Exam 2: Derivatives

The cost (in dollars) of producing x units of a certain commodity is . Find the average rate of change with respect to x when the production level is changed from to .

Plot the graph of the function f in an appropriate viewing window. f (x) =

The mass of part of a wire is kilograms, where x is measured in meters from one end of the wire. Find the linear density of the wire when x =16m .

A boat is pulled into a dock by a rope attached to the bow of the boat and passing through a pulley on the dock that is 1 m higher than the bow of the boat. If the rope is pulled in at a rate of 2 m/s how fast is the boat approaching the dock when it is 3 m from the dock? Round the result to the nearest hundredth if necessary.

Show that the function is continuous but not differentiable at the given value of x.

Plot the graph of the function f in an appropriate viewing window. f (x) =

An equation relating the variables x and y, the values of x and y, and the value of at a particular instant of time are given. Find the value of .

If is a differentiable function, find an expression for the derivative of

Two carts, A and B, are connected by a rope 39 ft long that passes over a pulley (see the figure below). The point Q is on the floor 12 ft directly beneath and between the carts. Cart A is being pulled away from Q at a speed of ft/s. How fast is cart B moving toward Q at the instant when cart A is 5 ft from Q?

Find the derivative of the function in two ways. F(x) = (x + 9)(x² - 3x + 7)

Find an equation of the tangent line to the curve at the point .

The graph shows the percentage of households in a certain city watching television during a 24-hr period on a weekday ( corresponds to 6 a.m.).By computing the slope of the respective tangent line, estimate the rate of change of the percentage of households watching television at a-12 p.m.Note that

In calm waters, the oil spilling from the ruptured hull of a grounded tanker spreads in all directions. Assuming that the polluted area is circular, determine how fast the area is increasing when the radius of the circle is 20 ft and is increasing at the rate of ft/sec. Round to the nearest tenth if necessary.

Use the Quotient Rule to find the derivative of the function.

Differentiate.

Newton's Law of Gravitation says that the magnitude F of the force exerted by a body of mass m on a body of mass M is . Find .

Find all points at which the tangent line is horizontal on the graph of the function.

Find by implicit differentiation.

Calculate .

Differentiate.