1. EGR 1101: Unit 9 Lecture #1 Applications of Derivatives: Electric Circuits (Section 8.4 of Rattan/Klingbeil text)

2. Review: Some Derivative Rules  where a, c, n, and  are constants.

3. Two New Derivative Rules  (Product rule) If f(g) is a function of g and g(t) is a function of t,If f(g) is a function of g and g(t) is a function of t, (Chain rule) (Chain rule)

4. Today’s Examples 1. Voltage, current, & power 2. Current & voltage in an inductor 3. Current & voltage in an inductor (graphical and working backwards) 4. Current & voltage in a capacitor 5. Current & voltage in a capacitor (graphical and working backwards)

5. Review: Some Electrical Quantities QuantitySymbolUnit Symbol for Unit Voltage V or v(t) voltV Current I or i(t) ampereA Charge Q or q(t) coulombC Energy W or w(t) jouleJ Power P or p(t) wattW Timetseconds

6. Review: More Electrical Quantities QuantitySymbolUnit Symbol for Unit ResistanceRohm InductanceLhenryH CapacitanceCfaradF

7. Voltage-versus-Current Relations  For resistors, For inductors,For inductors, For capacitors,For capacitors,

8. EGR 1101: Unit 9 Lecture #2 Applications of Derivatives: Beams (Section 8.5 of Rattan/Klingbeil text)

9. Some Beam Terminology  Types of beams Simply supported Cantilever  Types of load on a beam Concentrated Distributed

10. More Beam Terminology  In addition to the type of beam and load, a beam’s behavior also depends on its geometry and the material it is made of.  Its geometry is summarized in a quantity called the second moment of area ( I).  Its material is summarized in a quantity called the modulus of elasticity (E).  The product of these two (E I ) is called the flexural rigidity.

11. Some Beam Quantities QuantitySymbolUnit Modulus of elasticity E lb/in 2 or N/m 2 Second moment of area I in 4 or m 4 Flexural rigidity EI lbin 2 or Nm 2 Deflectiony(x) in or m Slope (x) radians

12. Excellent Online Resource  University of Wisconsin’s online lessons on Strength of Materials: http://www3.uwstout.edu/faculty/ scotta/upload/Foley- StaticsStrengths.pdf http://www3.uwstout.edu/faculty/ scotta/upload/Foley- StaticsStrengths.pdf  See especially Topic 4 (Beams) and Topic 8.2 (Stress on Incline Planes).

13. Review  Given a function f(x), the function’s local maxima occur at values of x where and  Its local minima occur at values of x where and

14. Today’s Examples 1. Deflection of a cantilever beam with end load 2. Deflection of a simply supported beam with central load 3. Deflection of a simply supported beam with distributed load 4. Maximum stress under axial loading