3.6 Linearization: Tangent Planes and Differentials

Linearization: Tangent Planes and Differentials

Introduction

The Tangent Plane

Linearization

Differentiability and Local Linearity

We want to find the equation of the plane, using the form given in Proposition 3.6.1, that best describes the surface given by $z=f(x,y)=6-\frac{x^2}2 - y^2$ for input values around $(x_0,y_0) = (1,1)$. In particular, we will need to find how the values of $z_0$, $a$, and $b$ are related to $f(x,y)$.

Find $f(1,1)$, $f_x(1,1)$, and $f_y(1,1)$.

Find the equation of the tangent plane to $f(x,y) = x^2y$ at the point $(1,2)$.

Suppose that the tangent plane to the graph of a continuously differentiable function $z=g(x,y)$ is given in the form

. Use the equation of the tangent plane to identify a point on the graph as well as a value of $g_x$ and a value of $g_y$. Be sure to identify at what point(s) you have found the values of the partial derivatives.

Work through this exercise and explain your reasoning step by step.

Find the linearization $L(x,y)$ for the function $g$ defined by

at the point $(1,2)$. Use the linearization to estimate the value of $g(0.8, 2.3)$.

The table below provides a collection of values of the wind chill $w(v,T)$, in degrees Fahrenheit, as a function of wind speed, in miles per hour, and temperature, also in degrees Fahrenheit.

$v \backslash T$ | $-20$ | $-15$ | $-10$ | $-5$ | $0$ | $5$ | $10$

$10$ | $-41$ | $-35$ | $-28$ | $-22$ | $-16$ | $-10$ | $-4$

$15$ | $-45$ | $-39$ | $-32$ | $-26$ | $-19$ | $-13$ | $-7$

$20$ | $-48$ | $-42$ | $-35$ | $-29$ | $-22$ | $-15$ | $-9$

$25$ | $-51$ | $-44$ | $-37$ | $-31$ | $-24$ | $-17$ | $-11$

$30$ | $-53$ | $-46$ | $-39$ | $-33$ | $-26$ | $-19$ | $-12$

$35$ | $-55$ | $-48$ | $-41$ | $-34$ | $-27$ | $-21$ | $-14$

• Use the data to estimate the appropriate partial derivatives of $w(v,T)$ at the point $(20,-10)$. - Then find the linearization $L(v,T)$ at the point $(20,-10)$. - Use the linearization to estimate $w(10,-10)$, $w(20,-12)$, and $w(18,-12)$. - Compare your results to what you obtained in Activity 3.4.4.

The image below gives a contour plot of a continuously differentiable function $f$. After estimating appropriate partial derivatives, determine the linearization $L(x,y)$ at the point $(2,1)$, and use it to estimate $f(2.2, 1)$, $f(2, 0.8)$, and $f(2.2, 0.8)$.

Work through this exercise and explain your reasoning step by step.

Suppose that the elevation of a plot of land is given by the function $h$, where we additionally know that $h(3,1) = 4.35$, $h_x(3,1) = 0.27$, and $h_y(3,1) = -0.19$. Assume that $x$ and $y$ are measured in miles in the east and north directions, respectively, from $(0,0)$.

Your GPS device says that you are currently at the point $(3,1)$. However, you know that the coordinates are only accurate to within $0.2$ miles; that is, $dx = \Delta x = 0.2$ and $dy= \Delta y = 0.2$. Estimate the uncertainty in your elevation using the linearization of $h$ for the input (3,1).

The pressure, volume, and temperature of an ideal gas are related by the equation

where $P$ is measured in kilopascals, $V$ in liters, and $T$ in kelvin. Find the pressure when the volume is 12 liters and the temperature is 310 K. Use the linearization of $P$ at the point $(310,12)$ to estimate the change in the pressure when the volume increases to 12.3 liters and the temperature decreases to 305 K.

Use the table of values for the wind chill $w(v,T)$, in degrees Fahrenheit, as a function of temperature, also in degrees Fahrenheit, and wind speed, in miles per hour provided in for this part. Suppose your anemometer（An instrument for measuring wind speed.） says the wind is blowing at $25$ miles per hour and your thermometer（An instrument for measuring the temperature.） shows a reading of $-15^\circ$ degrees. However, you know your thermometer is only accurate to within $2^\circ$ degrees and your anemometer is only accurate to within $3$ miles per hour. What is the wind chill based on your measurements? Estimate the uncertainty in your measurement of the wind chill.

Work through this exercise and explain your reasoning step by step.

Suppose that a function $f = f(x,y)$ is differentiable at a point $(x_0,y_0)$. Let $L = L(x,y) = f(x_0,y_0) + m(x-x_0) + n(y-y_0)$ as in the conditions of Definition. Show that $m = f_x(x_0,y_0)$ and $n = f_y(x_0,y_0)$. (Hint: Calculate the limits of the relative errors when $h = 0$ and $k = 0$.)

We know that if a function of a single variable is differentiable at a point, then that function is also continuous at that point. In this exercise we determine that the same property holds for functions of two variables. A function $f$ of the two variables $x$ and $y$ is continuous at a point $(x_0,y_0)$ in its domain if

or (letting $x=x_0+h$ and $y = y_0 + k$,

Show that if $f$ is differentiable at $(x_0,y_0)$, then $f$ is continuous at $(x_0,y_0)$. (Hint: Multiply both sides of the equality that comes from differentiability by $\lim_{(h,k) \to (0,0)} \sqrt{h^2+k^2}$.)