In each of examples 1 and 2, use the process of implicit ...
Common Errors!!
• Error 1 - labeling pieces that change over time with a constant value
• Error 2 - labeling pieces that are changing but which are irrelevant to the problem. Variable names are assigned to pieces that are changing if and only if you are given their rate of change or you are explicitly asked to find their rate of change.
• Error 3- assigning positive derivative values to pieces that are decreasing over time
Find the formula for the first derivative of the function [pic] two ways.
1. Use Explicit Differentiation. (i.e. differentiate the explicitly stated function.)
2. Use Implicit Differentiation. (i.e., differentiate the implicit equation [pic])
Find the slope of the curve in Figure 2 at the point [pic].
1) Differentiate both sides of the equation with respect to x.
• Remember to treat y as a function of x and use the chain rule. For example: [pic]
2) Algebraically solve for [pic].
3) Replace x and y with their values.
Figure 2: [pic]
Find the slope of the curve in Figure 3 at the point [pic].
1) Differentiate both sides of the equation with respect to x.
• Remember to treat y as a function of x and use the chain rule. For example: [pic]
2) Algebraically solve for [pic].
3) Replace x and y with their values.
Figure 3: [pic]
Solve each Related Rates problem.
In Figure 4 x, y, and z represent the lengths of the indicated sides measured in inches. The 3 lengths are changing with respect to time, t, measured in seconds. The side labeled y is increasing at the constant rate of 4 in/s and the side labeled x is decreasing at the constant rate of 3 in/s. Find the rate at which the side labeled z is changing at the instant [pic] and [pic]. What is the instantaneous rate of change in z at the instant [pic] and [pic]? In each case make sure that you clearly communicate whether z is increasing or decreasing.
In Figure 5 x represents the length of the indicated side (cm) and [pic] represents the measurement of the indicated angle (rad); both variables are changing with respect to time, t, measured in seconds. The lengths of the other two sides are fixed at 4 cm and 7 cm. [pic] is increasing at a constant rate of 4 deg/min. Find the rate at which x is changing at the instant[pic]. Make sure that you clearly communicate whether x is increasing or decreasing.
In Figure 6 x and y represent the lengths of the indicated sides measured in feet. The 2 lengths are changing with respect to time, t, measured in seconds. The side labeled y is increasing at the constant rate of .2 ft/s. Find the rate at which the side labeled x is changing at the instant[pic]. Make sure that you clearly communicate whether x is increasing or decreasing.
At noon one day a truck is 250 miles due east of a car. The truck is travelling west at a constant speed of 25 mph and the car is travelling due north at a constant speed of 50 mph. At what rate is the distance between the two vehicles changing 15 minutes after noon?
A light house sits on an island of the coast of Maine. The beam source is exactly two miles from the coast along the perpendicular line from the beam source to the coast. The beam makes one complete revolution every 10 seconds. Find the rate at which the beam moves along the coast at the instant the angle between the beam and the perpendicular line to the coast is 50(.
To solve this problem you must assume that the coastline is perfectly straight. You must also assume that the light beam is laser like and at ground level.
A tank filled with water is in the shape of an inverted cone 20 feet high with a circular base (on top) whose radius is 5 feet. Water is running out the bottom of the tank at the constant rate of 2 ft3/min. How fast is the water level falling when the water is 8 feet deep?
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x
y
x
y
x
r
y
Figure 4: find related rates equation
Figure 1: [pic]
4
7
x
[pic]
Figure 5: find related rates equation
Figure 6: find related rates equation
x
y
The total length of these two sides is 20 ft
3 ft
Related Rates Algorithm
The following algorithm - when executed correctly - will lead to success with related rates problems.
1. Draw a picture of the described situation.
2. Label any piece of the picture for which you are given the rate of change with a variable name.
3. Label the piece of the picture for which you are asked to find the rate of change with a variable name.
4. Label any piece that does not change with respect to time and for which you are given the constant value with that constant value.
5. Explicitly define your variables
6. Find the related rates equation without introducing any additional variables.
7. Your related rates equation contains variables and rates. Somewhere in the problem you are given values for each of the variables at a certain point in time and you are given values for each of the rates except the one you are trying to find. Make a list stating each of the values and rates. If you assign a value of zero to any rate that piece of the picture should not have been assigned a variable name. If there are two or more pieces of the picture for which you were not told the rate of change you have labeled a piece of the picture that should not have been labeled! (See footnote).
8. Plug the information from step 7 into your rate equation and solve for the unknown rate.
9. State your conclusion using a complete sentence and proper units.
footnote
Occasionally a “scratch work” picture must be drawn to eliminate an unwanted variable. This occurs when you are using “textbook” formulas like volume of a cone ([pic]). If you are told to find the rate of change in volume given the rate of change in height you will need to use additional information in the problem to eliminate r from the volume equation.
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