MATH 251 Lecture 11

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Test 1 in 1 week

Multidimensional Limits

We only have a way to show that limits do not exist.

"Functions are continuous except when they are not continuous."

Chain Rule

Recall from Calc 2 that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\mathrm{d}}{\mathrm{d}x} \left( f \left( g \left( x \right) \right) \right) = f' \left( g \left( x \right) \right) \, g' \left( x \right)}

The multivariable version:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \frac{\mathrm{d}}{\mathrm{d}t} \left( f \left( x \left( t \right),y \left( t \right) \right) \right) &= f_{x} \left( x \left( t \right) ,\, y \left( t \right) \right) \, x' \left( t \right) + f_{y} \left( x \left( t \right) ,\, y \left( t \right) \right) \, y' \left( t \right) \\ &=\frac{\partial f}{\partial x} \, \frac{\mathrm{d} x}{\mathrm{d} t} + \frac{\partial f}{\partial y} \, \frac{\mathrm{d} y}{\mathrm{d} t} \end{align}}

Example

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} f(x,y) &= e^x \cos{y} \\ x(t) &= t \\ y(t) &= -2t^2 \end{align}}

Calculate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\mathrm{d}f}{\mathrm{d}t}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \frac{\mathrm{d}f}{\mathrm{d}t} &= f_x \cdot 1 + f_y \cdot \left( -4t \right) \\ f_x &= e^x \cos{y} \\ f_y &= -e^x \sin{y} \\ \frac{\mathrm{d}f}{\mathrm{d}t} &= e^t \left[ \cos{2t^2} - 4t\sin{2t^2} \right] \end{align}}

Evil Example

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} f &= f(x, y, t) \\ x &= x(y,t) \\ y &= y(t) \\ \therefore & f(x(y(t), t), y(t), t) \end{align}}

Create a Tree: f |-- x | |-- y | | `--t | `-- t |-- y | `-- t `-- t

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\mathrm{d}f}{\mathrm{d}t} = \frac{\partial f}{\partial x} \, \left( \frac{\partial x}{\partial y} \, y' + \frac{\partial x}{\partial t} \right) + \frac{\partial f}{\partial y} \, y' + \frac{\partial f}{\partial t}}