ISH The height of a fish compared to sea level as it jumps out of the ocean water can be represented by a function. Sketch a nonlinear 9 rimp hat shows the height of a fish \( y \), in inches, as a function of time \( x \), in seconds. Interpret the key features. Positive: between 2 seconds and 8 seconds Negative: for time less than 2 seconds and greater than 8 seconds Increasing: for time less than 5 seconds Decreasing: for time greater 5 seconds Relative Maximumi at 5 seconds, when the height of the fish is 9 inches End Behavion: As time increases or decreases, the height of the fish decreases. Symmetry: The height of the fish for time less than 5 seconds is the same as the height for time greater than 5 seconds.

Use L'Hospital to determine the following limit. Use exact values. \( \lim _{x \rightarrow 0^{+}} x \ln x=\square \)

Let \( y=\sin (4 x) \), If \( \Delta x=0.1 \) at \( x=0 \), use linear approximation to estimate \( \Delta y \) \( \Delta y \approx=0.4 \) Find the percentage error error \( =\square \% \)

The equation of line \( L_{1} \) is \( y=3 x-4 \) The equation of line \( L_{2} \) is \( 9 y+k x-18=0 \) \( L_{1} \) is perpendicular to \( L_{2} \) Find the value of \( k \). You must show all your working. Show your working

\( \left. \begin{array} { l } { x _ { i } = 1,1,1,1,1,1,1,1,3,18,45,63,53,47,42,39,37,32,28,23,14,4,4 } \\ { n = } \end{array} \right. \)

\( P(x)=-x^{3}+\frac{27}{2} x^{2}-60 x+100, x \geq 5 \) is an approximation to the total profit (in thousands of dollars) from the sale of \( x \) hundred thousand tires. Find the number of hundred thousands of tires that must be sold to maximize profit. A. 4.5 hundred thousand B. 4 hundred thousand C. 5.5 hundred thousand D. 5 hundred thousand