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Chapter 13: Problem 33

Find and \(|\) ar when \(r=1, s=-1\) if \(w=(x+y+z)^{2}\) \(x=r-s, y=\cos (r+s), z=\sin (r+s)\)

Short Answer

Expert verified
The value of \(|w|\) is 9.

Step by step solution

01

Define the values of given variables

We begin by identifying the given values for the variables. From the problem, we know that \( r = 1 \) and \( s = -1 \). These will be used in our subsequent calculations.
02

Substitute values into expressions for x, y, z

Substitute the values of \( r \) and \( s \) into the expressions for \( x \), \( y \), and \( z \). This gives: \( x = r - s = 1 - (-1) = 2 \), \( y = \cos(r+s) = \cos(1 + (-1)) = \cos(0) = 1 \), and \( z = \sin(r+s) = \sin(1 + (-1)) = \sin(0) = 0 \).
03

Substitute expressions into the formula for w

Now, substitute the values of \( x \), \( y \), and \( z \) into the formula \( w = (x + y + z)^{2} \). This results in \( w = (2 + 1 + 0)^{2} = 3^{2} = 9 \).
04

Calculate |a| and |w|

We need to calculate the magnitude of the values. The length \(|a|\) is equivalent to \(|w|\) since \(w\) itself represents the magnitude in this context. Therefore, \(|w| = 9\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Algebra
Algebra is a branch of mathematics that uses variables to represent numbers in equations and expressions. In this problem, we are using algebra to manipulate given expressions with substitutions. Let's break this down:
  • We have expressions involving variables: \( x = r-s \), \( y = \cos(r+s) \), and \( z = \sin(r+s) \).
  • Given \( r = 1 \) and \( s = -1 \), algebra allows us to substitute these values into our expressions directly.
  • Substituting \( r \) and \( s \) means calculating \( x = 1 - (-1) = 2 \), which is a straightforward algebraic operation using substitution and simplification.
Algebra simplifies expressions and solves equations, integral for finding solutions in calculus problems. Here, by systematically substituting and simplifying, we established the necessary components \( x \), \( y \), and \( z \) to determine \( w \).
Hence, algebra's role is crucial in breaking down the problem into manageable calculations. Always start by assigning values properly and performing operations step-by-step.
Trigonometry
Trigonometry is the study of angles and the relationships between the lengths and angles of triangles. In this exercise, trigonometry helps calculate values of \( y \) and \( z \) using standard trigonometric functions: cosine and sine.
  • We use \( y = \cos(r+s) \), which simplifies to \( \cos(0) \) since \( r+s = 1+(-1)=0 \).
  • The cosine of zero degrees or radians, \( \cos(0) \), is equal to 1. Similarly, \( z = \sin(r+s) \) simplifies to \( \sin(0) \).
The sine of zero is 0, hence \( z=0 \). These trigonometric values are essential in further calculations of \( w = (x+y+z)^2 \).Understanding these basic trigonometric values is crucial as they often appear in calculus problems. Remember, cosine and sine start at 1 and 0 respectively when the angle is 0, and this knowledge supports solving many mathematical challenges.
Functions
Functions play a central role in calculus and algebra. They offer a way to describe relationships between varying quantities. In this example, the function \( w=(x+y+z)^2 \) tells how \( w \) changes based on inputs \( x \), \( y \), and \( z \).
  • Functions let us substitute specific input values, enabling calculation of outputs 鈥� here we calculated \( w \) using determined values.
  • With \( x = 2 \), \( y = 1 \), and \( z = 0 \), we find \( w = (2 + 1 + 0)^2 \), simplifying to \( 3^2 = 9 \).
This function represents a simple quadratic relationship that determines magnitude, showing how output varies with summed inputs. By understanding these relationships, functions allow predictions and modeling of real-world scenarios in mathematics.Getting comfortable with functions and their evaluation forms helps make calculus problems less daunting. Recognizing patterns and substituting effectively leads to clearer solutions.

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Most popular questions from this chapter

Use a CAS to perform the following steps implementing the method of Lagrange multipliers for finding constrained extrema:In Exercises \(49-54,\) use a CAS to perform the following steps implementing the method of Lagrange multipliers for finding constrained extrema: a. Form the function \(h=f-\lambda_{1} g_{1}-\lambda_{2} g_{2},\) where \(f\) is the function to optimize subject to the constraints \(g_{1}=0\) and \(g_{2}=0\) b. Determine all the first partial derivatives of \(h,\) including the partials with respect to \(\lambda_{1}\) and \(\lambda_{2},\) and set them equal to 0 c. Solve the system of equations found in part (b) for all the unknowns, including \(\lambda_{1}\) and \(\lambda_{2}\) d. Evaluate \(f\) at each of the solution points found in part (c) and select the extreme value subject to the constraints asked for in the exercise. Minimize \(f(x, y, z, w)=x^{2}+y^{2}+z^{2}+w^{2}\) subject to the constraints \(\quad 2 x-y+z-w-1=0 \quad\) and \(\quad x+y-z+\) \(w-1=0.\)

Find the linearizations \(L(x, y, z)\) of the functions at the given points. \(f(x, y, z)=(\sin x y) / z\) at a. \((\pi / 2,1,1)\) b. (2,0,1)

a. Maximum on a sphere Show that the maximum value of \(a^{2} b^{2} c^{2}\) on a sphere of radius \(r\) centered at the origin of a Cartesian \(a b c\) -coordinate system is \(\left(r^{2} / 3\right)^{3}\) b. Geometric and arithmetic means Using part (a), show that for nonnegative numbers \(a, b,\) and \(c,\) $$(a b c)^{1 / 3} \leq \frac{a+b+c}{3}$$ that is, the geometric mean of three nonnegative numbers is less than or equal to their arithmetic mean.

Find the minimum distance from the cone \(z=\sqrt{x^{2}+y^{2}}\) to the point (-6,4,0).

If a function \(f(x, y)\) has continuous second partial derivatives throughout an open region \(R,\) must the first-order partial derivatives of \(f\) be continuous on \(R ?\) Give reasons for your answer.
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