144
Describe the graphs of:
Describe some appropriate level surfaces and sections of the graphs of:
Compute the derivative \({\bf D}\! f ({\bf x})\) of each of the following functions:
Suppose \(f (x,y) = f(y,x)\) for all \((x,y)\). Prove that \[ ( \partial f / \partial x) (a,b)= ( \partial f / \partial y) (b,a). \]
Let \(f(u, v)=(\cos u, v+\sin u)\) and \(g(x, y, z)=(x^2+\pi y^2, xz)\). Compute \(D(f \circ g)\) at (0, 1, 1) using the chain rule.
Use the chain rule to find \(D(f \circ g)(-2, 1)\) for \(f(u, v, w)=(v^2+uw, u^2+w^2, u^2v-w^3)\) and \(g(x, y)=(xy^3, x^2-y^2, 3x+5y)\).
Use the chain rule to find \(D(f \circ g)(-1, 2)\) for \(f(u, v, w)=(v^2+w^2, u^3-vw, u^2v+w)\) and \(g(x, y)=(3x+2y, x^3y, y^2-x^2)\).
Let \(f(x, y)=(xy, \frac{x}{y}, x+y)\) and \(g(w, s, t)=(we^s, se^{wt})\). Find \(D(f \circ g)(3, 1, 0)\).
Let \(\textbf{r}(t)=(t\cos(\pi t), t\sin(\pi t), t)\) be a path. Where will the tangent line to \(\textbf{r}\) at \(t=5\) intersect the \(xy\) plane?
Let \(f(x, y)=x^2e^{-xy}\).
Let \(f(x,y) = (1- x^2 - y^2)^{1/2}\). 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 \((x_0, y_0 ,f (x_0, y_0))\). Interpret geometrically.
Let \(F(u,v)\) and \(u=h(x,y,z)\), \(v=k(x,y,z)\) be given (differentiable) real-valued functions and let \(f(x,y,z)\) be defined by \(f(x,y,z)=F(h(x,y,z)\), \(k(x,y,z))\). Write a formula for the gradient of \(f\) in terms of the partial derivatives of \(F, h\), and \(k\).
Find an equation for the tangent plane of the graph of \(f\) at the point \((x_0, y_0, f (x_0, y_0))\) for:
Compute an equation for the tangent planes of the following surfaces at the indicated points.
Draw some level curves for the following functions:
Consider a temperature function \(T (x,y)= x \sin y\). Plot a few level curves. Compute \(\nabla T\) and explain its meaning.
Find the following limits if they exist:
Compute the first partial derivatives and gradients of the following functions:
Compute \(\displaystyle\frac{\partial}{\partial x}[x\exp\,(1+x^2+y^2)]\)
145
Let \(f \colon \mathbb{R}^2 \to \mathbb{R}^4\) and \(g \colon \mathbb{R}^2 \to \mathbb{R}^2\) be given by \(f(x, z)=(x^2-y^2, 0, \sin(xy), 1)\) and \(g(x, y)=(ye^{x^2}, xe^{y^2})\). Compute \(D(f \circ g)(1, 2)\).
Let \(f(x, y)=(x^2+y^2)e^{-(x^2+y^2+10)}\). Find the rate of change of \(f\) at (2, 1) in the direction pointing toward the origin.
Let \(y (x)\) be a differentiable function defined implicitly by \(F (x,y(x)) =0\). From Exercise 19(a), Section 2.6, we know that \[ \frac{dy}{dx} = - \frac{ \partial F / \partial x}{ \partial F/ \partial y}. \]
Consider the surface \(z = F (x,y)\), and suppose \(F\) is increasing as a function of \(x\) and as a function of \(y\); that is, \(\partial F/ \partial x > 0\) and \(\partial F / \partial y > 0\). By considering the graph and the plane \(z=0\), show that for \(z\) fixed at \(z=0\), \(y\) should decrease as \(x\) increases and \(x\) should decrease as \(y\) increases. Does this agree with the minus sign in the formula for \(dy/dx\)?
Consider the graph of a function \(f (x,y)\) [Figure 2.56]. Let \((x_0, y_0)\) lie on a level curve \(C\), so \(\nabla \! f (x_0, y_0)\) is perpendicular to this curve. Show that the tangent plane of the graph is the plane that (i) contains the line perpendicular to \(\nabla \! f (x_0, y_0)\) and lying in the horizontal plane \(z = f (x_0, y_0)\), and (ii) has slope \( \| \nabla \! f (x_0, y_0) \|\) relative to the \(xy\) plane.
(By the slope of a plane \(P\) relative to the \(xy\) plane we mean the tangent of the angle \(\theta, 0 \le \theta \le \pi\), between the upward-pointing normal \({\bf p}\) to \(P\) and the unit vector \({\bf k}\).)
Find the plane tangent to the surface \(z = x^2 + y^2\) at the point \((1, -2,5)\). Explain the geometric significance, for this surface, of the gradient of \(f (x,y) = x^2 + y^2\) (see Exercise 23).
In which direction is the directional derivative of \(f (x,y) = (x^2 - y^2) / (x^2 + y^2)\) at (1, 1) equal to zero?
Find the directional derivative of the given function at the given point and in the direction of the given vector.
Find the tangent plane and normal to the hyperboloid \(x^2 + y^2 - z^2 = 18\) at \((3,5,-4)\).
Let \((x (t), y(t))\) be a path in the plane, \(0 \le t \le 1\), and let \(f(x,y)\) be a \(C^1\) function of two variables. Assume that \((dx/dt)f_x+(dy/dt)f_y \le 0\). Show that \(f (x (1), y (1)) \le f (x (0), y(0))\).
146
A bug finds itself in a toxic environment. The toxicity level is given by \(T (x,y) = 2x^2 - 4y^2\). The bug is at \((-1,2)\). In what direction should it move to lower the toxicity the fastest?
Find the direction in which the function \(w = x^2 + xy\) increases most rapidly at the point \((-1,1)\). What is the magnitude of \(\nabla w\) at this point? Interpret this magnitude geometrically.
Let \(f\) be defined on an open set \(S\) in \({\mathbb R}^n\). We say that \(f\) is homogeneous of degree \(p\) over \(S\) if \(f ( \lambda {\bf x})= \lambda^p f({\bf x})\) for every real \(\lambda\) and for every \({\bf x}\) in \(S\) for which \(\lambda {\bf x} \in S\).
If \(z= [ f(x-y) ]/y\) (where \(f\) is differentiable and \(y\neq 0\)), show that the identity \(z + y( \partial z / \partial x) + y ( \partial z / \partial y) =0\) holds.
Given \(z= f((x +y)/ (x-y))\) for \(f\) a \(C^1\) function, show that \[ x \frac{\partial z}{\partial x} + y \frac{\partial z}{ \partial y} =0. \]
Let \(f\) have partial derivatives \(\partial f ( {\bf x}) / \partial x_i\), where \(i=1,2, \ldots , n\), at each point \({\bf x}\) of an open set \(U\) in \({\mathbb R}^n\). If \(f\) has a local maximum or a local minimum at the point \({\bf x}_0\) in \(U\), show that \(\partial f ({\bf x}_0) / \partial x_i =0\) for each \(i\).
Consider the functions defined in \({\mathbb R}^2\) by the following formulas:
Compute the gradient vector \(\nabla \! f(x,y)\) at all points \((x,y)\) in \({\mathbb R}^2\) for each of the following functions:
Find the directional derivatives of the following functions at the point \((1, 1)\) in the direction \(({\bf i} + {\bf j})/ \sqrt{2}\):
Let \(h (x,y) = 2 e^{-x^2} + e^{-3y^2}\) denote the height on a mountain at position \((x,y)\). In what direction from \((1, 0)\) should one begin walking in order to climb the fastest?
Compute an equation for the plane tangent to the graph of \[ f(x,y) = \frac{e^x}{x^2 + y^2} \] at \(x=1, y=2\).
At time \(t=0\), a particle is ejected from the surface \(x^2 + 2 y^2 + 3 z^2 =6\) at the point \((1, 1, 1)\) in a direction normal to the surface at a speed of 10 units per second. At what time does it cross the sphere \(x^2 + y^2 + z^2 =103\)?
At what point(s) on the surface in Exercise 43 is the normal vector parallel to the line \(x=y=z\)?
Compute \(\partial z / \partial x\) and \(\partial z / \partial y\) if \[ z = \frac{u^2 + v^2}{u^2 - v^2}, \qquad u = {\mathop e}^{-x-y}, \qquad v = e^{xy} \] (a) by substitution and direct calculation, and (b) by the chain rule.
147
Compute the partial derivatives as in Exercise 45 if \(z= uv, u = x+y,\) and \(v=x-y\).
What is wrong with the following argument? Suppose that \(w = f(x,y)\) and \(y = x^2\). By the chain rule, \[ \frac{\partial w}{\partial x} = \frac{\partial w}{\partial x} \frac{\partial x}{\partial x} + \frac{\partial w\, \partial y}{\partial y\, \partial x} = \frac{\partial w}{\partial x} + 2x \frac{\partial w}{\partial y} . \]
Hence, \(0 = 2x ( \partial w/ \partial y)\), and so \(\partial w / \partial y =0\). Choose an explicit example to really see that this is incorrect.
A boat is sailing northeast at 20 km/h. Assuming that the temperature drops at a rate of 0.2\(^\circ\)C/km in the northerly direction and 0.3\(^\circ\)C/km in the easterly direction, what is the time rate of change of temperature as observed on the boat?
Use the chain rule to find a formula for \((d /dt) \,{\exp}\, [f(t) g(t)].\)
Use the chain rule to find a formula for \((d/ dt) ( f (t)^{g(t)})\).
Verify the chain rule for the function \(f(x,y,z) = [ \ln\, (1+ x^2 + 2 z^2)] /(1 + y^2)\) and the path \({\bf c} (t) = (t, 1- t^2, \cos t)\).
Verify the chain rule for the function \(f(x,y) = x^2 / (2 + \cos y)\) and the path \(x = e^t,\) \(y= e^{-t}\).
Suppose that \(u (x,t)\) satisfies the differential equation \(u_t + uu_x =0\) and that \(x\), as a function \(x= f(t)\) of \(t\), satisfies \(dx/dt = u(x,t)\). Prove that \(u ( f(t),t)\) is constant in \(t\).
The displacement at time \(t\) and horizontal position on a line \(x\) of a certain violin string is given by \(u = \sin\, ( x- 6t) + \sin\, ( x+ 6t)\). Calculate the velocity of the string at \(x=1\) when \(t = \frac{1}{3}\).
The ideal gas law \(PV = nRT\) involves a constant \(R\), the number \(n\) of moles of the gas, the volume \(V\), the Kelvin temperature \(T\), and the pressure \(P\).
The potential temperature \(\theta\) is defined in terms of temperature \(T\) and pressure \(p\) by \[ \theta = T \left( \frac{1000}{p} \right)^{0.286}. \]
The temperature and pressure may be thought of as functions of position \((x,y,z)\) in the atmosphere and also of time \(t\).
The specific volume \(V\), pressure \(P\), and temperature \(T\) of a van der Waals gas are related by \(P = RT / ( V- \beta ) - \alpha / V^2\), where \(\alpha, \beta,\) and \(R\) are constants.
The height \(h\) of the Hawaiian volcano Mauna Loa is (roughly) described by the function \(h (x,y) = 2.59 - 0.00024y^2 - 0.00065 x^2\), where \(h\) is the height above sea level in miles and \(x\) and \(y\) measure east–west and north–south distances in miles from the top of the mountain. At \((x,y) = (-2, -4)\):
148
Suppose that \(f\) is a differentiable function of one variable and that a function \(u = g (x,y)\) is defined by \[ u = g (x,y)= xyf \left( \frac{x+y}{xy} \right). \]
Show that \(u\) satisfies a (partial) differential equation of the form \[ x^2 \frac{ \partial u}{ \partial x} - y^2 \frac{\partial u}{ \partial y}= G (x,y) u \] and find the function \(G (x,y)\).
1Some mathematicians would write such an \(f\) in boldface, using the notation \({\bf f}({\bf x})\), because the function is vector-valued. We did not do so, as a matter of personal taste. We use boldface primarily for mappings that are vector fields, introduced later. The notion of function was developed over many centuries, with the definition extended to cover more cases as they arose. For example, in 1667 James Gregory defined a function as “a quantity obtained from other quantities by a succession of algebraic operations or by any other operation imaginable.” In 1755 Euler gave the following definition: “If some quantities depend on others in such a way as to undergo variation when the latter are varied then the former are called functions of the latter.”
2It turns out that we need to postulate the existence of only some matrix giving the best linear approximation near \({\bf x}_0\in {\mathbb R}^n\), because in fact this matrix is necessarily the matrix whose \(ij\)th entry is \(\partial f_i/\partial x_j\) (see the Internet supplement for Chapter 2).
3If \(t\) lies at the endpoint of an interval, we should, as in one-variable calculus, take right- or left-handed limits.
4See S. M. Binder, “Mathematical Methods in Elementary Thermodynamics,” J. Chem. Educ., 43 (1966): 85–92. A proper understanding of partial differentiation can be of significant use in applications; for example, see M. Feinberg, “Constitutive Equation for Ideal Gas Mixtures and Ideal Solutions as Consequences of Simple Postulates,” Chem. Eng. Sci., 32 (1977): 75–78.
5This discussion assumes that one walks at the same speed in all directions. Of course, hikers know that this is not necessarily realistic.