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question 1d says f(x) = x^2 - 3x-8 and a=0

Let's solve question 1(d) for the given function \( f(x) = x^2 - 3x - 8 \) with \( a = 0 \). The task is divided into three parts:

### Step (i): Find the derivative function \( f'(x) \)

The derivative of the function \( f(x) = x^2 - 3x - 8 \) can be found using the power rule.

\[
f'(x) = \frac{d}{dx}(x^2) - \frac{d}{dx}(3x) - \frac{d}{dx}(8)
\]

Using the power rule:
\[
\frac{d}{dx}(x^2) = 2x, \quad \frac{d}{dx}(3x) = 3, \quad \frac{d}{dx}(8) = 0
\]

Thus:
\[
f'(x) = 2x - 3
\]

### Step (ii): Find \( f'(a) \) at the given point \( a = 0 \)

Now we substitute \( a = 0 \) into the derivative function \( f'(x) = 2x - 3 \):

\[
f'(0) = 2(0) - 3 = -3
\]

### Step (iii): Find the equation of the tangent line at \( x_0 = a = 0 \)

The equation of the tangent line can be written in the form \( y = mx + b \), where:
- \( m = f'(a) \) is the slope of the tangent line,
- \( b \) is the y-intercept, which can be found using the point \( (a, f(a)) \).

We already know that \( f'(0) = -3 \), so the slope \( m = -3 \).

Now, to find \( b \), we need to find the value of \( f(0) \):

\[
f(0) = (0)^2 - 3(0) - 8 = -8
\]

Thus, the point of tangency is \( (0, -8) \). Substituting this point into the tangent line equation \( y = -3x + b \):

\[
-8 = -3(0) + b \quad \Rightarrow \quad b = -8
\]

Therefore, the equation of the tangent line is:

\[
y = -3x - 8
\]

### Summary:
- The derivative function is \( f'(x) = 2x - 3 \),
- \( f'(0) = -3 \),
- The equation of the tangent line is \( y = -3x - 8 \).

Let me know if you would like any further details or clarifications!

Here are 5 related questions:
1. How do you find the derivative of a more complex function?
2. What is the significance of the tangent line in calculus?
3. How do you use the limit definition of the derivative to verify the result?
4. What happens to the tangent line equation if we choose a different point \( a \)?
5. Can a function have more than one tangent line at a given point?

**Tip**: Always double-check the values of \( f(a) \) and \( f'(a) \) to ensure accurate slope and point for the tangent line.

question 1d says f(x) = x^2 - 3x - 8 and a=0

Learn how to find the derivative of f(x) = x^2 - 3x - 8 using the power rule and apply it to determine the equation of the tangent line at x = 0. This problem covers key calculus concepts such as differentiation and tangent lines, suitable for high school calculus students.

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