Let f (x) = x3 + 10 and a = 3.
find the slope ma of the line tangent to the graph of f at the point (a, f (a))

To find the slope \( m_a \) of the tangent line to the graph of the function \( f(x) \) at the point \( (a, f(a)) \), we need to compute the derivative of \( f(x) \) and then evaluate it at \( x = a \).

Given:
\[
f(x) = x^3 + 10
\]
and \( a = 3 \).

### Step 1: Find the derivative \( f'(x) \)
The derivative of \( f(x) \) with respect to \( x \) is:
\[
f'(x) = \frac{d}{dx}(x^3 + 10)
\]
The derivative of \( x^3 \) is \( 3x^2 \), and the derivative of the constant \( 10 \) is \( 0 \). So:
\[
f'(x) = 3x^2
\]

### Step 2: Evaluate the derivative at \( x = a = 3 \)
Now, substitute \( x = 3 \) into the derivative to find the slope at that point:
\[
m_a = f'(3) = 3(3)^2 = 3(9) = 27
\]

### Final Answer:
The slope \( m_a \) of the tangent line to the graph of \( f(x) \) at the point \( (3, f(3)) \) is \( 27 \).

Would you like more details or have any questions?

Here are 5 related questions to explore:
1. How do you find the equation of the tangent line at \( x = 3 \)?
2. What is the second derivative \( f''(x) \) and what does it tell you about the graph?
3. How does the tangent line relate to the concept of local linearity?
4. What happens to the slope of the tangent as \( x \) increases for the function \( f(x) = x^3 + 10 \)?
5. How would the process differ if \( f(x) \) had a higher degree term, such as \( x^4 \)?

**Tip:** The derivative gives the slope of the tangent line at any point on a curve, which is a fundamental concept in calculus for understanding how a function behaves locally.

Learn how to find the slope of the tangent line to the function f(x) = x^3 + 10 at the point (3, 37). Detailed step-by-step solution using calculus, including derivative calculation and evaluation. Suitable for college-level calculus students.

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