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/Optimiza
tion.aspx
– http://www.qcalculus.com/cal08.htm
• Differentials
– http://tutorial.math.lamar.edu/Classes/CalcI/Differenti
als.aspx
– http://www.cliffsnotes.com/study_guide/Differentials.
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