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### Given:
- \( f(3) = 4 \) and \( f'(3) = 3 \)
- \( g(x) = x^2 \cdot f(x) \)
- \( h(x) = \frac{f(x)}{x - 4} \)

### Questions:
a. Find an equation of the line tangent to \( y = g(x) \) at \( x = 3 \).  
b. Find an equation of the line tangent to \( y = h(x) \) at \( x = 3 \).

#### Tangent Line Equation:
The general form for the tangent line to a curve \( y = f(x) \) at \( x = a \) is:
\[
y - f(a) = f'(a)(x - a)
\]

#### Part a: Tangent to \( g(x) \)
To find the tangent line for \( g(x) = x^2 \cdot f(x) \) at \( x = 3 \):
1. **Find \( g(3) \)**:
\[
g(3) = 3^2 \cdot f(3) = 9 \cdot 4 = 36
\]

2. **Find \( g'(x) \)** using the product rule:
\[
g'(x) = \frac{d}{dx} [x^2 \cdot f(x)] = 2x \cdot f(x) + x^2 \cdot f'(x)
\]
Evaluate at \( x = 3 \):
\[
g'(3) = 2 \cdot 3 \cdot f(3) + 3^2 \cdot f'(3) = 6 \cdot 4 + 9 \cdot 3 = 24 + 27 = 51
\]

3. **Equation of the tangent line** at \( x = 3 \):
Using \( y - f(a) = f'(a)(x - a) \):
\[
y - 36 = 51(x - 3)
\]
Simplify:
\[
y = 51(x - 3) + 36
\]
This is the equation of the tangent line to \( g(x) \) at \( x = 3 \).

#### Part b: Tangent to \( h(x) \)
For \( h(x) = \frac{f(x)}{x - 4} \):
1. **Find \( h(3) \)**:
\[
h(3) = \frac{f(3)}{3 - 4} = \frac{4}{-1} = -4
\]

2. **Find \( h'(x) \)** using the quotient rule:
The quotient rule is:
\[
h'(x) = \frac{(x - 4) \cdot f'(x) - f(x)}{(x - 4)^2}
\]
Evaluate at \( x = 3 \):
\[
h'(3) = \frac{(3 - 4) \cdot f'(3) - f(3)}{(3 - 4)^2} = \frac{(-1) \cdot 3 - 4}{(-1)^2} = \frac{-3 - 4}{1} = -7
\]

3. **Equation of the tangent line** at \( x = 3 \):
\[
y - (-4) = -7(x - 3)
\]
Simplify:
\[
y + 4 = -7(x - 3)
\]
\[
y = -7(x - 3) - 4
\]
This is the equation of the tangent line to \( h(x) \) at \( x = 3 \).

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### Related questions:
1. How would the tangent line equation change if \( f'(3) \) was a different value?
2. What if the point of tangency was at \( x = 4 \)? How would you calculate the tangent line for \( g(x) \)?
3. How does the product rule apply to more complex functions like \( g(x) \)?
4. How does the quotient rule handle functions with higher-order polynomials?
5. Can you explain the impact of singularities or discontinuities in functions like \( h(x) \) at points like \( x = 4 \)?

### Tip:
Always verify the conditions of the function at the given point before differentiating, especially when dealing with products or quotients!

Suppose f(3) = 4 and f'(3) = 3. Let g(x) = x^2 * f(x) and h(x) = f(x) / (x - 4). a. Find an equation of the line tangent to y = g(x) at x = 3. b. Find an equation of the line tangent to y = h(x) at x = 3.

Find the tangent lines for g(x) = x^2 * f(x) and h(x) = f(x) / (x - 4) at x = 3. This solution involves using the product rule and quotient rule of differentiation, calculating f(3) = 4 and f'(3) = 3.

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