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To solve this problem, we need to calculate the difference quotients for the function \( H(x) = 3 \ln x + 7 \), using the values \( h = 0.1 \), \( 0.01 \), and \( 0.001 \), and then use these to approximate the slope of the tangent line at the point \( (e, 10) \).

### The formula for the difference quotient:
\[
\frac{H(e + h) - H(e)}{h}
\]
This gives an approximation of the derivative of \( H(x) \) at \( x = e \).

First, let's calculate the function \( H(x) \) at \( x = e \). Since \( H(x) = 3 \ln x + 7 \), we have:
\[
H(e) = 3 \ln e + 7 = 3 \times 1 + 7 = 10
\]
So, \( H(e) = 10 \).

Now, we will calculate the difference quotients for the three given values of \( h \):

### For \( h = 0.1 \):
\[
H(e + 0.1) = 3 \ln (e + 0.1) + 7
\]
Using this, we can calculate:
\[
\frac{H(e + 0.1) - H(e)}{0.1}
\]

### For \( h = 0.01 \):
\[
H(e + 0.01) = 3 \ln (e + 0.01) + 7
\]
Again, we calculate:
\[
\frac{H(e + 0.01) - H(e)}{0.01}
\]

### For \( h = 0.001 \):
\[
H(e + 0.001) = 3 \ln (e + 0.001) + 7
\]
And then:
\[
\frac{H(e + 0.001) - H(e)}{0.001}
\]

Finally, these difference quotients will approximate the slope of the tangent line at the point \( (e, 10) \). Let me compute these values.The difference quotients for the given values of \( h \) are:

- For \( h = 0.1 \), the difference quotient is approximately \( 1.083823 \)
- For \( h = 0.01 \), the difference quotient is approximately \( 1.101613 \)
- For \( h = 0.001 \), the difference quotient is approximately \( 1.103435 \)

As \( h \) gets smaller, the difference quotients approach the slope of the tangent line at \( (e, 10) \), which is approximately **1.103435**.

Would you like further explanation or details on any part of this process?

Here are 5 related questions to deepen understanding:

1. What is the geometric meaning of the slope of the tangent line?
2. How does the difference quotient relate to the derivative?
3. Why does the difference quotient become more accurate as \( h \) gets smaller?
4. How would we calculate the exact derivative of \( H(x) = 3 \ln x + 7 \) without using difference quotients?
5. How does the natural logarithm function behave as \( x \) increases or decreases?

**Tip:** When calculating derivatives numerically, smaller values of \( h \) generally provide better approximations but can be prone to round-off errors in computation.

Calculate the difference quotients for H(x) = 3 ln x + 7 at (e, 10) using h = 0.1, 0.01, 0.001.

Learn how to calculate the slope of the tangent line to the function H(x) = 3 ln x + 7 at the point (e, 10) using difference quotients with h values of 0.1, 0.01, and 0.001. Explore how logarithmic functions and calculus concepts like derivatives are applied to approximate slopes.

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