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Section 3.6 Recap. Derivatives of Inverse Functions. Our Main Focus . Additional 3.6 Example. Find . What We’ve Learned Thus Far. Section 3.7. Related Rates.

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Presentation Transcript
Section 3 6 recap

Section 3.6 Recap

Derivatives of Inverse Functions





Section 3 7

Section 3.7

Related Rates


Section 3 6 recap

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.


Section 3 6 recap

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 1
Example 1

A 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 2
Example 2

Air 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 3
Example 3

Assume that and are both differentiable functions of t and find the required values.


Example 4
Example 4

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


Example 5
Example 5

The 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 6
Example 6

All 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 7
Example 7

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


Example 8
Example 8

A 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 9
Example 9

A 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?



Concept check
Concept Check

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


Concept check1
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
Questions?

  • Remember to be working the practice problems.