7.2 – Exponential Change and Separable Differential Equations © 2010 Pearson Education, Inc. All rights reserved Separable Differential Equations.

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7.2 – Exponential Change and Separable Differential Equations © 2010 Pearson Education, Inc. All rights reserved Separable Differential Equations

7.2 – Exponential Change and Separable Differential Equations © 2010 Pearson Education, Inc. All rights reserved Separable Differential Equations

© 2010 Pearson Education, Inc. All rights reserved 7.2 – Exponential Change and Separable Differential Equations Uninhibited Exponential Growth Uninhibited Exponential Decay

© 2010 Pearson Education, Inc. All rights reserved 7.2 – Exponential Change and Separable Differential Equations Example The population of the United States was approximately 227 million in 1980 and 282 million in Estimate the population in the year Find k2010

© 2010 Pearson Education, Inc. All rights reserved 7.2 – Exponential Change and Separable Differential Equations Example A radioactive material has a half-life of 700 years. If there were ten grams initially, how much would remain after 300 years? When will the material weigh 7.5 grams? Find k or 300 years or 7.5 grams

© 2010 Pearson Education, Inc. All rights reserved 7.2 – Exponential Change and Separable Differential Equations Newton’s Law of Cooling

© 2010 Pearson Education, Inc. All rights reserved 7.2 – Exponential Change and Separable Differential Equations Newton’s Law of Cooling Example A pizza pan is removed at 3:00 PM from an oven whose temperature is fixed at 450  F into a room that is a constant 70  F. After 5 minutes, the pizza pan is at 300  F. How long will it take for the pan to cool to 135  F? Find k 135  F

© 2010 Pearson Education, Inc. All rights reserved 7.3 – Hyperbolic Functions Hyperbolic functions have similar properties to the trigonometric functions. Uses: Catenary cables Specific and general relativity Magnetic polarizationVoltage input/output for transistors and amplifiers

Hyperbolic Functions © 2010 Pearson Education, Inc. All rights reserved 7.3 – Hyperbolic Functions

Hyperbolic Functions © 2010 Pearson Education, Inc. All rights reserved 7.3 – Hyperbolic Functions

Hyperbolic Functions © 2010 Pearson Education, Inc. All rights reserved 7.3 – Hyperbolic Functions

Inverse Hyperbolic Functions © 2010 Pearson Education, Inc. All rights reserved 7.3 – Hyperbolic Functions

Inverse Hyperbolic Functions © 2010 Pearson Education, Inc. All rights reserved 7.3 – Hyperbolic Functions

Inverse Hyperbolic Functions © 2010 Pearson Education, Inc. All rights reserved 7.3 – Hyperbolic Functions

Inverse Hyperbolic Functions © 2010 Pearson Education, Inc. All rights reserved 7.3 – Hyperbolic Functions

Inverse Hyperbolic Functions © 2010 Pearson Education, Inc. All rights reserved 7.3 – Hyperbolic Functions