142
Show that the directional derivative of \(f(x,y,z)=z^2x+y^3\) at \((1, 1, 2)\) in the direction \((1/\sqrt{5}){\bf i}+(2/\sqrt{5}){\bf j}\) is \(2\sqrt{5}\).
Compute the directional derivatives of the following functions at the indicated points in the given directions:
Compute the directional derivatives of the following functions along unit vectors at the indicated points in directions parallel to the given vector:
You are walking on the graph of \(f(x,y)= y\cos(\pi x) -x\cos(\pi y) +10\), standing at the point (2, 1, 13). Find an \(x,y\)-direction you should walk in to stay at the same level.
Find a vector which is normal to the curve \(x^3+xy+y^3=11\) at (1, 2).
Find the rate of change of \(f(x,y,z)=xyz\) in the direction normal to the surface \(yx^2+xy^2+yz^2=3\) at (1, 1, 1).
Find the planes tangent to the following surfaces at the indicated points:
Find the equation for the plane tangent to each surface \(z=f(x,y)\) at the indicated point:
Compute the gradient \(\nabla \! f\) for each of the following functions:
For the functions in Exercise 10, what is the direction of fastest increase at \((1, 1, 1)\)? [The solution to part (c) only is in the Study Guide to this text.]
Show that a unit normal to the surface \(x^3y^3+y-z+2=0\) at (0, 0, 2) is given by \({\bf n}=(1/\sqrt{2})(\,{\bf j}\,-\,{\bf k})\).
Find a unit normal to the surface \(\cos (xy) = e^z-2\) at \((1,\pi,0)\).
Verify Theorems 13 and 14 for \(f(x,y,z)=x^2+y^2+z^2\).
Show that the definition following Theorem 14 yields, as a special case, the formula for the plane tangent to the graph of \(f(x,y)\) by regarding the graph as a level surface of \(F(x,y,z)=f(x,y)-z\) (see Section 2.3).
Let \(f(x,y)=-(1-x^2-y^2)^{1/2}\) for \((x,y)\) such that \(x^2+y^2<1\). Show that the plane tangent to the graph of \(f\) at \((x_0,y_0,f(x_0,y_0))\) is orthogonal to the vector with components \((x_0,y_0,f(x_0,y_0))\). Interpret this geometrically.
For the following functions \(f\colon\, {\mathbb R}^3\to {\mathbb R}\) and \({\bf g}\colon\, {\mathbb R}\to {\mathbb R}^3\), find \(\nabla \! f\) and \({\bf g}'\) and evaluate \((f\circ {\bf g})'(1)\).
143
Compute the directional derivative of \(f\) in the given directions \({\bf v}\) at the given points P.
You are standing on the graph of \(f(x, y)=100-2x^2-3y^2\) at the point (2, 3, 65).
Find the two points on the hyperboloid \(x^2+4y^2-z^2=4\), where the tangent plane is parallel to the plane \(2x+2y+z=5\).
Let \({\bf r}=x{\bf i}+y{\bf j}+z{\bf k}\) and \(r= \| {\bf r} \| .\) Prove that \[ \nabla \Big(\frac{1}{r}\Big)=-\frac{{\bf r}}{r^3}. \]
Captain Ralph is in trouble near the sunny side of Mercury. The temperature of the ship’s hull when he is at location \((x,y,z)\) will be given by \(T(x,y,z)=e^{-x^2-2y^2-3z^2}\), where \(x,y\), and \(z\) are measured in meters. He is currently at \((1, 1, 1)\).
A function \(f\colon\, {\mathbb R}^2\to {\mathbb R}\) is said to be independent of the second variable if there is a function \(g\colon\, {\mathbb R}\to {\mathbb R}\) such that \(f(x,y)=g(x)\) for all \(x\) in \({\mathbb R}\). In this case, calculate \(\nabla \! f\) in terms of \(g'\).
Let \(f\) and \(g\) be functions from \({\mathbb R}^3\) to \({\mathbb R}\). Suppose \(f\) is differentiable and \(\nabla \! f({\bf x})=g({\bf x}){\bf x}\). Show that spheres centered at the origin are contained in the level sets for \(f\); that is, \(f\) is constant on such spheres.
A function \(f\colon\, {\mathbb R}^n\to {\mathbb R}\) is called an even function if \(f({\bf x})=f(-{\bf x})\) for every \({\bf x}\) in \({\mathbb R}^n\). If \(f\) is differentiable and even, find \({\bf D}f\) at the origin.
Suppose that a mountain has the shape of an elliptic paraboloid \(z=c-ax^2-by^2\), where \(a,b\), and \(c\) are positive constants, \(x\) and \(y\) are the east–west and north–south map coordinates, and \(z\) is the altitude above sea level \((x,y,z\) are all measured in meters). At the point (1, 1), in what direction is the altitude increasing most rapidly? If a marble were released at (1, 1), in what direction would it begin to roll?
An engineer wishes to build a railroad up the mountain of Exercise 26. Straight up the mountain is much too steep for the power of the engines. At the point (1, 1), in what directions may the track be laid so that it will be climbing with a 3&percent; grade—that is, an angle whose tangent is 0.03? (There are two possibilities.) Make a sketch of the situation indicating the two possible directions for a 3&percent; grade at (1, 1).
In electrostatics, the force P of attraction between two particles of opposite charge is given by \({\bf P}=k({\bf r}/ \| {\bf r} \| ^3)\) (Coulomb’s law), where \(k\) is a constant and \({\bf r}=x{\bf i}+y{\bf j}+z{\bf k}\). Show that P is the gradient of \(f=-k/ \| {\bf r} \|\).
The electrostatic potential \(V\) due to two infinite parallel filaments with linear charge densities \(\lambda\) and \(-\lambda\) is \(V=(\lambda/2\pi\varepsilon_0)\ln \,(r_2/r_1)\), where \(r_1^2=(x-x_0)^2+y^2\) and \(r_2^2=(x+x_0)^2+y^2\). We think of the filaments as being in the \(z\)-direction, passing through the \(xy\) plane at \((-x_0,0)\) and \((x_0,0)\). Find \(\nabla V(x,y)\).
For each of the following, find the maximum and minimum values attained by the function \(f\) along the path \({\bf c} (t)\):
Suppose that a particle is ejected from the surface \(x^2+y^2-z^2=-1\) at the point \((1,1,\sqrt{3})\) along the normal directed toward the \(xy\) plane to the surface at time \(t=0\) with a speed of 10 units per second. When and where does it cross the \(xy\) plane?
Let \(f\colon\, {\mathbb R}^3\to {\mathbb R}\) and regard \({\bf D}\! f(x,y,z)\) as a linear map of \({\mathbb R}^3\) to \({\mathbb R}\). Show that the kernel (that is, the set of vectors mapped to zero) of \({\bf D}\! f\) is the plane in \({\mathbb R}^3\) orthogonal to \(\nabla \! f\).