A
Let's address each part step by step.
### (a) Write the product function
The product function is the product of \( g(x) \) and \( h(x) \). Given:
\[ g(x) = 6e^{9.5x} \]
\[ h(x) = 6(9.5x) \]
The product function \( f(x) \) is:
\[ f(x) = g(x) \cdot h(x) \]
Substitute \( g(x) \) and \( h(x) \):
\[ f(x) = (6e^{9.5x}) \cdot (6 \cdot 9.5x) \]
Simplify the expression:
\[ f(x) = 36 \cdot 9.5 \cdot x \cdot e^{9.5x} \]
\[ f(x) = 342x e^{9.5x} \]
### (b) Write the rate-of-change function
To find the rate-of-change function, we need to find the derivative of \( f(x) = 342x e^{9.5x} \). We'll use the product rule, which states that if \( f(x) = u(x)v(x) \), then \( f'(x) = u'(x)v(x) + u(x)v'(x) \).
Let \( u(x) = 342x \) and \( v(x) = e^{9.5x} \).
1. Compute \( u'(x) \):
\[ u(x) = 342x \]
\[ u'(x) = 342 \]
2. Compute \( v'(x) \):
\[ v(x) = e^{9.5x} \]
\[ v'(x) = 9.5e^{9.5x} \]
Now apply the product rule:
\[ f'(x) = u'(x)v(x) + u(x)v'(x) \]
Substitute \( u(x) \), \( u'(x) \), \( v(x) \), and \( v'(x) \):
\[ f'(x) = 342 \cdot e^{9.5x} + 342x \cdot 9.5e^{9.5x} \]
Factor out \( 342e^{9.5x} \):
\[ f'(x) = 342e^{9.5x} (1 + 9.5x) \]
Thus, the rate-of-change function is:
\[ f'(x) = 342e^{9.5x} (1 + 9.5x) \]
