 14.6.1: A contour map of barometric pressure (in millibars) is shown for 7:...
 14.6.2: The contour map shows the average annual snowfall (in inches) near ...
 14.6.3: A table of values for the windchill index is given in Exercise 3 o...
 14.6.4: Find the directional derivative of at the given point in the direct...
 14.6.5: Find the directional derivative of at the given point in the direct...
 14.6.6: Find the directional derivative of at the given point in the direct...
 14.6.7: (a) Find the gradient of . (b) Evaluate the gradient at the point ....
 14.6.8: (a) Find the gradient of . (b) Evaluate the gradient at the point ....
 14.6.9: (a) Find the gradient of . (b) Evaluate the gradient at the point ....
 14.6.10: (a) Find the gradient of . (b) Evaluate the gradient at the point ....
 14.6.11: Find the directional derivative of the function at the given point ...
 14.6.12: Find the directional derivative of the function at the given point ...
 14.6.13: Find the directional derivative of the function at the given point ...
 14.6.14: Find the directional derivative of the function at the given point ...
 14.6.15: Find the directional derivative of the function at the given point ...
 14.6.16: Find the directional derivative of the function at the given point ...
 14.6.17: Find the directional derivative of the function at the given point ...
 14.6.18: Use the figure to estimate .
 14.6.19: Find the directional derivative of at in the direction of .
 14.6.20: Find the directional derivative of at in the direction of the origin.
 14.6.21: Find the maximum rate of change of at the given point and the direc...
 14.6.22: Find the maximum rate of change of at the given point and the direc...
 14.6.23: Find the maximum rate of change of at the given point and the direc...
 14.6.24: Find the maximum rate of change of at the given point and the direc...
 14.6.25: Find the maximum rate of change of at the given point and the direc...
 14.6.26: Find the maximum rate of change of at the given point and the direc...
 14.6.27: (a) Show that a differentiable function decreases most rapidly at i...
 14.6.28: Find the directions in which the directional derivative of at the p...
 14.6.29: Find all points at which the direction of fastest change of the fun...
 14.6.30: Near a buoy, the depth of a lake at the point with coordinates is ,...
 14.6.31: The temperature in a metal ball is inversely proportional to the di...
 14.6.32: The temperature at a point is given by where is measured in and , ,...
 14.6.33: Suppose that over a certain region of space the electrical potentia...
 14.6.34: Suppose you are climbing a hill whose shape is given by the equatio...
 14.6.35: Let be a function of two variables that has continuous yz lnx z 0, ...
 14.6.36: For the given contour map draw the curves of steepest ascent starti...
 14.6.37: Show that the operation of taking the gradient of a function has th...
 14.6.38: Sketch the gradient vector for the function whose level curves are ...
 14.6.39: Find equations of (a) the tangent plane and (b) the normal line to ...
 14.6.40: Find equations of (a) the tangent plane and (b) the normal line to ...
 14.6.41: Find equations of (a) the tangent plane and (b) the normal line to ...
 14.6.42: Find equations of (a) the tangent plane and (b) the normal line to ...
 14.6.43: Find equations of (a) the tangent plane and (b) the normal line to ...
 14.6.44: Find equations of (a) the tangent plane and (b) the normal line to ...
 14.6.45: Use a computer to graph the surface, the tangent plane, and the nor...
 14.6.46: Use a computer to graph the surface, the tangent plane, and the nor...
 14.6.47: If , find the gradient vector and use it to find the tangent line t...
 14.6.48: If , find the gradient vector and use it to find the tangent line t...
 14.6.49: Show that the equation of the tangent plane to the ellipsoid at the...
 14.6.50: Find the equation of the tangent plane to the hyperboloid at and ex...
 14.6.51: Show that the equation of the tangent plane to the elliptic parabol...
 14.6.52: Find the points on the ellipsoid where the tangent plane is paralle...
 14.6.53: Find the points on the hyperboloid where the normal line is paralle...
 14.6.54: Show that the ellipsoid and the sphere are tangent to each other at...
 14.6.55: Show that every plane that is tangent to the cone passes through th...
 14.6.56: Show that every normal line to the sphere passes through the center...
 14.6.57: Show that the sum of the , , and intercepts of any tangent plane...
 14.6.58: Show that the product of the , , and intercepts of any tangent p...
 14.6.59: Find parametric equations for the tangent line to the curve of inte...
 14.6.60: (a) The plane intersects the cylinder in an ellipse. Find parametri...
 14.6.61: (a) Two surfaces are called orthogonal at a point of intersection i...
 14.6.62: (a) Show that the function is continuous and the partial derivative...
 14.6.63: Suppose that the directional derivatives of are known at a given po...
 14.6.64: Show that if is differentiable at then [Hint: Use Definition 14.4.7...
Solutions for Chapter 14.6: Directional Derivatives and the Gradient Vector
Full solutions for Calculus,  5th Edition
ISBN: 9780534393397
Solutions for Chapter 14.6: Directional Derivatives and the Gradient Vector
Get Full SolutionsThis textbook survival guide was created for the textbook: Calculus,, edition: 5. Chapter 14.6: Directional Derivatives and the Gradient Vector includes 64 full stepbystep solutions. Calculus, was written by and is associated to the ISBN: 9780534393397. This expansive textbook survival guide covers the following chapters and their solutions. Since 64 problems in chapter 14.6: Directional Derivatives and the Gradient Vector have been answered, more than 47516 students have viewed full stepbystep solutions from this chapter.

Angle of depression
The acute angle formed by the line of sight (downward) and the horizontal

Arcsecant function
See Inverse secant function.

Complex conjugates
Complex numbers a + bi and a  bi

Compounded monthly
See Compounded k times per year.

Convergence of a sequence
A sequence {an} converges to a if limn: q an = a

Difference of functions
(ƒ  g)(x) = ƒ(x)  g(x)

Focal axis
The line through the focus and perpendicular to the directrix of a conic.

Focus, foci
See Ellipse, Hyperbola, Parabola.

Higherdegree polynomial function
A polynomial function whose degree is ? 3

Natural logarithm
A logarithm with base e.

Probability distribution
The collection of probabilities of outcomes in a sample space assigned by a probability function.

Real number line
A horizontal line that represents the set of real numbers.

Right triangle
A triangle with a 90° angle.

Sum of complex numbers
(a + bi) + (c + di) = (a + c) + (b + d)i

Symmetric about the origin
A graph in which (x, y) is on the the graph whenever (x, y) is; or a graph in which (r, ?) or (r, ? + ?) is on the graph whenever (r, ?) is

Transpose of a matrix
The matrix AT obtained by interchanging the rows and columns of A.

Tree diagram
A visualization of the Multiplication Principle of Probability.

Xmin
The xvalue of the left side of the viewing window,.

xzplane
The points x, 0, z in Cartesian space.

Zero matrix
A matrix consisting entirely of zeros.