Degree

Taylor Polynomial

1

$f\left(a\+h\,b\+k\right)\=f\left(a\,b\right)\+f_x\left(a\,b\right)h\+f_y\left(a\,b\right)k$

2

$f\left(a\+h\,b\+k\right)\=f\left(a\,b\right)\+f_x\left(a\,b\right)h\+f_y\left(a\,b\right)k \+\frac{1}{2}\left(f_{\mathrm{xx}}\left(a\,b\right)h^2\+2 f_{\mathrm{xy}}\left(a\,b\right)h k\+f_{\mathrm{yy}}\left(a\,b\right)k^2\right)$ 

Table 4.7.1   Taylor polynomials of degrees one and two

Example 4.7.1

If $x$ changes from 3 to 3.04 and $y$ changes from $-2$ to $-2.2$, compare $\mathrm{\Δ}f$ and $\mathrm{df}$ for the function $f\left(x\,y\right)\=3 x^2\+4 x y-5 y^2\+7$.

Example 4.7.2

Approximate $\sqrt{4{\left(3.03\right)}^3\+5{\left(6.2\right)}^2}$ by using the total differential for some appropriate function $f\left(x\,y\right)$.

Example 4.7.3

Approximate $\frac{1}{2.03}\+\frac{1}{3.02}\+\frac{1}{4.97}$ by using the total differential for some appropriate function $f\left(x\,y\,z\right)$.

Example 4.7.4

Use the total differential to approximate the value of $f\left(x\,y\right)\=x^2{e}^{2 {\mathrm{xy}}^2}$ at $\left(x\,y\right)\=\left(1.03\,0.96\right)$.

Example 4.7.5

Use the total differential to approximate the value of $f\left(x\,y\,z\right)\=x \ln \left(\frac{y\+ z}{x^2\+y z}\right)$ at $\left(x\,y\,z\right)\=\left(2.03\,3.12\,1.48\right)$.

Example 4.7.6

Use differentials to estimate the maximum error in determining the area of a rectangle measured (in cm) to be $4\times 5$, if each measurement is accurate to .1 cm.

Example 4.7.7

Use differentials to estimate the maximum error in determining the surface area of a closed rectangular box measured (in inches) to be $15\times 6\times 21$, if each measurement is accurate to .2 in.

Example 4.7.8

A closed box is constructed from lumber that is $3\/4$ in thick. The outside measurements are $9\times 7\times 5$. Use differentials to estimate the interior volume of the box.

Example 4.7.9

Show that the plane tangent to $f\left(x\,y\right)\=3 x^2\+5 x y-7 y^2\+4$ at $\left(x\,y\right)\=\left(2\,3\right)$ is the first-degree Taylor polynomial constructed at the point of contact. Show further that approximating $\mathrm{\Δ}f$ by the total differential $\mathrm{df}$ amounts to a tangent-plane approximation.

Example 4.7.10

At $\left(x\,y\right)\=\left(4\,1\right)$, construct the second-degree Taylor polynomial for $f\left(x\,y\right)\=\sqrt{3 x^2\+y^3}$.