Todays Agenda Upcoming Homework Section 2.6: Implicit - PowerPoint PPT Presentation

Today’s Agenda • Upcoming Homework • Section 2.6: Implicit Differentiation and Section 2.7: Related Rates • Return Tests Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Monday, 28 September 2015 1 / 7

Upcoming Homework • Written HW E (Sections 2.5 and 2.6), due 9/30 • WeBWorK HW #10 (Section 2.7), due 10/2 • WeBWorK HW #11 (Section 2.8), due 10/5 Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Monday, 28 September 2015 2 / 7

Section 2.6 Example 2.6.2 1 The van der Waals equation for n moles of gas is P + n 2 a � � ( V − nb ) = nRT , V 2 where P is the pressure, V is the volume, and T is the temperature of the gas. The constant R is the universal gas constant and a and b are positive constants that are characteristic of a particular gas. If T remains constant, use implicit differentiation to find dV / dP . 2 Find the rate of change of volume with respect to pressure of 1 mole of carbon dioxide at a volume of V = 10 L and a pressure of P = 2 . 5 atm. Use a = 3 . 592 L 2 -atm/mole 2 and b = 0 . 04267 L/mole. Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Monday, 28 September 2015 3 / 7

Section 2.6 Example 2.6.3 Where does the normal line to the ellipse x 2 − xy + y 2 = 3 at the point ( − 1 , 1) intersect the ellipse a second time? Example 2.6.4 Find y ′′ by implicit differentiation: 1 9 x 2 + y 2 = 9 2 x 3 + y 3 = 1 Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Monday, 28 September 2015 4 / 7

Section 2.7 First, a warning: Related Rates problems can be very lengthy. I strongly recommend that you at least take a look at some of the WeBWorK questions for Section 2.7 before Wednesday so that you can come to office hours for help if needed. Example 2.7.1 A water tank has the shape of an inverted circular cone with base radius 2 m and height 4 m. If water is being pumped into the tank at a rate of 2 m 3 /min, find the rate at which the water level is rising when the water is 3 m deep. Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Monday, 28 September 2015 5 / 7

Section 2.7 Your textbook suggests the following steps for solving Related Rates problems (page 131): 1 Read the problem carefully. 2 Draw a diagram if possible. 3 Introduce notation. Assign symbols to all quantities that are functions of time. 4 Express the given information and the required rate in terms of derivatives. 5 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. 6 Use the Chain Rule to differentiate both sides of the equation with respect to t . 7 Substitute the given information into the resulting equation and solve for the unknown rate. Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Monday, 28 September 2015 6 / 7

Section 2.7 Example 2.7.2 A ladder 10 ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1 ft/s, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 ft from the wall? Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Monday, 28 September 2015 7 / 7