Numerical Methods for Engineers
7th Edition
ISBN: 9780073397924
Author: Steven C. Chapra Dr., Raymond P. Canale
Publisher: McGraw-Hill Education
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Chapter 14, Problem 11P
Develop a one-dimensional equation in the pressure gradient direction at the point
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Find the gradient of the function at the given point.
z = x?y,
(7, 1)
Vz(7, 1) =
Find the maximum value of the directional derivative at the given point.
flx.y)=X
P=(-3,;4h
V=2-23
A. Find the gradient of f.
Vf = 1/y
i+ -x/y^2
Note: Your answers should be expressions of x and y; e.g. "3x - 4y"
B. Find the gradient of f at the point P.
(Vf) (P)
-1/4
i+ 3/16
Note: Your answers should be numbers
C. Find the directional derivative of f at P in the direction of v.
Duf =
(-7sqrt2)/32
Note: Your answer should be a number
D. Find the maximum rate of change of f at P.
5/16
Note: Your answer should be a number
E. Find the (unit) direction vector in which the maximum rate of change occurs at P.
i+
Note: Your answers should be numbers
When you cough, you are using a high-speed stream of air to clear your trachea (windpipe). During a cough, your trachea contracts, forcing the air to move faster, but also increasing the friction. If a trachea contracts from a normal radius of 3 centimeters to a radius of r centimeters, the velocity of the airstream is V(r) = c(3-r)r2 where c is a positive constant depending on the length and the elasticity of the trachea. Find the radius r that maximizes this velocity. (X-ray pictures verify that the trachea does indeed contract to this radius.)
Chapter 14 Solutions
Numerical Methods for Engineers
Ch. 14 - 14.1 Find the directional derivative of
at in...Ch. 14 - Repeat Example 14.2 for the following function at...Ch. 14 - 14.3 Given
Construct and solve a system of...Ch. 14 - (a) Start with an initial guess of x=1 and y=1 and...Ch. 14 - 14.5 Find the gradient vector and Hessian matrix...Ch. 14 - Prob. 6PCh. 14 - Perform one iteration of the steepest ascent...Ch. 14 - Perform one iteration of the optimal gradient...Ch. 14 - Develop a program using a programming or macro...Ch. 14 - 14.10 The grid search is another brute force...
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- Find the constant of proportionality. y is directly proportional to x. If x=30, then y=15.arrow_forwardSuppose f (x, y) = , P= (-3, –1) and v = -2i + 3j. y' A. Find the gradient of f. Vƒ = 1/y i+ -x/y^2 j Note: Your answers should be expressions of x and y; e.g. "3x - 4y" B. Find the gradient of f at the point P. (Vf) (P) i+ j Note: Your answers should be numbers C. Find the directional derivative of f at P in the direction of v. Duf = Note: Your answer should be a number D. Find the maximum rate of change of f at P. Note: Your answer should be a number E. Find the (unit) direction vector in which the maximum rate of change occurs at P. u = i+ Note: Your answers should be numbersarrow_forwardfind the gradient of the function at the given point. Then sketch the gradient together with the level curve that passes through the point f(x,y) = y - x, (2, I) Need Solution through 15minarrow_forward
- Find the gradient of f (x, y) = 2xy? - 3.x°y.arrow_forward× Find the maximum rate of change of f at the given point and the direction in which it occurs. f(x, y) = y²/x, (2, 4) direction of maximum rate of change (in unit vector) = < maximum rate of change =arrow_forwardc) The equation of a curve is y = 5x + 4 i. Calculate the gradient of the curve at the point where x = 1 A point with coordinates (x,y) moves along the curve in such a way that the rate of increase of x has the constant value 0.03 units per second. ii. Find the rate of increase of y ath the instant when x = 1. ii. Find the area enclosed by the curve, the x - axis, the y - axis and the line y = 1.arrow_forward
- Explain why the function is differentiable at the given point. Then find the linearization L(x,y) of the function at that point. f(x,y)=y+sin(x/y),(0,3)arrow_forwardFind the gradient of the function f(x,y) = 4x+ 2y at the point (- 1,4). Then sketch the gradient together with the level curve that passes through the point.arrow_forwardc) The equation of a curve is y = V5x+4 i. Calculate the gradient of the curve at the point where x = 1 ii. A point with coordinates (x,y) moves along the curve in such a way that the rate of increase of x has the constant value 0.03 units per second. Find the rate of increase of y ath the instant when x = 1. iii. Find the area enclosed by the curve, the x – axis, the y – axis and the line y = 1.arrow_forward
- Find the gradient of the function at the given point. Function Point f(x, y) = x2 + 2xy (2, 4) Vf(2, 4) = Find the maximum value of the directional derivative at the given point.arrow_forwardUse the gradient to find the directional derivative of the function at P in the direction of Q. (Give your answer correct to 2 decimal places.) f(x, y) = 3x2 - y² + 4, P(7, 2), Q(3, 8)arrow_forward
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