Question 2.78

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 2.79

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 2.80

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 2.81

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 2.82

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

Question 2.83

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 2.84

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 2.85

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 2.86

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 2.87

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 2.88

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 2.89

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 2.90

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

Question 2.91

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 2.92

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

Question 2.93

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 2.94

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 2.95

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 2.96

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 2.97

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 2.98

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

Question 2.99

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 2.100

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 2.101

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

Question 2.102

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

Question 2.103

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

Question 2.104

Describe all Hölder-continuous functions with \(\alpha >1\) (see Exercise 33, Section 2.2). (HINT: What is the derivative of such a function?)

Question 2.105

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