SECTION 3.6 RECAPDerivatives of Inverse Functions

OUR MAIN FOCUS

ADDITIONAL 3.6 EXAMPLE

Find .a.

b.

WHAT WE’VE LEARNED THUS FAR

SECTION 3.7Related Rates

When water is drained out of a conical tank the volume V, the radius r, and the height h of the water level are all functions of time t.

Let’s say the circle given by the equation

is changing with respect to time. Hence, both and are functions of time, . What is ?

EXAMPLE 1A pebble is dropped into a calm pond, causing ripples in the form of concentric circles. The radius r of the outer ripple is increasing at a constant rate of 1 foot per second. When the radius is 4 feet, at what rate is the total area A of the disturbed water changing.

EXAMPLE 2Air is being pumped into a spherical balloon at a rate of 4.5 cubic feet per minute. Find the rate of change of the radius when the radius is 2 feet.

EXAMPLE 3Assume that and are both differentiable functions of t and find the required values.Equation Find Given

a. when

b. when 3

EXAMPLE 4A point is moving along the graph such that is centimeters per second. Find for the given x value.Equation x-value

EXAMPLE 5The radius r of a sphere is increasing at a rate of 3 inches per minute. Find the rate of change of the volume when inches.

EXAMPLE 6All edges of a cube are expanding at a rate of 6 centimeters per second. How fast is the volume changing when each edge is 10 centimeters?

EXAMPLE 7Using example 6, how fast is the surface area changing when each edge is 10 cm?

EXAMPLE 8A conical tank with vertex down is 10 ft. across the top and 12 feet deep. If water is flowing into the tank at a rate of 10 cubic ft. per minute, find the rate of change of the depth of the water when the water is 8 ft. deep.

EXAMPLE 9A fish is reeled in at a rate of 1 foot per second from a point 10 feet above the water. At what rate is the angle between the line and the water changing when there is a total of 25 feet of line from the end of the rod to the water?

SUMMARY

CONCEPT CHECK

Let be the area of a circle of radius that is changing with respect to time. If is constant, is constant?

CONCEPT CHECK

Consider the linear function . If changes at a constant rate, does change at a constant rate? If so, does it change at the same rate as ?

QUESTIONS?

Remember to be working the practice problems.