4952 A function f is given along with a local linear approximationL to f at a point P

Assume that \(f (x, y)\) is differentiable at \(\left(x_{0}, y_{0}\right)\) and let \(\Delta f\) denote the change in f from its value at \(\left(x_{0}, y_{0}\right)\) to its value at \(\left(x_{0}+\Delta x, y_{0}+\Delta y\right)\). 1. \(\Delta \mathrm{f} \approx\) _______ 2. The limit that guarantees the error in the approximation in part (a) is very small when both \(\Delta \mathrm{x}\) and \(\Delta \mathrm{y}\) are close to 0 is _______ . Equation Transcription: Text Transcription: f (x, y) (x_0, y_0) delta f approx f (x_0 + delta x, y_0 + delta y) delta x delta y

Compute the differential of each function. (a) \(z=x e^{y^{2}}\) (b) \(\mathrm{w}=\mathrm{x} \sin (\mathrm{y} \mathrm{z})\) Equation Transcription: Text Transcription: z = xe^y2 w = x sin (yz)

If \(f\) is differentiable at \(\left(x_{0}, y_{0}\right)\), then the local linear approximation to \(f\) at \(\left(x_{0}, y_{0}\right)\) is \(L(x)\) = _______ . Equation Transcription: Text Transcription: (x_0, y_0) f L(x)

Assume that \(f(1,-2)=4\) and \(f (x, y)\) is differentiable at \((1, ?2)\) with \(f_{x}(1,-2)=2\) and \(f_{y}(1,-2)=-3\). Estimate the value of \(f(0.9,-1.950)\). Equation Transcription: Text Transcription: f(1, -2) = 4 f(x,y) (1, -2) f_x (1, -2) = 2 F_y (1, -2) = -3 f(0.9, -1.950)

Use Definitions 13.4.1 and 13.4.2 to prove that a constant function of two or three variables is differentiable everywhere.

Use Definitions 13.4.1 and 13.4.2 to prove that a linear function of two or three variables is differentiable everywhere.

Use Definition 13.4.2 to prove that \(f(x, y, z)=x^{2}+y^{2}+z^{2}\) is differentiable at \((0, 0, 0)\). Equation Transcription: Text Transcription: f (x,y,z) = x^2 + y^2 + z^2 (0,0,0)

Use Definition 13.4.2 to determine all values of \(r\) such that \(f(x, y, z)=\left(x^{2}+y^{2}+z^{2}\right) r\) is differentiable at \((0, 0, 0)\). Equation Transcription: Text Transcription: r f(x,y,z) = (x^2 + y^2 + z^2) r (0, 0, 0)

Compute the differential dz or dw of the function. \(z=7 x-2 y\) Equation Transcription: Text Transcription: z = 7x - 2y

Compute the differential dz or dw of the function. \(z=e^{x y}\) Equation Transcription: Text Transcription: z = e^xy

Compute the differential dz or dw of the function. \(z=x^{3} y\) Equation Transcription: Text Transcription: z = x^3 y

Compute the differential dz or dw of the function. \(z=5 x^{2} y^{5}-2 x+4 y+7\) Equation Transcription: Text Transcription: z = 5x_2 y_5 ? 2x + 4y + 7

Compute the differential dz or dw of the function. \(z=\tan ^{-1} x y\) Equation Transcription: Text Transcription: z = tan^-1 xy

Compute the differential dz or dw of the function. \(z=e^{-3 x} \cos 6 y\) Equation Transcription: Text Transcription: z = e^-3x cos 6y

Compute the differential dz or dw of the function. \(w = 8x ? 3y + 4z\) Equation Transcription: Text Transcription: w = 8x ? 3y + 4z

Compute the differential dz or dw of the function. \(w=e^{x y z}\) Equation Transcription: Text Transcription: w = e^xyz

Compute the differential dz or dw of the function. \(w=x^{3} y^{2} z\) Equation Transcription: Text Transcription: w = x^3 y^2 z

Compute the differential dz or dw of the function. \(w=4 x^{2} y^{3} z^{7}-3 x y+z+5\) Equation Transcription: Text Transcription: W = 4x^2 y^3 z^7 - 3xy + z + 5

Compute the differential dz or dw of the function. \(\mathrm{w}=\tan ^{-1}(\mathrm{xyz})\) Equation Transcription: Text Transcription: W = tan^-1 (xyz)

Compute the differential dz or dw of the function. \(\mathrm{w}=\sqrt{x}+\sqrt{y}+\sqrt{z}\) Equation Transcription: Text Transcription: w = sqrt x + sqrt y + sqrt z

Use a total differential to approximate the change in the values of \(f\) from \(P\) to \(Q\). Compare your estimate with the actual change in \(f\). \(\mathrm{f}(\mathrm{x}, \mathrm{y})=\mathrm{x}^{2}+2 \mathrm{xy}-4 \mathrm{x} ; \mathrm{P}(1,2), \mathrm{Q}(1.01,2.04)\) Equation Transcription: Text Transcription: f P Q f(x, y) = x^2 + 2xy ? 4x; P (1, 2), Q(1.01, 2.04)

Use a total differential to approximate the change in the values of \(f\) from \(P\) to \(Q\). Compare your estimate with the actual change in \(f\). \(f(x, y)=x^{1 / 3} y^{1 / 2} ; P(8,9), Q(7.78,9.03)\) Equation Transcription: Text Transcription: f P Q f(x, y) = x^1/3 y^1/2 ; P (8, 9), Q(7.78, 9.03)

Use a total differential to approximate the change in the values of \(f\) from \(P\) to \(Q\). Compare your estimate with the actual change in \(f\). \(f(x, y)=\frac{x+y}{x y} ; P(-1,-2), Q(-1.02,-2.04)\) Equation Transcription: Text Transcription: f P Q f(x, y) = x+y /xy ; P (?1, ?2), Q(?1.02, ?2.04)

Use a total differential to approximate the change in the values of \(f\) from \(P\) to \(Q\). Compare your estimate with the actual change in \(f\). \(\mathrm{f}(\mathrm{x}, \mathrm{y})=\ln \sqrt{1+x y} ; \mathrm{P}(0,2), \mathrm{Q}(-0.09,1.98)\) Equation Transcription: Text Transcription: f P Q f(x, y) = ln sqrt 1+xy ; P (0, 2), Q(?0.09, 1.98)

Use a total differential to approximate the change in the values of \(f\) from \(P\) to \(Q\). Compare your estimate with the actual change in \(f\). \(f(x, y, z)=2 x y^{2} z^{3} ; P(1,-1,2), Q(0.99,-1.02,2.02)\) Equation Transcription: Text Transcription: f P Q f(x, y, z) = 2xy^2 z^3; P (1, ?1, 2), Q(0.99, ?1.02, 2.02)

Use a total differential to approximate the change in the values of \(f\) from \(P\) to \(Q\). Compare your estimate with the actual change in \(f\). f (x, y, z) = ; P (?1, ?2, 4), Q(?1.04, ?1.98, 3.97) Equation Transcription: Text Transcription: f P Q f (x, y, z) = xyz/ x+y+z ; P (?1, ?2, 4), Q(?1.04, ?1.98, 3.97)

Determine whether the statement is true or false. Explain your answer. By definition, a function \(f(x, y)\) is differentiable at \(\left(x_{0}, y_{0}\right)\) provided both \(f x\left(x_{0}, y_{0}\right)\) and \(\text { fy }\left(x_{0}, y_{0}\right)\) are defined. Equation Transcription: Text Transcription: f(x, y) (x_0, y_0) fx (x_0, y_0) fy (x_0, y_0)

Determine whether the statement is true or false. Explain your answer. For any point \(\left(x_{0}, y_{0}\right)\) in the domain of a function \(f(x, y)\), we have \(\lim _\limits{(\Delta x, \Delta y) \rightarrow(0,0)} \Delta f=0\) Where \(\Delta f=f\left(x_{0}+\Delta x, y_{0}+y\right)-f\left(x_{0}, y_{0}\right)\) Equation Transcription: Text Transcription: (x_0, y_0) f(x, y) lim_(Delta x, Delta y) right arrow (0,0) Delta f = 0 Delta f = f(x_0 + delta x, y_0 + y) - f(x_0, y_0)

Determine whether the statement is true or false. Explain your answer. If \(fx\) and \(fy\) are both continuous at \(\left(x_{0}, y_{0}\right)\), then so is \(f\) . Equation Transcription: Text Transcription: fx fy (x_0, y_0) f

Determine whether the statement is true or false. Explain your answer. The graph of a local linear approximation to a function \(f(x, y)\) is a plane. Equation Transcription: Text Transcription: f(x,y)

In the accompanying figure a rectangle with initial length \(x_{0}\) and initial width \(\mathrm{y}_{0}\) has been enlarged, resulting in a rectangle with length \(x_{0}+\Delta x\) and width \(\mathrm{y}_{0}+\Delta \mathrm{y}\). What portion of the figure represents the increase in the area of the rectangle? What portion of the figure represents an approximation of the increase in area by a total differential? Equation Transcription: Text Transcription: x_0 y_0 x_0 + Delta x y_0 + Delta y

The volume \(V\) of a right circular cone of radius \(r\) and height \(h\) is given by \(V=\frac{1}{3} \pi r^{2} h\). Suppose that the height decreases from \(20 in\) to \(19.95 in\) and the radius increases from \(4 in\) to \(4.05 in\). Compare the change in volume of the cone with an approximation of this change using a total differential. Equation Transcription: Text Transcription: V r h V = 1/3 pi r^2 h 20 in 19.95 in 4 in 4.05 in

(a) Find the local linear approximation \(L\) to the specified function \(f\) at the designated point \(P\). (b) Compare the error in approximating \(f\) by \(L\) at the specified point \(Q\) with the distance between \(P\) and \(Q\). \(f(x, y)=\frac{1}{\sqrt{x^{2}+y 2}} ; P(4,3), Q(3.92,3.01)\) Equation Transcription: Text Transcription: L f P Q f (x, y) = 1/sqrt x2 + y2 ; P (4, 3), Q(3.92, 3.01)

(a) Find the local linear approximation \(L\) to the specified function \(f\) at the designated point \(P\). (b) Compare the error in approximating \(f\) by \(L\) at the specified point \(Q\) with the distance between \(P\) and \(Q\). \(f(x, y)=x^{0.5} y^{0.3} ; P(1,1), Q(1.05,0.97)\) Equation Transcription: Text Transcription: L f P Q f(x,y) = x^0.5 y^0.3; P(1,1), Q(1.05, 0.97)

(a) Find the local linear approximation \(L\) to the specified function \(f\) at the designated point \(P\). (b) Compare the error in approximating \(f\) by \(L\) at the specified point \(Q\) with the distance between \(P\) and \(Q\). \(f(x, y)=x \sin y ; P(0,0), Q(0.003,0.004)\) Equation Transcription: Text Transcription: L f P Q f (x,y) = x sin y; P (0,0), Q (0.003, 0.004)

(a) Find the local linear approximation \(L\) to the specified function \(f\) at the designated point \(P\). (b) Compare the error in approximating \(f\) by \(L\) at the specified point \(Q\) with the distance between \(P\) and \(Q\). \(\mathrm{f}(\mathrm{x}, \mathrm{y})=\ln \mathrm{xy} ; \mathrm{P}(1,2), Q(1.01,2.02)\) Equation Transcription: Text Transcription: L f P Q f (x,y) = ln xy; P (1,2), Q (1.01, 2.02)

(a) Find the local linear approximation \(L\) to the specified function \(f\) at the designated point \(P\). (b) Compare the error in approximating \(f\) by \(L\) at the specified point \(Q\) with the distance between \(P\) and \(Q\). \(\mathrm{f}(\mathrm{x}, \mathrm{y}, \mathrm{z})=\mathrm{xyz} ; \mathrm{P}(1,2,3), Q(1.001,2.002,3.003)\) Equation Transcription: Text Transcription: L f P Q f (x, y, z) = xyz; P (1, 2, 3), Q(1.001, 2.002, 3.003)

(a) Find the local linear approximation \(L\) to the specified function \(f\) at the designated point \(P\). (b) Compare the error in approximating \(f\) by \(L\) at the specified point \(Q\) with the distance between \(P\) and \(Q\). \(f(x, y, z)=\frac{x+y}{y+z} ; P(-1,1,1), Q(-0.99,0.99,1.01)\) Equation Transcription: Text Transcription: L f P Q f (x, y, z) = x+y / y+z; P (?1, 1, 1), Q(?0.99, 0.99, 1.01)

(a) Find the local linear approximation \(L\) to the specified function \(f\) at the designated point \(P\). (b) Compare the error in approximating \(f\) by \(L\) at the specified point \(Q\) with the distance between \(P\) and \(Q\). \(f(x, y, z)=x e^{y z} ; P(1,-1,-1), Q(0.99,-1.01,-0.99)\) Equation Transcription: Text Transcription: L f P Q f(x,y,z) = xe^yz; P (1, -1, -1), Q(0.99, -1.01, 0.99)

(a) Find the local linear approximation \(L\) to the specified function \(f\) at the designated point \(P\). (b) Compare the error in approximating \(f\) by \(L\) at the specified point \(Q\) with the distance between \(P\) and \(Q\). \(\mathrm{f}(\mathrm{x}, \mathrm{y}, \mathrm{z})=\ln (\mathrm{x}+\mathrm{yz}) ; \mathrm{P}(2,1,-1), \mathrm{Q}(2.02,0.97,-1.01)\) Equation Transcription: Text Transcription: L f P Q f (x, y, z) = ln(x + yz); P (2, 1, ?1), Q(2.02, 0.97, ?1.01)

In each part, confirm that the stated formula is the local linear approximation at \((0, 0)\). (a) \(\mathrm{e}^{\mathrm{x}} \sin \mathrm{y} \approx \mathrm{y}\) (b) \(\frac{2 x+1}{y+1} \approx 1+2 x-y\) Equation Transcription: Text Transcription: (0, 0) e^x sin y approx y 2x + 1 / y + 1 approx 1 + 2x - y

Show that the local linear approximation of the function \(f(x, y)=x^{\alpha} y^{\beta} \text { at }(1,1) \text { is }\) \(x^{a} y_{n}^{\beta} \approx 1+\alpha(x-1)+\beta(y-1)\) Equation Transcription: Text Transcription: f(x,y) = x^alpha y^beta at (1,1) is x^alpha y^beta approx 1 + alpha(x-1) + beta(y-1)

In each part, confirm that the stated formula is the local linear approximation at \((1,1,1)\). (a) \(x y z+2 \approx x+y+z\) (b) \(\frac{4 x}{y+z} \approx 2 x-y-z+2\) Equation Transcription: Text Transcription: (1,1,1) xyz + 2 approx x + y + z 4x / y+z approx 2x - y - z + 2

Based on Exercise 42, what would you conjecture is the local linear approximation to \(\mathrm{x}^{\alpha} \mathrm{y}^{\beta} \mathbf{z}^{\mathrm{\gamma}}\) at \((1, 1, 1)\)? Verify your conjecture by finding this local linear approximation. Equation Transcription: Text Transcription: x^alpha y^beta z^gamma

Suppose that a function \(f(x, y)\) is differentiable at the point \((1, 1)\) with \(f_{x}(1,1)=2\) and \(f(1,1)=3\). Let \(L(x, y)\) denote the local linear approximation of \(f\) at \((1, 1)\). If \(L(1.1,0.9)=3.15\), find the value of \(f_{y}(1,1)\). Equation Transcription: Text Transcription: f(x, y) (1, 1) f_x (1, 1) = 2 f(1, 1) = 3 L(x, y) f (1, 1) L(1.1, 0.9) = 3.15 f_y (1, 1)

Suppose that a function \(f(x, y)\) is differentiable at the point \((0, ?1)\) with \(f_{y}(0,-1)=-2\) and \(f(0,-1)=3\). Let \(L(x, y)\) denote the local linear approximation of \(f\) at \((0,-1)\). If \(L(0.1,-1.1)\), find the value of \(f_{x}(0,-1)\). Equation Transcription: Text Transcription: f(x,y) (0,-1) f_y (0, -1) = -2 f(0,-1) = 3 L(x,y) f L(0.1, -1.1) = 3.3 f_x(0,-1)

Suppose that a function \(\mathrm{f}(\mathrm{x}, \mathrm{y}, \mathrm{z})\) is differentiable at the point \((3,2,1)\) and \(L(x, y, z)=x-y+2 z-2\) is the lo- cal linear approximation to \(f\) at \((3,2,1)\). Find \(f(3,2,1), f_{x}(3,2,1), f_{y}(3,2,1), \text { and } f_{z}(3,2,1)\). Equation Transcription: Text Transcription: f(x,y,z) (3, 2, 1) L(x,y,z) = x - y + 2z - 2 F f(3,2,1), f_x (3,2,1), f_y (3,2,1), f_z (3,2,1)

Suppose that a function \(\mathrm{f}(\mathrm{x}, \mathrm{y}, \mathrm{z})\) is differentiable at the point \((0,-1,-2)\) and \(L(x, y, z)=x+2 y+3 z+4\) is the local linear approximation to \(f\) at \((0, ?1, ?2)\). Find \(f(0,-1,-2), f_{x}(0,-1,-2), f_{y}(0,-1,-2), \text { and } f_{z}(0,-1,-2)\). Equation Transcription: Text Transcription: f(x,y,z) (0, -1, -2) L(x,y,z) = x + 2y +3z +4 F f(0, -1, -2), f_x(0, -1, -2), f_y(0, -1, -2), f_z(0, -1, -2)

A function \(f\) is given along with a local linear approximation \(L\) to \(f\) at a point \(P\). Use the information given to determine point \(P\). \(f(x, y)=x^{2}+y^{2} ; L(x, y)=2 y-2 x-2\) Equation Transcription: Text Transcription: f L P f(x, y) = x^2 + y^2; L(x, y) = 2y ? 2x ? 2

A function \(f\) is given along with a local linear approximation \(L\) to \(f\) at a point \(P\). Use the information given to determine point \(P\). \(f(x, y)=x^{2} y ; L(x, y)=4 y-4 x+8\) Equation Transcription: Text Transcription: f L P f(x,y) = x^2 y; L(x,y) = 4y - 4x + 8

A function \(f\) is given along with a local linear approximation \(L\) to \(f\) at a point \(P\). Use the information given to determine point \(P\). \(f(x, y, z)=x y+z^{2} ; L(x, y, z)=y+2 z-1\) Equation Transcription: Text Transcription: f L P f(x,y,z) = xy + z^2 ; L(x,y,z) = y + 2z - 1

A function \(f\) is given along with a local linear approximation \(L\) to \(f\) at a point \(P\). Use the information given to determine point \(P\). \(f(x, y, z)=x y z ; L(x, y, z)=x-y-z-2\) Equation Transcription: Text Transcription: f L P f(x,y,z) = xyz; L(x,y,z) = x-y-z-2

The length and width of a rectangle are measured with errors of at most \(3 \%\) and \(5 \%\), respectively. Use differentials to approximate the maximum percentage error in the calculated area. Equation Transcription: Text Transcription: 3% 5%

The radius and height of a right circular cone are measured with errors of at most \(1 \%\) and \(4 \%\), respectively. Use differentials to approximate the maximum percentage error in the calculated volume. Equation Transcription: Text Transcription: 1% 4%

The length and width of a rectangle are measured with errors of at most \(r \%\), where \(r\) is small. Use differentials to approximate the maximum percentage error in the calculated length of the diagonal. Equation Transcription: Text Transcription: r% r

The legs of a right triangle are measured to be \(3 \mathrm{~cm}\) and \(4 \mathrm{~cm}\), with a maximum error of \(0.05 \mathrm{~cm}\) in each measurement. Use differentials to approximate the maximum possible error in the calculated value of (a) the hypotenuse and (b) the area of the triangle. Equation Transcription: Text Transcription: 3cm 4cm 0.05cm

The period \(T\) of a simple pendulum with small oscillations is calculated from the formula \(\mathrm{T}=2 \pi \quad \sqrt{L / g}\), where \(L\) is the length of the pendulum and \(g\) is the acceleration due to gravity. Suppose that measured values of \(L\) and g have errors of at most \(0.5 \%\) and \(0.1\%\), respectively. Use differentials to approximate the maximum percentage error in the calculated value of \(T\) . Equation Transcription: Text Transcription: T T = 2pi sqrt L/g L g 0.5% 0.1%

According to the ideal gas law, the pressure, temperature, and volume of a confined gas are related by \(P=k T / V\) , where \(k\) is a constant. Use differentials to approximate the percentage change in pressure if the temperature of a gas is increased \(3 \%\) and the volume is increased \(5\%\). Equation Transcription: Text Transcription: P = kT /V K 3% 5%

Suppose that certain measured quantities \(x\) and \(y\) have errors of at most \(r \%\) and \(s \%\) respectively. For each of the following formulas in \(x\) and \(y\), use differentials to approximate the maximum possible error in the calculated result. (a) \(xy\) (b) \(x / y\) (c) \(x^{3} y^{3}\) (d) \(x^{3} \sqrt{y}\) Equation Transcription: Text Transcription: x y r% s% x y xy x/y x^3y^3 x^3 sqrt y

The total resistance \(R\) of three resistances \(R_{1}\), \(R_{2}\), and \(R_{3}\), connected in parallel, is given by \(\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}+\frac{1}{R_{3}}\) Suppose that \(R_{1}\), \(R_{2}\), and \(R_{3}\) are measured to be 100 ohms, 200 ohms, and 500 ohms, respectively, with a maximum error of 10% in each. Use differentials to approximate the maximum percentage error in the calculated value of \(R\). Equation Transcription: Text Transcription: R R_1 R_2 R_3 frac{1}{R} = frac{1}{R_1} + frac{1}{R_2} + frac{1}{R_3} R_1 R_2 R_3 R

The area of a triangle is to be computed from the formula \(A=\frac{1}{2} a b \sin \theta\), where \(a\) and \(b\) are the lengths of two sides and \(\theta\) is the included angle. Suppose that \(a\), \(b\), and \(\theta\) are measured to be 40 ft, 50 ft, and \(30^{\circ}\), respectively. Use differentials to approximate the maximum error in the calculated value of \(A\) if the maximum errors in \(a\), \(b\), and \(\theta\) are \(\frac{1}{2}\) ft, \(\frac{1}{4}\) ft, and \(2^{\circ}\), respectively. Equation Transcription: 30° 2° Text Transcription: A = frac{1}{2} ab sin theta a b theta a b theta 30 degrees A a b theta frac{1}{2} frac{1}{4} 2 degrees

The length, width, and height of a rectangular box are measured with errors of at most \(r\)% (where \(r\) is small). Use differentials to approximate the maximum percentage error in the computed value of the volume. Equation Transcription: Text Transcription: r r

Use Theorem 13.4.4 to prove that \(f(x, y)=x^{2} \sin y\) is differentiable everywhere. Equation Transcription: Text Transcription: f (x, y) = x^2 sin y

Use Theorem 13.4.4 to prove that \(f(x, y, z)=x y \sin z\) is differentiable everywhere. Equation Transcription: Text Transcription: f (x, y, z) = xy sin z

Suppose that \(f(x, y)\) is differentiable at the point \(\left(x_{0}, y_{0}\right)\) and let \(z_{0}=f\left(x_{0}, y_{0}\right)\). Prove that \(g(x, y, z)=z-f(x, y)\) is differentiable at \(\left(x_{0}, y_{0}, z_{0}\right)\). Equation Transcription: Text Transcription: f (x, y) (x_0, y_0) z_0 = f(x_0, y_0) g(x, y, z) = z - f(x, y) (x_0, y_0, z_0)

Suppose that \(\Delta f\) satisfies an equation in the form of (5), where \(\epsilon(\Delta x, \Delta y)\) is continuous at \((\Delta x, \Delta y)=(0,0)\) with \(\epsilon(0,0)=0\). Prove that \(f\) is differentiable at \(\left(x_{0}, y_{0}\right)\). Equation Transcription: Text Transcription: Delta f (epsilon(Delta x, Delta y) (Delta x, Delta y) = (0,0) (epsilon(0,0) = 0 f (x_0, y_0)

Writing Discuss the similarities and differences between the definition of differentiability for a function of a single variable and the definition of differentiability for a function of two variables.

Writing Discuss the use of differentials in the approximation of increments and in the estimation of errors.