Unit 5 Summary

1D Optimization

I can …

  • Find the local maxima, local minima, global maximum and global minimum for the graph of a function.
  • Find the critical points of \(f(x)\) by solving \(f'(x)=0\).
  • Use the first derivative test to determine whether a critical point is a local minimum, a local maximum or a point of inflection.
  • Use the second derivative test to determine whether a critical point is a local minimum or a local maximum.

2D Optimization

I can …

  • Find the 2D critical points of \(f(x,y)\) by (simultaneously) solving \(f_x(x,y)=0\) and \(f_y(x,y)=0\).
  • Characterize a 2D critical point \((a,b)\) as a local minimimum, local maximum or saddle point by:
    • Looking at the values of \(f(a,b)-f(x,y)\) on a small circle around \((a,b)\).
    • Making a contour plot near \((a,b)\)
    • Using the 2D second derivative test at \((a,b)\)

Gradient Search

I can …

  • Find a local maximum of a 2D function \(f(x,y)\) using gradient search in RStudio.
  • Find a local minimum of a 2D function \(f(x,y)\) using gradient search in RStudio.
  • Simulate gradient search using the contour plot of a 2D function to find the trajectory that gradient search follows to reach an extreme point.
  • Explain why the extreme point found by gradient search depends upon the starting point that we use.

Constrained Optimization

I can …

  • Identify the objective function and the constaint function for a constrained optimization problem.
  • Explain why the extreme point for a constrained optimization problem occurs where the constraint contour is tangent to a contour of the objective function.
  • Use RStudio to create a contour plot to solve a constrained optimization problem.
  • Estimate the value of the Lagrange multiplier \(\lambda\) using a contour plot.
  • Interpret the Lagrange multiplier \(\lambda\) as a rate of change.