A3.2

verify the constant, constant multiple, sum, and difference rules graphically and numerically [e.g., by using the function $g(x)=kf(x)$ and comparing the graphs of $g^{\prime }(x)$ and $k{f}^{\prime }(x)$; by using a table of values to verify that $f^{\prime }(x)+g^{\prime }(x)=(f+g{)}^{\prime }(x)$, given $f(x)=x$ and $g(x)=3x$], and read and interpret proofs involving ${\lim }_{h\to 0}\frac{f(x+h)-f(x)}{h}$ of the constant, constant multiple, sum, and difference rules (student reproduction of the development of the general case is not required) Sample problem: The amounts of water flowing into two barrels are represented by the functions $f(t)$ and $g(t)$. Explain what $f^{\prime }(t)$, $g^{\prime }(t)$, $f^{\prime }(t)+g^{\prime }(t)$, and $(f+g{)}^{\prime }(t)$ represent. Explain how you can use this context to verify the sum rule, $f^{\prime }(t)+g^{\prime }(t)=(f+g{)}^{\prime }(t)$.