Optimization and Differentials(Sections 3.7 and 3.9)

By Brandan Jeter, Zach Young, and Daniel Cotton

Optimization

• Optimization is the process of using calculus to determine the maximum or minimum value of a given function.

• For example, you’re given the surface area of a cube, and you want to find the maximum volume of that cube.

Optimization Guidelines

• 1. Identify all given quantities and quantities to be determined.

• 2. Write a primary equation for the quantity that is to be maximized or minimized.

• 3. Reduce the primary equation to one having a single independent variable. This may involve using secondary equations.

• 4. Determine the desired maximum or minimum by finding the derivative.

Differentials

• dy=f’(x)dx• d(cu)=c du• d(u±v)=du±dv• D(uv)=udv+vdu• D(u/v)=(vdu-udv)/v2

Error Propagation

• For example, the radius of a ball is .7 inches with an error within .01 inches. Estimate the propagated error of the volume of the ball.

• V=(4/3) r3

• dV=4 r2dr• =4 (0.7)(±0.01)• =±0.0616 cubic inches

Additional Resources

• Optimization– http://tutorial.math.lamar.edu/Classes/CalcI/Opt

imization.aspx– http://www.qcalculus.com/cal08.htm

• Differentials– http://tutorial.math.lamar.edu/Classes/CalcI/Diff

erentials.aspx– http://www.cliffsnotes.com/study_guide/Differe

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