 14.6.1: Level curves for barometric pressure (in millibars) are shown for 6...
 14.6.2: The contour map shows the average maximum temperature for November ...
 14.6.3: A table of values for the windchill index is given in Exercise 3 o...
 14.6.4: 46 Find the directional derivative of at the given point in the dir...
 14.6.5: 46 Find the directional derivative of at the given point in the dir...
 14.6.6: 46 Find the directional derivative of at the given point in the dir...
 14.6.7: 710 (a) Find the gradient of . (b) Evaluate the gradient at the poi...
 14.6.8: 710 (a) Find the gradient of . (b) Evaluate the gradient at the poi...
 14.6.9: 710 (a) Find the gradient of . (b) Evaluate the gradient at the poi...
 14.6.10: 710 (a) Find the gradient of . (b) Evaluate the gradient at the poi...
 14.6.11: 1117 Find the directional derivative of the function at the given p...
 14.6.12: 1117 Find the directional derivative of the function at the given p...
 14.6.13: 1117 Find the directional derivative of the function at the given p...
 14.6.14: 1117 Find the directional derivative of the function at the given p...
 14.6.15: 1117 Find the directional derivative of the function at the given p...
 14.6.16: 1117 Find the directional derivative of the function at the given p...
 14.6.17: 1117 Find the directional derivative of the function at the given p...
 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 .
 14.6.21: 2126 Find the maximum rate of change of at the given point and the ...
 14.6.22: 2126 Find the maximum rate of change of at the given point and the ...
 14.6.23: 2126 Find the maximum rate of change of at the given point and the ...
 14.6.24: 2126 Find the maximum rate of change of at the given point and the ...
 14.6.25: 2126 Find the maximum rate of change of at the given point and the ...
 14.6.26: 2126 Find the maximum rate of change of at the given point and the ...
 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 coordi nates 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 partial deri...
 14.6.36: Shown is a topographic map of Blue River Pine Provincial Park in Br...
 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: The second directional derivative of is
 14.6.40: (a) If is a unit vector and has continuous second partial derivativ...
 14.6.41: 41 46 Find equations of (a) the tangent plane and (b) the normal li...
 14.6.42: 41 46 Find equations of (a) the tangent plane and (b) the normal li...
 14.6.43: 41 46 Find equations of (a) the tangent plane and (b) the normal li...
 14.6.44: 41 46 Find equations of (a) the tangent plane and (b) the normal li...
 14.6.45: 41 46 Find equations of (a) the tangent plane and (b) the normal li...
 14.6.46: 41 46 Find equations of (a) the tangent plane and (b) the normal li...
 14.6.47: 47 48 Use a computer to graph the surface, the tangent plane, and t...
 14.6.48: 47 48 Use a computer to graph the surface, the tangent plane, and t...
 14.6.49: If , find the gradient vector and use it to find the tangent line t...
 14.6.50: If , find the gradient vector and use it to find the tangent line t...
 14.6.51: Show that the equation of the tangent plane to the ellipsoid at the...
 14.6.52: Find the equation of the tangent plane to the hyperboloid at and ex...
 14.6.53: Show that the equation of the tangent plane to the elliptic parabol...
 14.6.54: At what point on the paraboloid is the tangent plane parallel to th...
 14.6.55: Are there any points on the hyperboloid where the tangent plane is ...
 14.6.56: Show that the ellipsoid and the sphere are tangent to each other at...
 14.6.57: Show that every plane that is tangent to the cone passes through th...
 14.6.58: Show that every normal line to the sphere passes through the center...
 14.6.59: Where does the normal line to the paraboloid at the point intersect...
 14.6.60: At what points does the normal line through the point on the ellips...
 14.6.61: Show that the sum of the , , and intercepts of any tangent plane...
 14.6.62: Show that the pyramids cut off from the first octant by any tangent...
 14.6.63: Find parametric equations for the tangent line to the curve of inte...
 14.6.64: (a) The plane intersects the cylinder in an ellipse. Find parametri...
 14.6.65: (a) Two surfaces are called orthogonal at a point of intersection i...
 14.6.66: (a) Show that the function is continuous and the partial derivative...
 14.6.67: Suppose that the directional derivatives of are known at a given po...
 14.6.68: . Show that if is differentiable at then [Hint: Use Definition 14.4...
Solutions for Chapter 14.6: Directional Derivatives and the Gradient Vector
Full solutions for Calculus: Early Transcendentals  7th Edition
ISBN: 9780538497909
Solutions for Chapter 14.6: Directional Derivatives and the Gradient Vector
Get Full SolutionsSince 68 problems in chapter 14.6: Directional Derivatives and the Gradient Vector have been answered, more than 33371 students have viewed full stepbystep solutions from this chapter. Calculus: Early Transcendentals was written by and is associated to the ISBN: 9780538497909. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Calculus: Early Transcendentals , edition: 7. Chapter 14.6: Directional Derivatives and the Gradient Vector includes 68 full stepbystep solutions.

Addition property of inequality
If u < v , then u + w < v + w

Arcsine function
See Inverse sine function.

Binomial probability
In an experiment with two possible outcomes, the probability of one outcome occurring k times in n independent trials is P1E2 = n!k!1n  k2!pk11  p) nk where p is the probability of the outcome occurring once

Boundary
The set of points on the “edge” of a region

Completing the square
A method of adding a constant to an expression in order to form a perfect square

Directed angle
See Polar coordinates.

Doubleblind experiment
A blind experiment in which the researcher gathering data from the subjects is not told which subjects have received which treatment

Expanded form
The right side of u(v + w) = uv + uw.

Horizontal Line Test
A test for determining whether the inverse of a relation is a function.

Identity matrix
A square matrix with 1’s in the main diagonal and 0’s elsewhere, p. 534.

Imaginary part of a complex number
See Complex number.

Linear factorization theorem
A polynomial ƒ(x) of degree n > 0 has the factorization ƒ(x) = a(x1  z1) 1x  i z 22 Á 1x  z n where the z1 are the zeros of ƒ

Modulus
See Absolute value of a complex number.

Natural exponential function
The function ƒ1x2 = ex.

Permutations of n objects taken r at a time
There are nPr = n!1n  r2! such permutations

Range of a function
The set of all output values corresponding to elements in the domain.

Shrink of factor c
A transformation of a graph obtained by multiplying all the xcoordinates (horizontal shrink) by the constant 1/c or all of the ycoordinates (vertical shrink) by the constant c, 0 < c < 1.

Standard form: equation of a circle
(x  h)2 + (y  k2) = r 2

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

Venn diagram
A visualization of the relationships among events within a sample space.