A coal-burning electrical generating plant emits sulfur dioxide into the surrounding air. The concentration \( C(x) \), in parts per million, is approximately given by the function \[ C(x)=\frac{0.4}{x^{2}} \] where \( x \) is the distance away from the plant in miles. The instantaneous rate of change of the sulfur dioxide concentration: (A) 2 miles from the plant = (B) 11 miles from the plant =

\( y=e^{2 \cos 2 x} \) deviuates

derivates \( y=\frac{1+x^{3}}{1-x^{3}} \)

Let \( f(x)=\frac{x}{x^{2}+13} \). Find (a) the intervals on which \( f \) is increasing, (b) the intervals on which \( f \) is decreasing, (c) the open intervals on which \( f \) is concave up, (d) the open intervals on which \( f \) is concave down, and (e) the \( x \)-coordinates of all inflection points. (a) \( f \) is increasing on the interval(s) \( [-\sqrt{13}, \sqrt{13}] \) (b) \( f \) is decreasing on the interval(s) \( (- \) infinity, 0\( ) \) (c) \( f \) is concave up on the open interval(s) (-infinity , 0\( ) \) (d) \( f \) is concave down on the open interval(s) \( (0 \), infinity) (e) the \( x \) coordinate(s) of the points of inflection are \( -\sqrt{39}, 0, \sqrt{39} \) Notes: In the first four boxes, your answer should either be a single interval, such as \( [0,1) \), a comma separated list of intervals, such as (-inf, 2), ( \( (3,4] \), or the word "none". In the last box, your answer should be a comma separated list of \( x \) values or the word "none".

(b) \( f(x)=-x^{6}-8 x^{5}-7 x^{4}+6 x^{2} \) Falls to the left and rises to the right Rises to the left and falls to the right Falls to the left and falls to the right

CHECKING COMPREMENSION (press pause for more time) Find the volume of the solid of revolution that results from rotating \( y=x^{2}+3 \) around the \( x \) axis, over the interval from 0 to 1 .