Question 13.84

Find \(\partial f/\partial x,\partial f /\partial y\) if

• (a) \(f(x,y)=xy\)
• (b) \(f(x,y)=e^{xy}\)
• (c) \(f(x,y)=x\cos x \cos y\)
• (d) \(f(x,y)=(x^2+y^2)\log\,(x^2+y^2)\)

Question 13.85

Evaluate the partial derivatives \(\partial z/\partial x,\partial z /\partial y\) for the given function at the indicated points.

• (a) \(z=\sqrt{a^2-x^2-y^2};(0,0),(a/2,a/2)\)
• (b) \(z=\log\,\sqrt{1+xy};(1,2),(0,0)\)
• (c) \(z=e^{ax}\cos\,(bx+y);(2\pi/b,0)\)

Question 13.86

In each case following, find the partial derivatives \(\partial w/\partial x,\partial w/\partial y\).

• (a) \(w=xe^{x^2+\,y^2}\)
• (b) \(w=\displaystyle \frac{x^2+y^2}{x^2-y^2}\)
• (c) \(w=e^{xy}\log\,(x^2+y^2)\)
• (d) \(w=x/y\)
• (e) \(w=\cos\, (ye^{xy})\sin x\)

Question 13.87

Decide which of the following functions are \(C^1\), which are just differentiable.

• (a) \(f(x,y) = \displaystyle \frac{2xy}{(x^2 + y^2)^2}\)
• (b) \(f(x,y) = \displaystyle \frac{x}{y} + \frac{y}{x}\)
• (c) \(f(r,\theta) = {\textstyle \frac{1}{2}} r \sin 2 \theta, r > 0\)
• (d) \(f(x,y) =\displaystyle \frac{xy}{\sqrt{x^2 + y^2}}\)
• (e) \(f(x,y) = \displaystyle \frac{x^2 y}{x^4 + y^2}\)

Question 13.88

Find the equation of the plane tangent to the surface \(z= x^2 + y^3\) at \((3,1,10)\).

Question 13.89

Let \(f(x, y)=e^{x+y}\). Find the equation for the tangent plane to the graph of \(f\) at the point (0, 0).

Question 13.90

Let \(f(x, y)=e^{x-y}\). Find the equation for the tangent plane to the graph of \(f\) at the point (1, 1).

Question 13.91

Using the respective functions in Exercise 1, compute the plane tangent to the graphs at the indicated points.

• (a) \((0,0)\)
• (b) \((0,1)\)
• (c) \((0, \pi)\)
• (d) \((0, 1)\)

Question 13.92

Compute the matrix of partial derivatives of the following functions:

• (a) \(f\colon\, {\mathbb R}^2 \rightarrow {\mathbb R}^2 , f(x,y) = (x,y)\)
• (b) \(f\colon\, {\mathbb R}^2 \rightarrow {\mathbb R}^3 , f(x,y) = (xe^y + \cos y, x, x+ e^y)\)
• (c) \(f\colon\, {\mathbb R}^3 \rightarrow {\mathbb R}^2 , f(x,y,z) = (x + e^z + y,y x^2)\)
• (d) \(f\colon\, {\mathbb R}^2 \rightarrow {\mathbb R}^3 , f(x,y) = (xy e^{xy}, x \sin y, 5xy^2)\)

Question 13.93

Compute the matrix of partial derivatives of

• (a) \(f(x,y) = ( e^x, \sin xy)\)
• (b) \(f(x,y,z) = ( x- y , y + z)\)
• (c) \(f(x,y) = ( x+ y, x- y, xy)\)
• (d) \(f(x,y,z) = ( x+z, y - 5z, x-y)\)

Question 13.94

Find the equation of the tangent plane to \(f(x, y)=x^2-2xy+2y^2\) having slope 2 in the positive \(x\) direction and slope 4 in the positive \(y\) direction.

Question 13.95

Let \(f(x, y)=e^{(2x+3y)}\).

• (a) Find the tangent plane to \(f\) at (0, 0).
• (b) Use this to approximate \(f(.1, 0)\) and \(f(0, .1)\).
• (c) With a calculator, find the exact values of \(f(.1,0)\) and \(f(0,.1)\).

Question 13.96

Where does the plane tangent to \(z= e^{x-y}\) at \((1,1,1)\) meet the \(z\) axis?

Question 13.97

Why should the graphs of \(f(x,y)= x^2 + y^2\) and \(g(x,y) = - x^2 - y^2 + xy^3\) be called “tangent” at \((0, 0)\)?

Question 13.98

Let \(f(x,y) = e^{xy}\). Show that \(x ( \partial f/ \partial x) = y ( \partial f / \partial y)\).

Question 13.99

Use the linear approximation to approximate a suitable function \(f(x,y)\) and thereby estimate the following:

• (a) \((0.99 e^{0.02})^8\)
• (b) \((0.99)^3+ (2.01)^3 - 6 (0.99) (2.01)\)
• (c) \(\sqrt{(4.01)^2 + (3.98)^2 + (2.02)^2}\)

Question 13.100

Let \(P\) be the tangent plane to the graph of \(g(x,y)=8-2x^2-3y^2\) at the point (1, 2, \(-\)6). Let \(f(x,y)=4-x^2-y^2\). Find the point on the graph of \(f\) which has tangent plane parallel to \(P\).

Question 13.101

Let \(f(x, y)=xe^{y^2}-ye^{x^2}\).

• (a) Find the equation for the tangent plane to the graph of \(f\) at (1, 2).
• (b) What point on the surface \(z=x^2-y^2\) has a tangent plane parallel to the plane found in part (a)?

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Question 13.102

Compute the gradients of the following functions:

• (a) \(f(x,y,z) = x \exp\,( - x^2 - y^2 - z^2)\,\) (Note that \(\exp u = e^u.)\)
• (b) \(f(x,y,z) = \displaystyle \frac{xyz}{x^2 + y^2 + z^2}\)
• (c) \(f(x,y,z) = z^2 e^x \cos y\)

Question 13.103

Compute the tangent plane at (1, 0, 1) for each of the functions in Exercise 19. [The solution to part (c) only is in the Study Guide.]

Question 13.104

Find the equation of the tangent plane to \(z = x^2 + 2 y^3\) at \((1,1,3)\).

Question 13.105

Let \[ f(x,y)= \left\{ \begin{array}{cc} \frac{x^2y^4}{x^4+6y^8} & \hbox{if } (x,y)\neq (0,0) \\ 0 & \hbox{if } (x,y)=(0,0) \end{array} \right. \]

• (a) Show that \(\frac{\partial f}{\partial x}(0,0)\) and \(\frac{\partial f}{\partial y}(0,0)\) exist.
• (b) Show that \(f\) is not differentiable at (0,0) by showing that \(f\) is not continuous at (0,0).

Question 13.106

Let \(P\) be the tangent plane to \(f(x,y)= x^2y^3\) at (1, 2, 8). Let \(l\) be the line contained in \(P\) which passes through the point (1, 3, 20) and passes directly above (2, 1). That is, \(l\) contains the point (1, 3, 20) and a point of the form \((2, 1, z)\). Find a parametrization for \(l\).

Question 13.107

Calculate \({\nabla} h (1,1,1)\) if \(h (x,y,z) = (x +z ) e^{x-y}.\)

Question 13.108

Let \(f(x,y,z) = x^2 + y^2 - z^2\). Calculate \({\nabla} f (0,0,1)\).

Question 13.109

Evaluate the gradient of \(f(x,y,z) = \log\,( x^2 + y^2 + z^2)\) at \((1,0,1)\).

Question 13.110

Describe all Hölder-continuous functions with \(\alpha >1\) (see Exercise 33, Section 13.2). (HINT: What is the derivative of such a function?)

Question 13.111

Suppose \(f\colon\, {\mathbb R}^n \rightarrow {\mathbb R}^m\) is a linear map. What is the derivative of \(f\)?