8 Calculus Questions

1.

The height (in meters) of a projectile shot vertically upward from a point 3 m above ground level with an initial velocity of 21.5 m/s is h = 3 + 2151' 4.91\"2 after t seconds. (Round your answers to two decimal places.) (a) Find the velocity after 2 s and after 4 5. V0) = m/s v(4) = m/s (b) When does the projectile reach its maximum height? 5 (c) What is the maximum height? m (d) When does it hit the ground? 5 (e) With what velocity does it hit the ground? m/s Sodium chlorate crystals are easy to grow in the shape of cubes by allowing a solution of water and sodium chlorate to evaporate slowly. If V is the volume of such a cube with side length x, calculate the derivative when x = 4 mm. V'(4) = mm3/mm What does V'(4) mean in this situation? O V(4) represents the rate at which the volume is increasing as x reaches 12 mm. O V(4) represents the rate at which the side length is increasing with respect to the volume as x reaches 4 mm. O V(4) represents the rate at which the volume is increasing with respect to the side length as V reaches 12 mm3. O V(4) represents the volume as the side length reaches 4 mm. O V(4) represents the rate at which the volume is increasing with respect to the side length as x reaches 4 mm.If a tank holds 3500 gallons of water, which drains from the bottom of the tank in 50 minutes, then Toricelli's Law gives the volume V of water remaining in the tank after t minutes as 2 v: 3500(1 510t) as t s 50. Find the rate at which water is draining from the tank after the following amounts of time. (Remember that the rate must be negative because the amount of water in the tank is decreasing.) (a) 5min gal/min (b) 10 min gal/min (c) 20 min gal/min (d) 50min gal/min At what time is the water flowing out the fastest? t = min At what time is the water flowing out the slowest? t = min Newton's Law of Gravitation says that the magnitude Fof the force exerted by a body of mass m on a body of mass M is = GmM ,2 where G is the gravitational constant and r is the distance between the bodies. (a) Find dF/dr. LS o'r What is the meaning of dF/dr? dF/o'r represents the rate of change of the distance between the bodies with respect to the force. F dF/dr represents the rate of change of the mass with respect to the force. dF/dr represents the rate of change of the mass with respect to the distance between the bodies. dF/dr represents the amount of force per distance. dF/dr represents the rate of change of the force with respect to the distance between the bodies. What does the minus sign indicate? ' The minus sign indicates that as the distance between the bodies decreases, the magnitude of the force remains constant. ' The minus sign indicates that the force between the bodies is decreasing. The minus sign indicates that as the distance between the bodies increases, the magnitude of the force decreases. The minus sign indicates that as the distance between the bodies increases, the magnitude of the force increases. ' The minus sign indicates that the bodies are being forced in the negative direction. (b) Suppose it is known that the earth attracts an object with a force that decreases at the rate of 4 N/km when r = 20,000 km. How fast does this force change when r = 10,000 km? N/km The number of yeast cells in a laboratory culture increases rapidly initially but levels off eventually. The population is modeled by the function ; 1 + line0'3t where t is measured in hours. At time t = 0 the population is 20 cells and is increasing at a rate of 4 cells/hour. Find the values of a and b. a: b: n=f(t) = According to this model, what happens to the yeast population in the long run? ' The yeast population will grow without bound. ' The yeast population will stabilize at 2 cells. The yeast population will stabilize at 30 cells. ' The yeast population will stabilize at 60 cells. The yeast population will shrink to 0 cells. Invasive species often display a wave of advance as they colonize new areas. Mathematical models based on random dispersal and reproduction have demonstrated that the speed with which such waves move is given by the expression 2'! D , where r is the reproductive rate of individuals and D is a parameter quantifying dispersal. Calculate v'(r), the derivative of the wave speed with respect to the reproductive rate r. Explain its meaning. ' v'(r) is the rate of change of the reproductive rate of a population with respect to the invasive species wave speed. v'(r) is the rate of change of the reproductive rate of a population with respect to the inverse of invasive species wave speed derivative. v'(r) is the rate of change of the reproductive rate of individuals with respect to invasive species wave speed. v'(r) is the rate of change of invasive species wave speed with respect to the reproductive rate of a population. v'(r) is the rate of change of invasive species wave speed with respect to the reproductive rate of individuals. The cost function for production of a commodity is C(x) = 318 + 26x 0.05x2 + 0.0005x3. (a) Find C'(100). Interpret C'(100). ' This is the rate at which the production level is decreasing with respect to the cost when x = 100. This is the cost of making 100 items. This is the rate at which costs are increasing with respect to the production level when x = 100. ' This is the amount of time, in minutes, it takes to produce 100 items. This is the number of items that must be produced before the costs reach 100. (b) Find the actual cost of producing the 1015t item. (Round your answer to the nearest cent.) $ The length of a rectangle is increasing at a rate of 9 cm/s and its width is increasing at a rate of 5 cm/s. When the length is 11 cm and the width is 6 cm, how fast is the area of the rectangle increasing? cmzls