MATH1011 - Lecture 7

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MATH1011 - Lecture 6
Skeleton Slides

Maximum Rate of Change

• We know that the directional derivative is given by $\nabla f(c)\cdot \hat{v}$
  • Another formula for dot product is: $|\nabla f(c)|\cdot |\hat{v}|\cdot \cos \theta$
    • The magnitude of a unit vector is always one
      • i.e. $|\hat{v}|=1$
    • $\theta$ is the angle formed between the two vectors
  • Using this, we can calculate in which direction does $f$ have the maximum or minimum rate of change
• The maximum value of $D_{v}f(c)$ is equal to $|\nabla f(c)|$
  • The minimum value of $D_{v}f(c)$ is equal to $-|\nabla f(c)|$
  • i.e. the max/min value of $D_{v}f(c)$ is when $\hat{v}$ points in the same/opposite direction of the gradient

Gradient is Normal to Level Surface

The gradient curve of $\nabla f(c)$ is perpendicular to the tangent line of any point on a level curve

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Maxima and Minima

Increasing and Decreasing Functions

• If $f^{\prime }(x)>0$ for all $x\in I$, then $f$ is increasing on $I$
• If $f^{\prime }(x)<0$ for all $x\in I$, then $f$ is decreasing on $I$
• If $f^{\prime }(x)>0$ for all $x\in I$, then $f$ is non-decreasing on $I$
• if $f^{\prime }(x)\ne 0$ for all $x\in I$, then $f$ is non-increasing on $I$

This is not true if the domain is not an interval

Critical Points

• Critical points are points where $f^{\prime }(c)=0$ or $f^{\prime }(c)$ does not exist
• Absolute extremum $\Rightarrow$ local extremum $\Rightarrow$ critical point/boundary point

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MATH1011 - Lecture 8
Annotated Slides