Related Rates - PDF

MODULE 9: RELATED RATES (TIME RATES) In a related rates problem, the idea is to compute the rate of change of one quantity in terms of the rate of change of another quantity (which may be more easily measured). The procedure is to find an equation that relates the two quantities and then differe...

MODULE 9: RELATED RATES (TIME RATES) In a related rates problem, the idea is to compute the rate of change of one quantity in terms of the rate of change of another quantity (which may be more easily measured). The procedure is to find an equation that relates the two quantities and then differentiate implicitly (or using Chain Rule) both sides with respect to time. Suggested steps in solving relate rates problems: 1. Read the problem carefully. Draw a diagram if possible. 2. Introduce notation. Assign symbols to all quantities that are functions of time. 3. Express the given information and the required rate in terms of derivatives. 4. Write an equation that relates the various quantities of the problem. If necessary, use the geometry of the situation to eliminate one of the variables by substitution. DO NOT substitute the values that changes as time passes until after the differentiation (step 5). 5. Use implicit differentiation (or Chain Rule) to differentiate both sides of the equation with respect to time 𝑡. 6. Substitute now the given values into the resulting equation and solve for the unknown rate. EXERCISES 9.1 𝑑𝑉 𝑑𝑥 1. If 𝑉 is the volume of a cube with edge length 𝑥 and the cube expands as time passes, find in terms of. 𝑑𝑡 𝑑𝑡 𝑑𝐴 𝑑𝑟 2. If 𝐴 is the area of a circle with radius 𝑟 and the circle expands as time passes, find in terms of. 𝑑𝑡 𝑑𝑡 3. Suppose oil spills from a ruptured tanker and spreads in a circular pattern. If the radius of the oil spill increases at a constant rate of 1 m/s , how fast is the area of the spill increasing when the radius is 30 m? 𝑑𝑥 𝑑𝑦 4. If 𝑦 = 𝑥 3 + 2𝑥 and = 5, find when 𝑥 = 2. 𝑑𝑡 𝑑𝑡 𝑑𝑦 𝑑𝑥 5. If 𝑥 2 + 𝑦 2 = 25 and = 6, find when 𝑦 = 4 𝑑𝑡 𝑑𝑡 ANSWERS: 𝑑𝑉 𝑑𝑥 1. = 3𝑥 2 𝑑𝑥 𝑑𝑡 𝑑𝐴 𝑑𝑟 2. = 2𝜋𝑟 𝑑𝑡 𝑑𝑡 3. 60𝜋 m2 /s 4. 70 5. ∓8