M408M Learning Module Pages

Main page

Chapter 10: Parametric Equations and Polar Coordinates

Chapter 12: Vectors and the Geometry of Space

Chapter 13: Vector Functions

Chapter 14: Partial Derivatives

Learning module LM 14.1: Functions of 2 or 3 variables:

Learning module LM 14.3: Partial derivatives:

Learning module LM 14.4: Tangent planes and linear approximations:

Learning module LM 14.5: Differentiability and the chain rule:

Learning module LM 14.6: Gradients and directional derivatives:

      Gradients
      Gradients and hill climbing
      Wind and weather
      Directional derivatives
      Worked problems

Learning module LM 14.7: Local maxima and minima:

Learning module LM 14.8: Absolute maxima and Lagrange multipliers:

Chapter 15: Multiple Integrals

Worked problems

Worked problems

We work the following example problems:
1. Let $f(x,y)=2x^2 + y^3$ .
    a) If we are at $(3,2)$ and are moving with unit speed in the $\langle 3,4 \rangle$ direction, how fast is $f(x,y)$ changing?
    b) What directions should we go to increase the fastest?
    c) What direction should we go to not change at all?
    d) What is the slope of a tangent line to the level curve $2x^2 + y^3 = 26$ at $(3,2)$ ?

2. Find the equation of the plane tangent to $\displaystyle{z=e^{xy^2-1}}$ at $(1,1,1)$ .