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:

      Differentiability
      Chain rule
      General chain rule
      Worked problems

Learning module LM 14.6: Gradients and directional derivatives:

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

In the following video we use the chain rule to work two problems:

A triangle is being squeezed so that its base $b$ is shrinking with $\frac{db}{dt}=-1$ and its height is increasing with $\frac{dh}{dt}=1$ . At what rate is the area of the triangle changing when $b=3$ and $h=5$ ?
Suppose that $w=xy+yz+xz$ and that $x = r\cos(\theta)$ , $y=r\sin(\theta)$ and $z=r\theta$ . Compute $\frac{\partial w}{\partial r}$ and $\frac{\partial w}{\partial \theta}$ when $r=2$ and $\theta=\pi/2$ .