?The radius of a right circular cylinder is measured as 2.5 ft, and the height is

The graph of a function f is shown. Find an equation of the tangent plane to the surface z = f(x, y) at the specified point.

The graph of a function f is shown. Find an equation of the tangent plane to the surface z = f(x, y) at the specified point.

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

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

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.

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

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

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

Find an equation of the tangent plane to the given surface at the specified point. z = ln(x - 2y), (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.

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.

Draw the graph of f and its tangent plane at the given point. (Use a computer to compute the partial derivatives.) Then zoom in until the surface and the tangent plane become indistinguishable.

Draw the graph of \(f\) and its tangent plane at the given point. (Use a computer to compute the partial derivatives.) Then zoom in until the surface and the tangent plane become indistinguishable. \(f(x, \ y)=e^{-x y / 10}(\sqrt{x}+\sqrt{y}+\sqrt{x y}),\left(1, \ 1, \ 3 e^{-0.1}\right)\)

Explain why the function is differentiable at the given point. Then find the linearization \(L(x, \ y)\) of the function at that point. \(f(x, \ y)=x^{3} y^{2}, \quad(-2, \ 1)\)

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

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

Explain why the function is differentiable at the given point. Then find the linearization \(L(x, \ y)\) of the function at that point. \(f(x, \ y)=\sqrt{x y}, \quad(1, \ 4)\)

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

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

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), (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), (0, 3)

Verify the linear approximation at (0, 0). ex cos(xy) ? x + 1

Verify the linear approximation at (0, 0). \(\frac{y-1}{x+1} \approx x+y-1\)

Given that f is a differentiable function with f(2, 5) = 6, fx(2, 5) = 1, and fy(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 ????y 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) = at (3, 2, 6) and use it to approximate the number .

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.

Use the table in Example 3 to find a linear approximation to the heat index function when the actual temperature is near 94°F and the relative humidity is near 80%. Then estimate the heat index when the actual temperature is 95°F and the relative humidity is 78%.

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°C and v is near 50 km/h. Then estimate the wind-chill index when the temperature is ?17°C and the wind speed is 55 km/h.

Find the differential of the function. m = p5q3

Find the differential of the function. z = x ln(y2 + 1)

Find the differential of the function. z = e?2x cos 2????t

Find the differential of the function. u =

Find the differential of the function. H = x2y4 + y3z5

Find the differential of the function. \(v=x z e^{-y^{2}-z^{2}}\)

Find the differential of the function. R = ???????? ? cos ????

Find the differential of the function.

If z = 5x2 + y2 and (x, y) changes from (1, 2d to s1.05, 2.1), compare the values of and dz.

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

The length and width of a rectangle are measured as 30 cm and 24 cm, respectively, with an error in measurement of at most 0.1 cm in each. Use differentials to estimate the maximum error in the calculated area of the rectangle.

Use differentials to estimate the amount of metal in a closed cylindrical can that is 10 cm high and 4 cm in diameter if the metal in the top and bottom is 0.1 cm thick and the metal in the sides is 0.05 cm thick.

Use differentials to estimate the amount of tin in a closed tin can with diameter 8 cm and height 12 cm if the tin is 0.04 cm thick.

The base and height of a triangle are measured as 28 inches and 16 inches, respectively. Suppose that each measurement has a possible error of at most \(\varepsilon\) inches. (a) Use differentials to estimate the maximum error in the calculated area of the triangle. (b) What is the estimated maximum error in the area of the triangle if the base and height are measured with errors at most \(\frac{1}{4}\) inch?

The radius of a right circular cylinder is measured as 2.5 ft, and the height is measured as 12 ft. Suppose that each measurement has a possible error of at most ???? feet. (a) Use differentials to estimate the maximum error in the calculated volume of the cylinder. (b) If the computed volume must be accurate to within one cubic foot, determine the largest allowable value of ????.

The wind-chill index is modeled by the function \(W=13.12+0.6215 T-11.37 v^{0.16}+0.3965 T v^{0.16}\) where \(T\) is the actual temperature (in \({ }^{\circ} \mathrm{C}\) ) and \(v\) is the wind speed (in km/h). The wind speed is measured as 26 km/h, with a possible error of \(\pm 2 \mathrm{km} / \mathrm{h}\), and the actual temperature is measured as \(-11^{\circ} \mathrm{C}\), with a possible error of \(\pm 1^{\circ} \mathrm{C}\). 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 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 R1, R2, R3, then If the resistances are measured in ohms as R1 = 25 ????, R2 = 40 ????, and R3 = 50 ????, 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 = 0.1091w0.425h0.725, 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 14.3.4, the body mass index of a person was defined as \(\mathrm{B}(\mathrm{m}, \mathrm{h})=\mathrm{m} / \mathrm{h}^{2}\), where m is the mass in kilograms and h is the height in meters. (a) What is the linear approximation of B(m, h) for a child with mass 23 kg and height 1.10 m? (b) 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 both lie on S. Find an equation of the tangent plane at P.

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

(a) The function was graphed in Figure 4. Show that fx(0, 0) and fy(0, 0) both exist but f is not differentiable at (0, 0). [Hint: Use the result of Exercise 53.] (b) Explain why fx and fy are not continuous at (0, 0).