 14.4.1: In Exercises 114, find the gradient of the function. Assumethe vari...
 14.4.2: In Exercises 114, find the gradient of the function. Assumethe vari...
 14.4.3: In Exercises 114, find the gradient of the function. Assumethe vari...
 14.4.4: In Exercises 114, find the gradient of the function. Assumethe vari...
 14.4.5: In Exercises 114, find the gradient of the function. Assumethe vari...
 14.4.6: In Exercises 114, find the gradient of the function. Assumethe vari...
 14.4.7: In Exercises 114, find the gradient of the function. Assumethe vari...
 14.4.8: In Exercises 114, find the gradient of the function. Assumethe vari...
 14.4.9: In Exercises 114, find the gradient of the function. Assumethe vari...
 14.4.10: In Exercises 114, find the gradient of the function. Assumethe vari...
 14.4.11: In Exercises 114, find the gradient of the function. Assumethe vari...
 14.4.12: In Exercises 114, find the gradient of the function. Assumethe vari...
 14.4.13: In Exercises 114, find the gradient of the function. Assumethe vari...
 14.4.14: In Exercises 114, find the gradient of the function. Assumethe vari...
 14.4.15: In Exercises 1522, find the gradient at the point.f(x, y) = x2y + 7...
 14.4.16: In Exercises 1522, find the gradient at the point.f(m, n)=5m2 + 3n4...
 14.4.17: In Exercises 1522, find the gradient at the point.f(r, h)=2rh + r2,...
 14.4.18: In Exercises 1522, find the gradient at the point.f(x, y) = esin y,...
 14.4.19: In Exercises 1522, find the gradient at the point.f(x, y) = sin (x2...
 14.4.20: In Exercises 1522, find the gradient at the point.f(x, y) = ln(x2 +...
 14.4.21: In Exercises 1522, find the gradient at the point.f(x, y)=1/(x2 + y...
 14.4.22: In Exercises 1522, find the gradient at the point.f(x, y) = tan x +...
 14.4.23: In Exercises 2326, find the directional derivative fu (1, 2)for the...
 14.4.24: In Exercises 2326, find the directional derivative fu (1, 2)for the...
 14.4.25: In Exercises 2326, find the directional derivative fu (1, 2)for the...
 14.4.26: In Exercises 2326, find the directional derivative fu (1, 2)for the...
 14.4.27: If f(x, y) = x2y and v = 4i 3j , find the directionalderivative at ...
 14.4.28: In Exercises 2829, find the differential df from the gradient.grad ...
 14.4.29: In Exercises 2829, find the differential df from the gradient.grad ...
 14.4.30: In Exercises 3031, find grad f from the differential.df = 2xdx + 10ydy
 14.4.31: In Exercises 3031, find grad f from the differential.df = (x + 1)ye...
 14.4.32: In Exercises 3237, use the contour diagram of f in Figure14.34 to d...
 14.4.33: In Exercises 3237, use the contour diagram of f in Figure14.34 to d...
 14.4.34: In Exercises 3237, use the contour diagram of f in Figure14.34 to d...
 14.4.35: In Exercises 3237, use the contour diagram of f in Figure14.34 to d...
 14.4.36: In Exercises 3237, use the contour diagram of f in Figure14.34 to d...
 14.4.37: In Exercises 3237, use the contour diagram of f in Figure14.34 to d...
 14.4.38: In Exercises 3845, use the contour diagram of f in Figure14.34 to f...
 14.4.39: In Exercises 3845, use the contour diagram of f in Figure14.34 to f...
 14.4.40: In Exercises 3845, use the contour diagram of f in Figure14.34 to f...
 14.4.41: In Exercises 3845, use the contour diagram of f in Figure14.34 to f...
 14.4.42: In Exercises 3845, use the contour diagram of f in Figure14.34 to f...
 14.4.43: In Exercises 3845, use the contour diagram of f in Figure14.34 to f...
 14.4.44: In Exercises 3845, use the contour diagram of f in Figure14.34 to f...
 14.4.45: In Exercises 3845, use the contour diagram of f in Figure14.34 to f...
 14.4.46: Let f(P) = 15 and f(Q) = 20 where P = (3, 4) andQ = (3.03, 3.96). A...
 14.4.47: (a) Give Q, the point at a distance of 0.1 from P =(4, 5) in the di...
 14.4.48: Find the directional derivative of f(x, y) = ex tan(y) +2x2y at the...
 14.4.49: Find the rate of change of f(x, y) = x2 + y2 at the point(1, 2) in ...
 14.4.50: (a) Let f(x, y)=(x+y)/(1+x2). Find the directionalderivative of f a...
 14.4.51: Let f(x, y) = x2y3. At the point (1, 2), find a vector(a) In the di...
 14.4.52: You are at the point (/4, 1) and start to move in thedirection of t...
 14.4.53: (a) Let f(x, y) = x2 + ln y. Find the average rate ofchange of f as...
 14.4.54: (a) What is the rate of change of f(x, y)=3xy + y2 atthe point (2, ...
 14.4.55: A student was asked to find the directional derivative off(x, y) = ...
 14.4.56: For 5660 use Figure 14.35, showing level curvesof f(x, y), to estim...
 14.4.57: For 5660 use Figure 14.35, showing level curvesof f(x, y), to estim...
 14.4.58: For 5660 use Figure 14.35, showing level curvesof f(x, y), to estim...
 14.4.59: For 5660 use Figure 14.35, showing level curvesof f(x, y), to estim...
 14.4.60: For 5660 use Figure 14.35, showing level curvesof f(x, y), to estim...
 14.4.61: In 6164, check that the point (2, 3) lies on thecurve. Then, viewin...
 14.4.62: In 6164, check that the point (2, 3) lies on thecurve. Then, viewin...
 14.4.63: In 6164, check that the point (2, 3) lies on thecurve. Then, viewin...
 14.4.64: In 6164, check that the point (2, 3) lies on thecurve. Then, viewin...
 14.4.65: The surface z = g(x, y) is in Figure 14.36. What is thesign of each...
 14.4.66: The table gives values of a differentiable functionf(x, y). At the ...
 14.4.67: Figure 14.37 represents the level curves f(x, y) = c ;the values of...
 14.4.68: In Figure 14.37, which is larger: f at P or f atQ? Explain how you ...
 14.4.69: In 6972, do the level curves of f(x, y) cross thelevel curves of g(...
 14.4.70: In 6972, do the level curves of f(x, y) cross thelevel curves of g(...
 14.4.71: In 6972, do the level curves of f(x, y) cross thelevel curves of g(...
 14.4.72: In 6972, do the level curves of f(x, y) cross thelevel curves of g(...
 14.4.73: (a) Sketch the surface z = f(x, y) = y2 in three dimensions.(b) Ske...
 14.4.74: You are standing above the point (1, 3) on the surfacez = 20 (2x2 +...
 14.4.75: Let P be a fixed point in the plane and let f(x, y) be thedistance ...
 14.4.76: The directional derivative of z = f(x, y) at (2, 1) inthe direction...
 14.4.77: Consider the function f(x, y). If you start at the point(4, 5) and ...
 14.4.78: (a) For g(x, y) = x2 + 3y + 3, find grad g(1, 4).(b) Find the best ...
 14.4.79: Find the directional derivative of z = x2y2 at the point(3, 1) in t...
 14.4.80: The temperature H in Fahrenheit y miles north of theCanadian border...
 14.4.81: At a certain point on a heated plate, the greatest rate oftemperatu...
 14.4.82: You are climbing a mountain by the steepest route at aslope of 20 w...
 14.4.83: Figure 14.39 is a graph of the directional derivative, fu ,at the p...
 14.4.84: You are standing at the point (1, 1, 3) on the hill whoseequation i...
 14.4.85: In this problem we see another way of obtaining the formulafu (a, b...
 14.4.86: Let L be a line tangent to the ellipse x2/2 + y2 = 1 atthe point (a...
 14.4.87: Let C be the contour C of f(x, y) through (a, b) andgrad f(a, b) = ...
 14.4.88: Let grad f(x, y) = grad g(x, y) at a point Pwhere these gradients a...
 14.4.89: In 8991, explain what is wrong with the statement.A function f has ...
 14.4.90: In 8991, explain what is wrong with the statement.A function f has ...
 14.4.91: In 8991, explain what is wrong with the statement.The gradient vect...
 14.4.92: In 9293, give an example of:A unit vector u such that fu (0, 0) < 0...
 14.4.93: In 9293, give an example of:A contour diagram of a function with tw...
 14.4.94: For the gradient f(P) of f at a point P, describe thegeometric inte...
 14.4.95: Are the statements in 95106 true or false? Givereasons for your ans...
 14.4.96: Are the statements in 95106 true or false? Givereasons for your ans...
 14.4.97: Are the statements in 95106 true or false? Givereasons for your ans...
 14.4.98: Are the statements in 95106 true or false? Givereasons for your ans...
 14.4.99: Are the statements in 95106 true or false? Givereasons for your ans...
 14.4.100: Are the statements in 95106 true or false? Givereasons for your ans...
 14.4.101: Are the statements in 95106 true or false? Givereasons for your ans...
 14.4.102: Are the statements in 95106 true or false? Givereasons for your ans...
 14.4.103: Are the statements in 95106 true or false? Givereasons for your ans...
 14.4.104: Are the statements in 95106 true or false? Givereasons for your ans...
 14.4.105: Are the statements in 95106 true or false? Givereasons for your ans...
 14.4.106: Are the statements in 95106 true or false? Givereasons for your ans...
 14.4.107: Assume that f(x, y) is a differentiable function. Are the statement...
 14.4.108: Assume that f(x, y) is a differentiable function. Are the statement...
 14.4.109: Assume that f(x, y) is a differentiable function. Are the statement...
 14.4.110: Assume that f(x, y) is a differentiable function. Are the statement...
 14.4.111: Assume that f(x, y) is a differentiable function. Are the statement...
Solutions for Chapter 14.4: GRADIENTS AND DIRECTIONAL DERIVATIVES IN THE PLANE
Full solutions for Calculus: Single and Multivariable  6th Edition
ISBN: 9780470888612
Solutions for Chapter 14.4: GRADIENTS AND DIRECTIONAL DERIVATIVES IN THE PLANE
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 14.4: GRADIENTS AND DIRECTIONAL DERIVATIVES IN THE PLANE includes 111 full stepbystep solutions. Since 111 problems in chapter 14.4: GRADIENTS AND DIRECTIONAL DERIVATIVES IN THE PLANE have been answered, more than 42184 students have viewed full stepbystep solutions from this chapter. Calculus: Single and Multivariable was written by and is associated to the ISBN: 9780470888612. This textbook survival guide was created for the textbook: Calculus: Single and Multivariable , edition: 6.

Average velocity
The change in position divided by the change in time.

Compounded monthly
See Compounded k times per year.

Directed line segment
See Arrow.

Exponential regression
A procedure for fitting an exponential function to a set of data.

Finite sequence
A function whose domain is the first n positive integers for some fixed integer n.

Horizontal asymptote
The line is a horizontal asymptote of the graph of a function ƒ if lim x: q ƒ(x) = or lim x: q ƒ(x) = b

Increasing on an interval
A function ƒ is increasing on an interval I if, for any two points in I, a positive change in x results in a positive change in.

Inductive step
See Mathematical induction.

Limaçon
A graph of a polar equation r = a b sin u or r = a b cos u with a > 0 b > 0

Logarithmic form
An equation written with logarithms instead of exponents

Onetoone function
A function in which each element of the range corresponds to exactly one element in the domain

Periodic function
A function ƒ for which there is a positive number c such that for every value t in the domain of ƒ. The smallest such number c is the period of the function.

Random variable
A function that assigns realnumber values to the outcomes in a sample space.

Rational function
Function of the form ƒ(x)/g(x) where ƒ(x) and g(x) are polynomials and g(x) is not the zero polynomial.

Second
Angle measure equal to 1/60 of a minute.

Semimajor axis
The distance from the center to a vertex of an ellipse.

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.

Speed
The magnitude of the velocity vector, given by distance/time.

Variation
See Power function.

yzplane
The points (0, y, z) in Cartesian space.