A volcanic cone spews fine particles of rock which land in the surrounding area and cause the size of the cone to increase. (a) Suppose that a volcanologist takes measurements that show that when the cone is 100 meters wide and 50 meters tall, its height is increasing at 0.5 meters/day. If the assumption is made that the cone will retain its height-width proportion, what is the rate at which the volume of rock is being released by the volcano? 12507! m3/day V (b) Suppose that the volcanologist takes more measurements and realizes the original assumption was poorly made, because the width is increasing at 0.9 meters/day. What would be the new nding for the rate at which the volume of rock is being released by the volcano? 11257! m3/day x Recall that two quantities X and y are related if there eXIsts a function fsuch that y = f(x). Ifx and y are also functions of time, t, then y(t) = f(x(t)). By the Chain Rule, the rates of change ofx and y are related as y'(t) = f'(x(t))x'(t). Sketch a picture that models the described situation. Use the given information to label the picture with variables and constants. Identify the units associated with each variable and write the variable quantities as functions of time. Recall that the volume of a cone is given by V = %m1h where r is the radius of the base of the cone and h is the height of the cone. Use this function to relate the rates at which the volume, height, and radius of the cone are changing. What are the units of the given quantities? What relationship is fixed among the height and radius in part (a)? What rule of differentiation is necessary if both height and radius are changing in part (b)? How does the radius change relative to the width of the cone? Find the following derivatives for the function f(x) = 9(In(x))2. (a) f'(x ) 18 In(x) (b) f" (x) 18 18 In (x) X4 x 2 (c ) f " ( x ) 324 36 In (x) + X Recall that the derivative of the natural logarithmic function is given by -(In(x)) = . What other rules of differentiation are necessary to find the first derivative? What rules of differentiation are necessary to find the second and third derivatives? Can the function be simplified using laws of logarithms before differentiating the function