slope change if the function were \( f(x) = e^{2x} \)?
3. What is the equation of the tangent line at \( x = -4 \) for \( f(x) = e^x \)?
4. How do you find the second derivative of \( f(x) = e^x \)?
5. What is the interpretation of the derivative of an exponential function?
6. How does the value of \( e^{-4} \) compare to \( e^{-2} \)?
7. Can you explain the significance of the number \( e \) in calculus?
8. How do exponential functions differ from polynomial functions in terms of their derivatives?

**Tip:** Remember that the derivative of \( e^x \) is unique because it is the same as the original function, making it a powerful tool in calculus.

Learn how to estimate the slope of the tangent line at x = -4 for the function f(x) = e^x. This problem involves calculating the derivative of an exponential function and applying it to find the slope at a specific point.

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