Chapter 4

Find the equation for the tangent plane to the surface at the indicated point. \(x^{2}+4 y^{2}=z^{2}, P(3,2,5)\)

Find the equation for the tangent plane to the surface at the indicated point. \(x^{3}+y^{3}=3 x y z, P\left(1,2, \frac{3}{2}\right)\)

Find the equation for the tangent plane to the surface at the indicated point. \(\quad z=\operatorname{axy}, P\left(1, \frac{1}{a}, 1\right)\)

Find the equation for the tangent plane to the surface at the indicated point. \(z=\sin x+\sin y+\sin (x+y), P(0,0,0)\)

Find the equation for the tangent plane to the surface at the indicated point. \(h(x, y)=\ln \sqrt{x^{2}+y^{2}}, P(3,4)\)

Find the equation for the tangent plane to the surface at the indicated point. \(z=x^{2}-2 x y+y^{2}, P(1,2,1)\)

Find parametric equations for the normal line to the surface at the indicated point. (Recall that to find the equation of a line in space, you need a point on the line, \(P_{0}\left(x_{0}, y_{0}, z_{0}\right),\) and a vector \(\mathbf{n}=\langle a, b, c\rangle\) that is parallel to the line. Then the equation of \(\quad\) the \(\quad\) line \(\quad\) is \(\left.x-x_{0}=a t, y-y_{0}=b t, z-z_{0}=c t .\right)\). \(z=5 x^{2}-2 y^{2}, P(2,1,18)\)

Find parametric equations for the normal line to the surface at the indicated point. (Recall that to find the equation of a line in space, you need a point on the line, \(P_{0}\left(x_{0}, y_{0}, z_{0}\right),\) and a vector \(\mathbf{n}=\langle a, b, c\rangle\) that is parallel to the line. Then the equation of \(\quad\) the \(\quad\) line \(\quad\) is \(\left.x-x_{0}=a t, y-y_{0}=b t, z-z_{0}=c t .\right)\). \(z=e^{4 x^{2}+6 y^{2}}, P(0,0,1)\)

Find parametric equations for the normal line to the surface at the indicated point. (Recall that to find the equation of a line in space, you need a point on the line, \(P_{0}\left(x_{0}, y_{0}, z_{0}\right),\) and a vector \(\mathbf{n}=\langle a, b, c\rangle\) that is parallel to the line. Then the equation of \(\quad\) the \(\quad\) line \(\quad\) is \(\left.x-x_{0}=a t, y-y_{0}=b t, z-z_{0}=c t .\right)\). \(z=x^{2}-2 x y+y^{2}\) at point \(P(1,2,1)\)

Complete each task. Show that \(f(x, y)=e^{x y} x\) is differentiable at point (1,0)

Complete each task. Find the total differential of the function \(w=e^{y} \cos (x)+z^{2}\)

Complete each task. Show that \(f(x, y)=x^{2}+3 y\) is differentiable at every point. In other words, show that \(\Delta z=f(x+\Delta x, y+\Delta y)-f(x, y)=f_{x} \Delta x+f_{y} \Delta y+\varepsilon_{1} \Delta x+\varepsilon_{2} \Delta y\) where both \(\varepsilon_{1}\) and \(\varepsilon_{2}\) approach zero as \((\Delta x, \Delta y)\) approaches (0,0) .

Complete each task. Find the total differential of the function \(z=\frac{x y}{y+x}\) where \(x\) changes from 10 to 10.5 and \(y\) changes from 15 to \(13 .\)

Complete each task. Let \(z=f(x, y)=x e^{y}\). Compute \(\Delta z\) from \(P(1,2)\) to \(Q(1.05,2.1)\) and then find the approximate change in \(z\) from point \(P\) to point Q. \(\quad\) Recall \(\Delta z=f(x+\Delta x, y+\Delta y)-f(x, y), \quad\) and \(d z\) and \(\Delta z\) are approximately equal.

Complete each task. The volume of a right circular cylinder is given by \(V(r, h)=\pi r^{2} h .\) Find the differential \(d V\). Interpret the formula geometrically.

Use the differential \(d z\) to approximate the change in \(z=\sqrt{4-x^{2}-y^{2}}\) as \((x, y)\) moves from point (1,1) to point (1.01,0.97) . Compare this approximation with the actual change in the function.

Let \(z=f(x, y)=x^{2}+3 x y-y^{2}\). Find the exact change in the function and the approximate change in the function as \(x\) changes from 2.00 to 2.05 and \(y\) changes from 3.00 to 2.96 .

The centripetal acceleration of a particle moving in a circle is given by \(a(r, v)=\frac{v^{2}}{r},\) where \(v\) is the velocity and \(r\) is the radius of the circle. Approximate the maximum percent error in measuring the acceleration resulting from errors of \(3 \%\) in \(v\) and \(2 \%\) in \(r\). (Recall that the percentage error is the ratio of the amount of error over the original amount. So, in this case, the percentage error in \(a\) is given by \(\frac{d a}{a} .\) )

The radius \(r\) and height \(h\) of a right circular cylinder are measured with possible errors of \(4 \%\) and \(5 \%\) respectively. Approximate the maximum possible percentage error in measuring the volume (Recall that the percentage error is the ratio of the amount of error over the original amount. So, in this case, the percentage error in \(V\) is given by \(\frac{d V}{V}\).)

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

The electrical resistance \(R\) produced by wiring resistors \(R_{1}\) and \(R_{2}\) in parallel can be calculated from the formula \(\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}\). If \(R_{1}\) and \(R_{2}\) are measured to be \(7 \Omega\) and \(6 \Omega,\) respectively, and if these measurements are accurate to within \(0.05 \Omega\), estimate the maximum possible error in computing \(R\). (The symbol \(\Omega\) represents an ohm, the unit of electrical resistance.)

The area of an ellipse with axes of length \(2 a\) and \(2 b\) is given by the formula \(A=\pi a b\). Approximate the percent change in the area when \(a\) increases by \(2 \%\) and \(b\) increases by \(1.5 \%\)

The period \(T\) of a simple pendulum with small oscillations is calculated from the formula \(T=2 \pi \sqrt{\frac{L}{g}}\), where \(L\) is the length of the pendulum and \(g\) is the acceleration resulting from gravity. Suppose that \(L\) 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 \(T\).

Electrical power \(P\) is given by \(P=\frac{V^{2}}{R},\) where \(V\) is the voltage and \(R\) is the resistance. Approximate the maximum percentage error in calculating power if 120 \(V\) is applied to a \(2000-\Omega\) resistor and the possible percent errors in measuring \(V\) and \(R\) are \(3 \%\) and \(4 \%\), respectively.

Find the linear approximation of each function at the indicated point. \(\quad f(x, y)=x \sqrt{y}, \quad P(1,4)\)

Find the linear approximation of each function at the indicated point. \(\quad f(x, y)=e^{x} \cos y ; P(0,0)\)

Find the linear approximation of each function at the indicated point. \(\quad f(x, y)=\arctan (x+2 y), P(1,0)\)

Find the linear approximation of each function at the indicated point. \(\quad f(x, y)=\sqrt{20-x^{2}-7 y^{2}}, \quad P(2,1)\)

Find the linear approximation of each function at the indicated point. \(\quad f(x, y, z)=\sqrt{x^{2}+y^{2}+z^{2}}, \quad P(3,2,6)\)

Find the equation for the tangent plane to the surface at the indicated point, and graph the surface and the tangent plane: \(z=\ln \left(10 x^{2}+2 y^{2}+1\right), P(0,0,0)\)

Find the equation of the tangent plane to the surface \(z=f(x, y)=\sin \left(x+y^{2}\right)\) at point \(\left(\frac{\pi}{4}, 0, \frac{\sqrt{2}}{2}\right),\) and graph the surface and the tangent plane.

Use the information provided to solve the problem. Let \(w(x, y, z)=x y \cos z, \quad\) where \(x=t, y=t^{2},\) and \(z=\arcsin t .\) Find \(\frac{d w}{d t}\)

Use the information provided to solve the problem. If \(w=5 x^{2}+2 y^{2}, x=-3 s+t,\) and \(y=s-4 t\), find \(\frac{\partial w}{\partial s}\) and \(\frac{\partial w}{\partial t}\).

Use the information provided to solve the problem. If \(w=x y^{2}, x=5 \cos (2 t), \quad\) and \(\quad y=5 \sin (2 t)\), find \(\frac{\partial w}{\partial t}\).

Use the information provided to solve the problem. If \(f(x, y)=x y, x=r \cos \theta, \quad\) and \(\quad y=r \sin \theta,\) find \(\frac{\partial f}{\partial r}\) and express the answer in terms of \(r\) and \(\theta\).

Find \(\frac{d f}{d t}\) using the chain rule and direct substitution. \(\quad f(x, y)=x^{2}+y^{2}, \quad x=t, y=t^{2}\)

Find \(\frac{d f}{d t}\) using the chain rule and direct substitution. \(\quad f(x, y)=\sqrt{x^{2}+y^{2}}, y=t^{2}, x=t\)

Find \(\frac{d f}{d t}\) using the chain rule and direct substitution. \(\quad f(x, y)=x y, x=1-\sqrt{t}, y=1+\sqrt{t}\)

Find \(\frac{d f}{d t}\) using the chain rule and direct substitution. \(\quad f(x, y)=\frac{x}{y}, x=e^{t}, y=2 e^{t}\)

Find \(\frac{d f}{d t}\) using the chain rule and direct substitution. \(\quad f(x, y)=\ln (x+y), \quad x=e^{t}, y=e^{t}\)

Find \(\frac{d f}{d t}\) using the chain rule and direct substitution. \(\quad f(x, y)=x^{4}, \quad x=t, y=t\)

Let \(w(x, y, z)=x^{2}+y^{2}+z^{2}\) \(x=\cos t, y=\sin t, \quad\) and \(z=e^{t} .\) Express \(w\) as a function of \(t\) and find \(\frac{d w}{d t}\) directly. Then, find \(\frac{d w}{d t}\) using the chain rule.

Let \(z=x^{2} y,\) where \(x=t^{2}\) and \(y=t^{3} .\) Find \(\frac{d z}{d t}\).

Let \(u=e^{x} \sin y,\) where \(x=t^{2}\) and \(y=\pi t\). Find \(\frac{d u}{d t}\) when \(x=\ln 2\) and \(y=\frac{\pi}{4}\).

Find \(\frac{d y}{d x}\) using partial derivatives. \(\sin (6 x)+\tan (8 y)+5=0\)

Find \(\frac{d y}{d x}\) using partial derivatives. \(x^{3}+y^{2} x-3=0\)

Find \(\frac{d y}{d x}\) using partial derivatives. \(\quad \sin (x+y)+\cos (x-y)=4\)

Find \(\frac{d y}{d x}\) using partial derivatives. \(x^{2}-2 x y+y^{4}=4\)

Find \(\frac{d y}{d x}\) using partial derivatives. \(x e^{y}+y e^{x}-2 x^{2} y=0\)

Find \(\frac{d y}{d x}\) using partial derivatives. \(x^{2 / 3}+y^{2 / 3}=a^{2 / 3}\)