Section 14.4: Tangent Planes

Suppose \(f\) has continuous partial derivatives. An equation of the tangent plane to the surface \(z = f(x, y)\) at the point \(P(x_0, y_0, z_0)\) is

\[ z - z_0 = f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0) \]

EXAMPLE 1
Find the tangent plane to the elliptic paraboloid \(z = 2x^2 + y^2\) at the point \((1, 1, 3)\).

If \(z = f(x,y)\), its differential is defined as

\[ dz = f_x(x, y)\,dx + f_y(x, y)\,dy = \frac{\partial z}{\partial x} dx + \frac{\partial z}{\partial y} dy \]

EXAMPLE
(a) If \(z = f(x, y) = x^2 + 3xy - y^2\), find the differential \(dz\).
(b) If \(x\) changes from 2 to 2.05 and \(y\) changes from 3 to 2.96, compare the values of \(\Delta z\) and \(dz\).

EXAMPLE
The base radius and height of a right circular cone are measured as 10 cm and 25 cm, respectively, with a possible error in measurement of as much as 0.1 cm in each. Use differentials to estimate the maximum error in the calculated volume of the cone.

The differential \(dw\) is defined in terms of the differentials \(dx\), \(dy\), and \(dz\) of the independent variables by

\[ dw = \frac{\partial w}{\partial x} dx + \frac{\partial w}{\partial y} dy + \frac{\partial w}{\partial z} dz \]

EXAMPLE
The dimensions of a rectangular box are measured to be 75 cm, 60 cm, and 40 cm, and each measurement is correct to within 0.2 cm. Use differentials to estimate the largest possible error when the volume of the box is calculated from these measurements.

Find an equation of the tangent plane to the given surface at the specified point.

\(z = 2x^2 + y^2 - 5y,\quad (1, 2, -4)\)

Find an equation of the tangent plane to the given surface at the specified point.

\(z = (x + 2)^2 - 2(y - 1)^2 - 5,\quad (2, 3, 3)\)

Find an equation of the tangent plane to the given surface at the specified point.

\(z = e^{x - y},\quad (2, 2, 1)\)

Find an equation of the tangent plane to the given surface at the specified point. \(z = \dfrac{x}{y^2},\quad (-4, 2, -1)\)

Find an equation of the tangent plane to the given surface at the specified point.

\(z = x \sin(x + y),\quad (-1, 1, 0)\)

Find an equation of the tangent plane to the given surface at the specified point.

\(z = \ln(x - 2y),\quad (3, 1, 0)\)

Graph the surface and the tangent plane at the given point. (Choose the domain and viewpoint so that you get a good view of both the surface and the tangent plane.) Then zoom in until the surface and the tangent plane become indistinguishable.

\(z = x^2 + xy + 3y^2,\quad (1, 1, 5)\)

Graph the surface and the tangent plane at the given point. (Choose the domain and viewpoint so that you get a good view of both the surface and the tangent plane.) Then zoom in until the surface and the tangent plane become indistinguishable.

\(z = \sqrt{9 + x^2y^2},\quad (2, 2, 5)\)

Draw the graph of \(f\) and its tangent plane at the given point. (Use your computer algebra system both to compute the partial derivatives and to graph the surface and its tangent plane.)

\(f(x, y) = \dfrac{1 + \cos^2(x - y)}{1 + \cos^2(x + y)},\quad \left(\dfrac{\pi}{3}, \dfrac{\pi}{6}, \dfrac{7}{4}\right)\)

Draw the graph of \(f\) and its tangent plane at the given point. (Use your computer algebra system both to compute the partial derivatives and to graph the surface and its tangent plane.)

\(f(x, y) = e^{-xy/10}\left(\sqrt{x} + \sqrt{y} + \sqrt{xy}\right),\quad (1, 1, 3e^{-0.1})\)

Explain why the function is differentiable at the given point. Then find the linearization \(L(x, y)\) of the function at that point.

\(f(x, y) = 1 + x \ln(xy - 5),\quad (2, 3)\)

Explain why the function is differentiable at the given point. Then find the linearization \(L(x, y)\) of the function at that point.

\(f(x, y) = \sqrt{xy},\quad (1, 4)\)

Explain why the function is differentiable at the given point. Then find the linearization \(L(x, y)\) of the function at that point.

\(f(x, y) = x^2 e^y,\quad (1, 0)\)

Explain why the function is differentiable at the given point. Then find the linearization \(L(x, y)\) of the function at that point.

\(f(x, y) = \dfrac{1 + y}{1 + x},\quad (1, 3)\)

Explain why the function is differentiable at the given point. Then find the linearization \(L(x, y)\) of the function at that point.

\(f(x, y) = 4 \arctan(xy),\quad (1, 1)\)

Explain why the function is differentiable at the given point. Then find the linearization \(L(x, y)\) of the function at that point.

\(f(x, y) = y + \sin(x/y),\quad (0, 3)\)

Verify the linear approximation at \((0, 0)\).

\(e^x \cos(xy) \approx x + 1\)

Verify the linear approximation at \((0, 0)\).

\(\dfrac{y - 1}{x + 1} \approx x + y - 1\)

Given that \(f\) is a differentiable function with \(f(2, 5) = 6\), \(f_x(2, 5) = 1\), and \(f_y(2, 5) = -1\), use a linear approximation to estimate \(f(2.2, 4.9)\).

Find the linear approximation of the function
\[ f(x, y) = 1 - xy \cos \pi y \text{ at } (1, 1) \]
and use it to approximate \(f(1.02, 0.97)\). Illustrate by graphing \(f\) and the tangent plane.

Find the linear approximation of the function
\[ f(x, y, z) = \sqrt{x^2 + y^2 + z^2} \text{ at } (3, 2, 6) \]
and use it to approximate the number \(\sqrt{(3.02)^2 + (1.97)^2 + (5.99)^2}\).

The wave heights \(h\) in the open sea depend on the speed \(v\) of the wind and the length of time \(t\) that the wind has been blowing at that speed. Values of the function \(h = f(v, t)\) are recorded in feet in the following table. Use the table to find a linear approximation to the wave height function when \(v\) is near 40 knots and \(t\) is near 20 hours. Then estimate the wave heights when the wind has been blowing for 24 hours at 43 knots.

The wind-chill index \(W\) is the perceived temperature when the actual temperature is \(T\) and the wind speed is \(v\), so we can write \(W = f(T, v)\). The following table of values is an excerpt from Table 1 in Section 14.1. Use the table to find a linear approximation to the wind-chill index function when \(T\) is near \(-15^\circ\)C and \(v\) is near 50 km/h. Then estimate the wind chill index when the temperature is \(-17^\circ\)C and the wind speed is 55 km/h.

Find the differential of the function

\[ z = e^{-2x} \cos(2\pi t) \]

Find the differential of the function
\[ u = \sqrt{x^2 + 3y^2} \]

Find the differential of the function
\[ m = p^5 q^3 \]

Find the differential of the function
\[ T = \dfrac{v}{1 + uvw} \]

Find the differential of the function

\[ R = \alpha \beta^2 \cos \gamma \]

Find the differential of the function

\[ L = xze^{-y^2 - z^2} \]

If \(z = 5x^2 + y^2\) and \((x, y)\) changes from \((1, 2)\) to \((1.05, 2.1)\), compare the values of \(\Delta z\) and \(dz\).

If \(z = x^2 - xy + 3y^2\) and \((x, y)\) changes from \((3, -1)\) to \((2.96, -0.95)\), compare the values of \(\Delta z\) and \(dz\).

The wind-chill index is modeled by the function
\[ W = 13.12 + 0.6215T - 11.37v^{0.16} + 0.3965Tv^{0.16} \]
where \(T\) is the temperature (in °C) and \(v\) is the wind speed (in km/h). The wind speed is measured as 26 km/h, with a possible error of \(\pm 2\) km/h, and the temperature is measured as \(-11^\circ\)C, with a possible error of \(\pm 1^\circ\)C. Use differentials to estimate the maximum error in the calculated value of \(W\) due to the measurement errors in \(T\) and \(v\).

The tension \(T\) in the string of the yo-yo in the figure is
\[ T = \frac{mgR}{2r^2 + R^2} \]
where \(m\) is the mass of the yo-yo and \(g\) is acceleration due to gravity. Use differentials to estimate the change in the tension if \(R\) is increased from 3 cm to 3.1 cm and \(r\) is increased from 0.7 cm to 0.8 cm. Does the tension increase or decrease?

The pressure, volume, and temperature of a mole of an ideal gas are related by the equation \(PV = 8.31T\), where \(P\) is measured in kilopascals, \(V\) in liters, and \(T\) in kelvins. Use differentials to find the approximate change in the pressure if the volume increases from 12 L to 12.3 L and the temperature decreases from 310 K to 305 K.

If \(R\) is the total resistance of three resistors, connected in parallel, with resistances \(R_1\), \(R_2\), \(R_3\), then
\[ \frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} \]
If the resistances are measured in ohms as \(R_1 = 25\,\Omega\), \(R_2 = 40\,\Omega\), and \(R_3 = 50\,\Omega\), with a possible error of 0.5% in each case, estimate the maximum error in the calculated value of \(R\).

A model for the surface area of a human body is given by
\[ S = 0.1091w^{0.425}h^{0.725} \]
where \(w\) is the weight (in pounds), \(h\) is the height (in inches), and \(S\) is measured in square feet. If the errors in measurement of \(w\) and \(h\) are at most 2%, use differentials to estimate the maximum percentage error in the calculated surface area.

In Exercise 14.1.39 and Example 14.3.3, the body mass index of a person was defined as \(B(m, h) = \frac{m}{h^2}\), where \(m\) is the mass in kilograms and \(h\) is the height in meters.

1. What is the linear approximation of \(B(m, h)\) for a child with mass 23 kg and height 1.10 m?

2. If the child’s mass increases by 1 kg and height by 3 cm, use the linear approximation to estimate the new BMI. Compare with the actual new BMI.

Suppose you need to know an equation of the tangent plane to a surface \(S\) at the point \(P(2, 1, 3)\). You don’t have an equation for \(S\) but you know that the curves
\[ \mathbf{r}_1(t) = \langle 2 + 3t,\ 1 - t^2,\ 3 - 4t + t^2 \rangle \\ \mathbf{r}_2(u) = \langle 1 + u^2,\ 2u^3 - 1,\ 2u + 1 \rangle \]
both lie on \(S\). Find an equation of the tangent plane at \(P\).

Show that the function is differentiable by finding values of \(\varepsilon_1\) and \(\varepsilon_2\) that satisfy Definition 7.

\(f(x, y) = x^2 + y^2\)

Show that the function is differentiable by finding values of \(\varepsilon_1\) and \(\varepsilon_2\) that satisfy Definition 7.

\(f(x, y) = xy - 5y^2\)

Prove that if \(f\) is a function of two variables that is differentiable at \((a, b)\), then \(f\) is continuous at \((a, b)\).

Hint: Show that
\[ \lim_{(\Delta x, \Delta y) \to (0, 0)} f(a + \Delta x,\ b + \Delta y) = f(a, b) \]