MATH 251 Lecture 9

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Differential of a Function

(Section 12.4)

The derivative of a function at a point $p$ is the best linear approximation of 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 f(\vec{x}) - f(p)}

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} \mathrm{d}f &= \frac{\partial f}{\partial x} \,\mathrm{d}x + \frac{\partial f}{\partial y} \,\mathrm{d}y \\ &= f_x \, \mathrm{d}x + f_y \, \mathrm{d}y \\ &= \vec\nabla f \cdot \left\langle \mathrm{d}x, \mathrm{d}y \right\rangle\\ &= \vec\nabla f \cdot \mathrm{d}\vec{s} \end{align}}

Example

f(x) is a single variable function.

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 f(x+h) = f(x) + f'(x)\cdot h + O(h^2)} , where 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 f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}}

The best approximation for 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 f(x+h)-f(x)} is 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 f'(x) \cdot h}

Multivariable

For 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 p = \left( x_0,\, y_0 \right)}

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\left( x_0+h, y_0+k \right) - f\left( x_0, y_0 \right) &\approx \frac{\partial f}{\partial x} \left( x_0, y_0 \right) \cdot h + \frac{\partial f}{\partial y} \left( x_0, y_0 \right) \cdot k \\ & \approx f_x\left(x_0, y_0\right) \, h + f_y(x_0, y_0) \, k \end{align}}

Example

Estimate 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 f(1.1, 1.2)} , where 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 f(x,y) = \mathrm{e}^{xy}\sin{\left(x^2-y^2\right)}}

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 \mathrm{d}f = \begin{cases} f_x = y\mathrm{e}^{xy}\sin{\left(x^2-y^2\right)} + 2x\mathrm{e}^{xy}\cos{\left(x^2-y^2\right)} \\ f_y = x\mathrm{e}^{xy}\sin{\left(x^2-y^2\right)} - 2y\mathrm{e}^{xy}\cos{\left(x^2-y^2\right)} \end{cases}}

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(1,1) &= 0 \\ f(1.1, 1.2) &\approx -0.2 \mathrm{e} \\ \mathrm{d}x = .1, \mathrm{d}y = .2 \\ \mathrm{d}f &= f(1,1) + f(1.1, 1.2) \approx -.2\mathrm{e} \end{align}}

Tangent Plane

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 z - z_0 = f_x(x_0, y_0) \cdot (x-x_0) + f_y(x_0, y_0) \cdot(y-y_0)}