For what values $a$ and $b$ is the line $2 x + y = b$ tangent to the curve $y = a x^{2}$ at the point with $x = 2$?

(A) Use the limit process, as we did in class, to find the slope of the tangent line to $y = a x^{2}$ when $x = 2$. (Your answer will contain an $a$.)

The slope of the tangent line to $y = a x^{2}$ when $x = 2$ is .

(B) What is the slope of the line $2 x + y = b$?

The slope of this line is .

(C) In order for the line $2 x + y = b$ to be tangent to the curve $y = a x^{2}$ at $x = 2$, we need for the slope of the tangent line to $y = a x^{2}$ at $x = 2$ to be the same as the slope of the line $2 x + y = b$.

Thus in comparing (A) and (B), we find that $a =$

(D) Lastly, in order for the line $2 x + y = b$ to be tangent to the curve $y = a x^{2}$ at $x = 2$, the graphs of $2 x + y = b$ and $y = a x^{2}$ must have the same $y$-coordinate at $x = 2$.

Comparing the $y$-coordinates of the two graphs tells us that $b =$

You can earn partial credit on this problem.