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 + n2a

V 2

• (V − nb) = nRT,

where P is the pressure, V is the volume, and T is the temperature

• f 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

• f carbon dioxide at a volume of V = 10 L and a pressure of

P = 2.5 atm. Use a = 3.592 L2-atm/mole2 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 x2 − xy + y2 = 3 at the point (−1, 1) intersect the ellipse a second time?

Example 2.6.4

Find y′′ by implicit differentiation:

1 9x2 + y2 = 9 2 x3 + y3 = 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 m3/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