Consider the function\(y = 4 x ^ { 2 } \). Find the slope of the tangent line at the point (2,16).

Extra Exercise: Consider the function\(f ( x ) = 4 x ^ { 2 } \). The derivative of\(f ( x ) \)is\(f ^ { \prime } ( x ) = 8 x\). Find the slope of the tangent line at the point (3,36).

The data pairs\( ( x , y ) \)represent the average annual tuition\( y \)(in dollars) for public colleges in the United States\( x \)years after 2005. Use the linear regression feature on a graphing calculator to find an equation of the line of best fit. Estimate the average annual tuition in 2020. Interpret the slope and\( y \)-intercept in this situation. (See Example 4.)

\( ( 0,11,386 ) , ( 1,11,731 ) , ( 2,11,848 ) \)

\( ( 3,12,375 ) , ( 4,12,804 ) , ( 5,13,297 ) \)

Compute\( r ^ { \prime \prime } ( t ) \)and\( r ^ { \prime \prime \prime } ( t ) \)for the following function.

\( r ( t ) = \langle \cos 2 t , \sin 6 t , \cos 5 t \rangle \)

\( r ^ { \prime \prime } ( t ) = \)

Let\( u ( t ) = 6 t ^ { 3 } i + ( t ^ { 2 } - 2 ) j - 8 k \). Compute the derivative of the following function.

\( u ( t ^ { 4 } - 2 t ) \)

Select the correct choice below and fill in the answer box(es) to complete your choice.

A. The derivative is the vector-valued function (\( \square i + \square ) j + \square k \).

B. The derivative is the scalar function .

Compute the following derivative.

\( \frac { d } { d t } [ t ^ { 2 } ( i + 4 j - 4 t k ) \cdot ( e ^ { t } i + 4 e ^ { t } j - 6 e ^ { - t } k ) ] \)