Questions: Use the definition (mtan =lim h rightarrow 0 fracf(a+h)-f(a)h) to find the slope of the line tangent to the graph of (f) at (P). Determine an equation of the tangent line at (P). (f(x)=7+4 x^2 ; P(0,7)) (mtan =) (square) (Type an integer or a fraction.)

Find the slope of the line tangent to the graph of \( f \) at point \( P \).

Calculate \( f(a+h) \) and \( f(a) \) for \( a = 0 \).

We have \( f(a+h) = f(0+h) = 7 + 4(h)^2 = 4h^2 + 7 \) and \( f(a) = f(0) = 7 \).

Use the limit definition to find the slope \( m_{\tan} \).

Using the limit definition, we find:
\[ m_{\tan} = \lim_{h \rightarrow 0} \frac{f(a+h) - f(a)}{h} = \lim_{h \rightarrow 0} \frac{(4h^2 + 7) - 7}{h} = \lim_{h \rightarrow 0} \frac{4h^2}{h} = \lim_{h \rightarrow 0} 4h = 0. \]

The slope of the tangent line is \( \boxed{0} \).

Determine an equation of the tangent line at point \( P \).

Use the point-slope form of the line equation.

The point-slope form is given by \( y - y_1 = m(x - x_1) \). Here, \( m = 0 \) and \( (x_1, y_1) = (0, 7) \). Thus, the equation becomes:
\[ y - 7 = 0(x - 0) \implies y = 7. \]

The equation of the tangent line is \( \boxed{y = 7} \).

The slope of the tangent line is \( \boxed{0} \).
The equation of the tangent line is \( \boxed{y = 7} \).