Problem $1.$ Let $f(x)=3x^2-1.$ Find the difference quotient $$ of $f(x)$ over the interval $x,x+h$ and use this to find ${}_{[2,6]},$ the average rate of change of $f$ over the interval $2,6.$

Problem $2.$ Use the definition of the derivative to find $f^{\text{'}}(x)$ for $f(x)$ above. By the Mean Value Theorem there exists a number $c$ in $2,6$ such that $f^{\text{'}}(c)={}_{[2,6]}$; find $c.$

Problem $3.$ Find the equation of the tangent line $T$ to $f(x)=3x^2-1$ at $x=3.$ Sketch the graph of $f(x)$ on $2,6$ and include $T($the graph does not have to be to scale$).$

Problem $4.$ Explain what the difference quotient is and what the derivative is$-$how do they relate to each other?