a. Use the definition $m_{\tan }=\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h}$ to find the slope of the line tangent to the graph of $f$ at $P$. b. Determine an equation of the tangent line at $P$. \[ f(x)=x^{4}, P(2,16) \] a. $m_{\tan }=32$ b. $y=$

Step 1 ：Given the function \(f(x) = x^4\) and the point \(P(2,16)\).

Step 2 ：The slope of the tangent line to the graph of the function at a point is given by the derivative of the function at that point.

Step 3 ：Find the derivative of the function \(f(x) = x^4\), which is \(f'(x) = 4x^3\).

Step 4 ：Evaluate the derivative at \(x = 2\) to find the slope of the tangent line, which is \(f'(2) = 4*2^3 = 32\).

Step 5 ：Use the point-slope form of the equation of a line, \(y - y_1 = m(x - x_1)\), to find the equation of the tangent line. Here, \(m = 32\) and \((x_1, y_1) = (2, 16)\).

Step 6 ：Substitute the values into the equation to get \(y - 16 = 32(x - 2)\).

Step 7 ：Simplify the equation to get \(y = 32x - 48\).

Step 8 ：Final Answer: The equation of the tangent line to the graph of the function \(f(x) = x^4\) at the point \(P(2,16)\) is \(\boxed{y = 32x - 48}\).