Related Rates in Calculus
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Related rate problems are differentiated with respect to time. This
Related rate problems are differentiated with respect to time. This
means every variable, except
means every variable, except
t
t
is a differentiated implicitly.
is a differentiated implicitly.
Related Rate Problems
Related Rate Problems
Frequently Used Formulas
Frequently Used Formulas
Ex.) A pebble is dropped into a calm pond, causing ripples in the form
Ex.) A pebble is dropped into a calm pond, causing ripples in the form
of concentric circles. The radius
of concentric circles. The radius
r
r
of the outer ripple is increasing at a
of the outer ripple is increasing at a
constant rate of 1 foot per second. When the radius is 4ft., what rate is
constant rate of 1 foot per second. When the radius is 4ft., what rate is
the total area
the total area
A
A
of the disturbed water increasing?
of the disturbed water increasing?
The Velocity of an Airplane Tracked by Radar
The Velocity of an Airplane Tracked by Radar
An airplane is flying at an elevation of 6 miles on a flight path that will take it
An airplane is flying at an elevation of 6 miles on a flight path that will take it
directly over a radar tracking station. Let
directly over a radar tracking station. Let
s
s
represent the distance (in miles)
represent the distance (in miles)
between the radar station and the plane. If
between the radar station and the plane. If
s
s
is decreasing at a rate of 400 miles
is decreasing at a rate of 400 miles
per hour when
per hour when
s
s
is 10 miles, what is the velocity of the plane?
is 10 miles, what is the velocity of the plane?
Ex.) A conical tank (with vertex down) is 10 ft. across the top and 12
Ex.) A conical tank (with vertex down) is 10 ft. across the top and 12
feet deep.
feet deep.
If water is flowing into the tank at a rate of 10 cubic feet per
If water is flowing into the tank at a rate of 10 cubic feet per
minute, find the rate of change of the depth of the water when the
minute, find the rate of change of the depth of the water when the
water is 8 feet deep.
water is 8 feet deep.
Related rates problems in calculus involve finding the rate of change of one quantity with respect to another, utilizing concepts of implicit differentiation, chain rule, and relevant formulas. This includes scenarios such as ripples in a pond, inflating balloons, tracking airplanes, and filling tanks, where rates of change are interconnected. By assigning symbols, writing equations, differentiating, and solving, one can determine the desired rates of change.
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Presentation Transcript
Chapter 2.6: Chapter 2.6: Related Rates Related Rates HONORS CALCULUS/CALCULUS HONORS CALCULUS/CALCULUS
Related rate problems are differentiated with respect to time. This means every variable, except t is a differentiated implicitly. Ex.) Given ? = ??+ ?, find ?? ?? when ? = ?, given that ?? ??= ?.
Related Rate Problems 1. Assign symbols to all given quantities and quantities to be determined. Make a sketch and label the quantities if possible. 2. Write an equation involving the variables whose rates of change are give or are to be determined. 3. Using the Chain Rule, implicitly differentiate both sides of the equation with respect to ?. 4. Substitute into the resulting equation all known values for the variables and their rates of change. 5. Solve for the required rate of change.
Frequently Used Formulas Pythagorean Theorem: ??+ ??= ?? Volume of a Cone: ? =? ????? Area of a Circle: ? = ??? Volume of a Sphere: ? =? ????
Ex.) 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 4ft., what rate is the total area A of the disturbed water increasing?
Ex.) Air if being pumped into a spherical balloon at the rate of ?.? ??? ??? ??????. Find the rate of change of the radius when the radius is 2 inches.
The Velocity of an Airplane Tracked by Radar An airplane is flying at an elevation of 6 miles on a flight path that will take it directly over a radar tracking station. Let s represent the distance (in miles) between the radar station and the plane. If s is decreasing at a rate of 400 miles per hour when s is 10 miles, what is the velocity of the plane?
Ex.) 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 feet per minute, find the rate of change of the depth of the water when the water is 8 feet deep.
?? ?????? ?? ??= ?? ?? ??=? ? = ? = ?
