Related Rate of Change - Lecture Notes PDF

Calculus I Lecture Notes-May 2018 Dr. Michael Munywoki Technical University of Mombasa Department of Mathemaics and Physics Copyright c 2020 Michael Munywoki. All rights reserved June 2, 2020  Outline for section 1 1 Applications o...

Calculus I Lecture Notes-May 2018 Dr. Michael Munywoki Technical University of Mombasa Department of Mathemaics and Physics Copyright c 2020 Michael Munywoki. All rights reserved June 2, 2020  Outline for section 1 1 Applications of Derivatives Related rate of change 2 Bibliography Dr. Michael Munywoki (Technical University Calculusof IMombasa) Lecture Notes-May 2018 June 2, 2020 2 / 7  Related rate of change dy dy dt The identity dx = dt × dx is useful in solving certain rate of change problems. Dr. Michael Munywoki (Technical University Calculusof IMombasa) Lecture Notes-May 2018 June 2, 2020 3 / 7  Related rate of change dy dy dt The identity dx = dt × dx is useful in solving certain rate of change problems. dy = Rate of change of y w.r.t t. dt Dr. Michael Munywoki (Technical University Calculusof IMombasa) Lecture Notes-May 2018 June 2, 2020 3 / 7  Related rate of change dy dy dt The identity dx = dt × dx is useful in solving certain rate of change problems. dy = Rate of change of y w.r.t t. dt dx = Rate of change of x w.r.t t. dt Dr. Michael Munywoki (Technical University Calculusof IMombasa) Lecture Notes-May 2018 June 2, 2020 3 / 7  Related rate of change dy dy dt The identity dx = dt × dx is useful in solving certain rate of change problems. dy = Rate of change of y w.r.t t. dt dx = Rate of change of x w.r.t t. dt Example 1.1 Dr. Michael Munywoki (Technical University Calculusof IMombasa) Lecture Notes-May 2018 June 2, 2020 3 / 7  Related rate of change dy dy dt The identity dx = dt × dx is useful in solving certain rate of change problems. dy = Rate of change of y w.r.t t. dt dx = Rate of change of x w.r.t t. dt Example 1.1 How fast does the radius of a spherical soap bubble change when air is blown into it at the rate of 10 cm3 /sec Dr. Michael Munywoki (Technical University Calculusof IMombasa) Lecture Notes-May 2018 June 2, 2020 3 / 7  Related rate of change dy dy dt The identity dx = dt × dx is useful in solving certain rate of change problems. dy = Rate of change of y w.r.t t. dt dx = Rate of change of x w.r.t t. dt Example 1.1 How fast does the radius of a spherical soap bubble change when air is blown into it at the rate of 10 cm3 /sec Solution: Dr. Michael Munywoki (Technical University Calculusof IMombasa) Lecture Notes-May 2018 June 2, 2020 3 / 7  Related rate of change dy dy dt The identity dx = dt × dx is useful in solving certain rate of change problems. dy = Rate of change of y w.r.t t. dt dx = Rate of change of x w.r.t t. dt Example 1.1 How fast does the radius of a spherical soap bubble change when air is blown into it at the rate of 10 cm3 /sec Solution: We need to find drdt. Dr. Michael Munywoki (Technical University Calculusof IMombasa) Lecture Notes-May 2018 June 2, 2020 3 / 7  Related rate of change dy dy dt The identity dx = dt × dx is useful in solving certain rate of change problems. dy = Rate of change of y w.r.t t. dt dx = Rate of change of x w.r.t t. dt Example 1.1 How fast does the radius of a spherical soap bubble change when air is blown into it at the rate of 10 cm3 /sec Solution: We need to find drdt. Dr. Michael Munywoki (Technical University Calculusof IMombasa) Lecture Notes-May 2018 June 2, 2020 3 / 7  Related rate of change dy dy dt The identity dx = dt × dx is useful in solving certain rate of change problems. dy = Rate of change of y w.r.t t. dt dx = Rate of change of x w.r.t t. dt Example 1.1 How fast does the radius of a spherical soap bubble change when air is blown into it at the rate of 10 cm3 /sec Solution: We need to find drdt. 4 3 V = 3 πr ; Dr. Michael Munywoki (Technical University Calculusof IMombasa) Lecture Notes-May 2018 June 2, 2020 3 / 7  Related rate of change dy dy dt The identity dx = dt × dx is useful in solving certain rate of change problems. dy = Rate of change of y w.r.t t. dt dx = Rate of change of x w.r.t t. dt Example 1.1 How fast does the radius of a spherical soap bubble change when air is blown into it at the rate of 10 cm3 /sec Solution: We need to find drdt. V = 3 πr ; dt = 4πr2 dr 4 3 dv dt ; Dr. Michael Munywoki (Technical University Calculusof IMombasa) Lecture Notes-May 2018 June 2, 2020 3 / 7  Related rate of change dy dy dt The identity dx = dt × dx is useful in solving certain rate of change problems. dy = Rate of change of y w.r.t t. dt dx = Rate of change of x w.r.t t. dt Example 1.1 How fast does the radius of a spherical soap bubble change when air is blown into it at the rate of 10 cm3 /sec Solution: We need to find drdt. V = 3 πr ; dt = 4πr2 dr 4 3 dv dt ; dv dt = 10 Dr. Michael Munywoki (Technical University Calculusof IMombasa) Lecture Notes-May 2018 June 2, 2020 3 / 7  Related rate of change dy dy dt The identity dx = dt × dx is useful in solving certain rate of change problems. dy = Rate of change of y w.r.t t. dt dx = Rate of change of x w.r.t t. dt Example 1.1 How fast does the radius of a spherical soap bubble change when air is blown into it at the rate of 10 cm3 /sec Solution: We need to find drdt. V = 3 πr ; dt = 4πr2 dr 4 3 dv dt ; dv dr 10 dt = 10 ⇒ dt = 4πr2 Dr. Michael Munywoki (Technical University Calculusof IMombasa) Lecture Notes-May 2018 June 2, 2020 3 / 7  Related rate of change (Cont...) Example 1.2 Dr. Michael Munywoki (Technical University Calculusof IMombasa) Lecture Notes-May 2018 June 2, 2020 4 / 7  Related rate of change (Cont...) Example 1.2 How fast does the water level drop when a cylindrical tank is drained at the rate of 3 litres/sec? Dr. Michael Munywoki (Technical University Calculusof IMombasa) Lecture Notes-May 2018 June 2, 2020 4 / 7  Related rate of change (Cont...) Example 1.2 How fast does the water level drop when a cylindrical tank is drained at the rate of 3 litres/sec? Solution: Dr. Michael Munywoki (Technical University Calculusof IMombasa) Lecture Notes-May 2018 June 2, 2020 4 / 7  Related rate of change (Cont...) Example 1.2 How fast does the water level drop when a cylindrical tank is drained at the rate of 3 litres/sec? Solution: The radius is constant but V and h change with time (since the water level is dropping). Dr. Michael Munywoki (Technical University Calculusof IMombasa) Lecture Notes-May 2018 June 2, 2020 4 / 7  Related rate of change (Cont...) Example 1.2 How fast does the water level drop when a cylindrical tank is drained at the rate of 3 litres/sec? Solution: The radius is constant but V and h change with time (since the water level is dropping). So V and h are differentiable functions of time (t). We need to find dhdt. V = πr2 h Dr. Michael Munywoki (Technical University Calculusof IMombasa) Lecture Notes-May 2018 June 2, 2020 4 / 7  Related rate of change (Cont...) Example 1.2 How fast does the water level drop when a cylindrical tank is drained at the rate of 3 litres/sec? Solution: The radius is constant but V and h change with time (since the water level is dropping). So V and h are differentiable functions of time (t). We need to find dhdt. V = πr2 h dv dh = πr2 dt dt Dr. Michael Munywoki (Technical University Calculusof IMombasa) Lecture Notes-May 2018 June 2, 2020 4 / 7  Related rate of change (Cont...) Example 1.2 How fast does the water level drop when a cylindrical tank is drained at the rate of 3 litres/sec? Solution: The radius is constant but V and h change with time (since the water level is dropping). So V and h are differentiable functions of time (t). We need to find dhdt. V = πr2 h dv dh = πr2 dt dt dv = −3 dt Dr. Michael Munywoki (Technical University Calculusof IMombasa) Lecture Notes-May 2018 June 2, 2020 4 / 7  Related rate of change (Cont...) Example 1.2 How fast does the water level drop when a cylindrical tank is drained at the rate of 3 litres/sec? Solution: The radius is constant but V and h change with time (since the water level is dropping). So V and h are differentiable functions of time (t). We need to find dhdt. V = πr2 h dv dh = πr2 dt dt dv = −3 dt dh −3 ⇒ = 2 dt πr Dr. Michael Munywoki (Technical University Calculusof IMombasa) Lecture Notes-May 2018 June 2, 2020 4 / 7  Related rate of change (Cont...) Exercise 1.3 Dr. Michael Munywoki (Technical University Calculusof IMombasa) Lecture Notes-May 2018 June 2, 2020 5 / 7  Related rate of change (Cont...) Exercise 1.3 A ladder 20f t long leans against a vertical wall. The bottom of the ladder slides away from the wall at the rate of 2f t/sec. How fast is the ladder sliding down when the top of the ladder is 12f t above the ground. Dr. Michael Munywoki (Technical University Calculusof IMombasa) Lecture Notes-May 2018 June 2, 2020 5 / 7  Related rate of change (Cont...) Exercise 1.3 A ladder 20f t long leans against a vertical wall. The bottom of the ladder slides away from the wall at the rate of 2f t/sec. How fast is the ladder sliding down when the top of the ladder is 12f t above the ground. Solution: Dr. Michael Munywoki (Technical University Calculusof IMombasa) Lecture Notes-May 2018 June 2, 2020 5 / 7  Related rate of change (Cont...) Exercise 1.3 A ladder 20f t long leans against a vertical wall. The bottom of the ladder slides away from the wall at the rate of 2f t/sec. How fast is the ladder sliding down when the top of the ladder is 12f t above the ground. Solution: 20 y ft x Dr. Michael Munywoki (Technical University Calculusof IMombasa) Lecture Notes-May 2018 June 2, 2020 5 / 7  Related rate of change (Cont...) Exercise 1.3 A ladder 20f t long leans against a vertical wall. The bottom of the ladder slides away from the wall at the rate of 2f t/sec. How fast is the ladder sliding down when the top of the ladder is 12f t above the ground. Solution: x2 + y 2 = 400 dx dy 2x + 2y = 0 20 dt dt y ft dy x dx = − dt y dt dx = 2 dt x Dr. Michael Munywoki (Technical University Calculusof IMombasa) Lecture Notes-May 2018 June 2, 2020 5 / 7  Related rate of change (Cont...) Exercise 1.3 A ladder 20f t long leans against a vertical wall. The bottom of the ladder slides away from the wall at the rate of 2f t/sec. How fast is the ladder sliding down when the top of the ladder is 12f t above the ground. Solution: When y = 12, p 2 2 x + y = 400 x = 202 − 122 √ dx dy = 400 − 144 2x + 2y = 0 √ 20 dt dt = 256 = 16 y ft dy x dx And = − dt y dt dy 16 dx = − ·2 = 2 dt 12 dt 8 = − f t/sec 3 x Dr. Michael Munywoki (Technical University Calculusof IMombasa) Lecture Notes-May 2018 June 2, 2020 5 / 7  Outline for section 2 1 Applications of Derivatives Related rate of change 2 Bibliography Dr. Michael Munywoki (Technical University Calculusof IMombasa) Lecture Notes-May 2018 June 2, 2020 6 / 7  Bibliography I J Stewart, Calculus, (Third Edition), CENGAGE Learning, Belmont, CA, 2012. D. F. Wright, S. P. Hurd and B. D. New, Essential Calculus, (Second Edition), Hawkes learning System, Charleston, SC, 2008. M. Sullivan, Algebra & Trigonometry, (Eighth Edition), Pearson Prentice Hall, upper Saddle River, New Jersey, NJ, 2008. D. Verberg, E. J. Purcell and S. E. Rigdon, Calculus, (Ninth Edition), Pearson Prentice Hall, upper Saddle River, New Jersey, NJ, 2007. Dr. Michael Munywoki (Technical University Calculusof IMombasa) Lecture Notes-May 2018 June 2, 2020 7 / 7