Applications of the Derivative. By Michele Bunch, Kimberly Duane and Mark Duane. Extreme Values. Critical values – any value along f(x) where f’(x) = 0 OR f’(x) is undefined These help you determine extreme values: Relative max/min → over ENTIRE FUNCTION
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By Michele Bunch, Kimberly Duane and Mark Duane
These help you determine extreme values:
Because x=4 is not in the interval, you don’t have to test it IN THIS CASE!!
Absolute max: x=2
Absolute min: x=-3
This is a test that uses the first derivative to help you find relative maxes and mins.
1. Find critical values and create a number line of f’(x) with these values.
x = 1 is a local max; x = 4 is a local min
2. Use f’(x) to determine if f’(x) is positive or negative.
-- If f’(x) is positive, f(x) is increasing
-- If f’(x) is negative, f(x) is decreasing
3. If f’(x) changes from positive to negative, the CV is a local max.
If f’(x) changes from negative to positive, the CV is a local min.
Still a test for determining local max and min!
1. Take the 2nd derivative, set it equal to zero, and solve.
2. Place these values on a number line for f”(x) and use f”(x) to determine if f”(x) is positive or negative.
3. For the critical values:
-- If f”(CV) is positive, then the CV is a local min
-- If f”(CV) is negative, then the CV is a local max
Looks like this:
Looks like this:
Given a differentiable function over an interval, there is at least one point on the curve where the derivative is equal to the average derivative of the entire interval
Find all values of “c” that satisfy the MVT.
2) y=x⁻¹ [1,4]
Using related formulas to find the maximum or minimum solution to a problem
Two pens are to be constructed using a total of 900 feet of fencing. One pen is to be a square X by X and the other is to be a rectangle with one side twice as long as the other (X by 2X). Determine the dimensions of the pens so that the enclosed areas are as large as possible.
So the dimensions with the maximum area are 105.9 ft by 105.9 ft and 79.4 ft by 158.8 ft.
Given the curve y=x² in the first quadrant and a vertical line x=3, determine the inscribed rectangle of maximum area which has a right side on line x=3.
The rectangle is 2 by 4, so the area of the rectangle is 8.
1) Draw a picture
2) State wanted and known rates
3) Write formula to compare rates
4) Take the derivative
5) Plug in the known rate
6) Solve for unknown rate
A 20 foot ladder is leaning against a building. The ladder is sliding down the wall at a constant rate of 2 ft/sec. At what rate is the angle between the ladder and the ground changing when the top of the ladder is 12 feet from the ground?
The angle is changing at a rate of -1/8 radians per second.
Water is flowing into and inverted cone at the rate of 5 in³/sec. If the cone has an altitude of 4 inches and a base radius of 3 inches, how fast is the water level rising when the water is 2 inches deep?
The water level is rising at 20/(9π) inches per second.
Given the following equation that gives the position of a particle, find A) the velocity equation, B) the velocity at time 15 minutes, and C) the speed at 15 minutes.
The acceleration at r=0.004 is -0.00008 centimeters per second squared.