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We are given a third-degree Taylor polynomial for the function \( f(x) = x^7 - 3x^2 + x - 5 \) about the point \( x = 1 \). The general form of the Taylor polynomial \( T_n(x) \) is:

\[
T_n(x) = a_0 + a_1(x - a) + a_2(x - a)^2 + a_3(x - a)^3
\]

where \( a = 1 \), and we need to find \( a_3 \), which is the coefficient of the \( (x - a)^3 \) term.

### Steps to calculate \( a_3 \):
The coefficient \( a_3 \) is given by the following formula for the Taylor polynomial expansion:
\[
a_3 = \frac{f^{(3)}(a)}{3!}
\]
where \( f^{(3)}(a) \) is the third derivative of \( f(x) \) evaluated at \( x = a = 1 \), and \( 3! \) is the factorial of 3.

Let's first compute the first few derivatives of \( f(x) = x^7 - 3x^2 + x - 5 \):

1. **First derivative:**
   \[
   f'(x) = 7x^6 - 6x + 1
   \]

2. **Second derivative:**
   \[
   f''(x) = 42x^5 - 6
   \]

3. **Third derivative:**
   \[
   f^{(3)}(x) = 210x^4
   \]

Now, substitute \( x = 1 \) into \( f^{(3)}(x) \):
\[
f^{(3)}(1) = 210(1)^4 = 210
\]

Finally, use the formula for \( a_3 \):
\[
a_3 = \frac{f^{(3)}(1)}{3!} = \frac{210}{6} = 35
\]

Thus, the value of \( a_3 \) is \( \boxed{35} \).

Would you like more details or have any questions? 

Here are 5 follow-up questions for deeper understanding:

1. How is the Taylor polynomial constructed for functions with different degrees?
2. What is the role of the factorial in the Taylor series expansion?
3. Why do higher-degree terms contribute less to the overall polynomial?
4. Can you calculate the second-degree coefficient \( a_2 \) for this function?
5. How would the Taylor polynomial change if we expanded about a different point \( x = a \)?

**Tip:** When computing higher derivatives, always simplify intermediate expressions to make final evaluations easier.

If the 3rd degree Taylor polynomial for f(x) = x^7 - 3x^2 + x - 5 about the point x = 1 is given as T_n(x) = a_0 + a_1(x - a) + a_2(x - a)^2 + a_3(x - a)^3, then what is the value of a_3?

Learn how to calculate the coefficient a_3 in the 3rd degree Taylor polynomial for the function f(x) = x^7 - 3x^2 + x - 5 about the point x = 1. This problem covers key calculus concepts, including polynomial functions and Taylor series expansions.

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