Exam 2: Derivatives

A car leaves an intersection traveling west. Its position 4 sec later is 26 ft from the intersection. At the same time, another car leaves the same intersection heading north so that its position 4 sec later is 26 ft from the intersection. If the speeds of the cars at that instant of time are 12 ft/sec and 10 ft/sec, respectively, find the rate at which the distance between the two cars is changing. Round to the nearest tenth if necessary.

If , find and .

Find the derivative of the function.

Find by implicit differentiation.

The height (in meters) of a projectile shot vertically upward from a point m above ground level with an initial velocity of 25.48 m/s is after t seconds. a) When does the projectile reach its maximum height? (b) What is the maximum height?

Two cars start moving from the same point. One travels south at mi/h and the other travels west at mi/h. At what rate is the distance between the cars increasing 2 hours later? Round the result to the nearest hundredth.

Find in terms of .

Determine the values of x for which the given linear approximation is accurate to within 0.07 at a = 0.

Find the derivative of the function.

Find the derivative of the function.

The top of a ladder slides down a vertical wall at a rate of 0.15 m/s . At the moment when the bottom of the ladder is 1.5 m from the wall, it slides away from the wall at a rate of 0.3 m/s. How long is the ladder?

Find the second derivative of the function.

Find the points on the curve where the tangent is horizontal.

Find an equation of the tangent line to the graph of at the point

Use the linear approximation of the function at to approximate the number .

A spherical balloon is being inflated. Find the rate of increase of the surface area with respect to the radius r when r = ft.

The mass of the part of a metal rod that lies between its left end and a point x meters to the right is . Find the linear density when x is m.

Differentiate.

Calculate .

If a snowball melts so that its surface area decreases at a rate of , find the rate at which the diameter decreases when the diameter is cm.