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Chapter 11: Problem 32

Sketch the surface given by the function. $$ f(x, y)=6-2 x-3 y $$

Short Answer

Expert verified
Sketching the surface of the given function, we get a plane with z-intercept at (0,0,6), x-intercept at (3,0,0) and y-intercept at (0,2,0). The slope along x axis is -2 and along y axis is -3.

Step by step solution

01

Identify the intercepts

Set each variable to 0 one at a time to find the intercepts. For example, setting x and y to zero, f(0,0) results in the z-intercept at (0,0,6). Likewise, setting y and z to zero results in x-intercept at (3,0,0) and setting x and z to zero results in y-intercept at (0,2,0).
02

Calculate the slopes

The formula of the function can be written as \(z = 6 - 2x - 3y\). Here, -2 and -3 are the slopes or gradients along x axis and y axis respectively.
03

Plot the intercepts and slopes

Draw the x, y, z-axis on a 3D space. Plot the intercepts obtained in step 1. Also, the slopes obtained in step 2 would give the direction in which the function is decreasing. You can then sketch the surface with the given decreasing direction from the intercepts.

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Key Concepts

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

Intercepts in 3D Graphs
Intercepts in 3D graphs are like the coordinates where the graph meets the three axes: x, y, and z. In simpler terms, intercepts show us where the surface would touch these axes if they could physically meet. To find them, we set the other variables to zero one at a time.
  • For a z-intercept, set both x and y to zero and solve for z.
  • For an x-intercept, set y and z to zero to solve for x.
  • For a y-intercept, set x and z to zero and solve for y.
Taking the function given, \(f(x, y) = 6 - 2x - 3y\), calculating these intercepts helps in understanding the surface's position in space.
  • The z-intercept is at the point (0, 0, 6).
  • The x-intercept can be found at the point (3, 0, 0).
  • The y-intercept occurs at (0, 2, 0).
These intercepts tell us crucial "anchor" points of the surface in the 3D coordinate system. By plotting these points, we can begin constructing the overall shape of the surface with respect to the axes.
Slopes in Multi-variable Functions
In the context of multi-variable functions, slopes represent the rate of change of the function as we move along the x or y axis. It's much like the slope of a line in 2D, but with an added dimension. For the function \( z = 6 - 2x - 3y \), the coefficients of x and y, which are -2 and -3 respectively, act as slopes.
  • The slope with respect to the x-axis tells us how quickly the function is changing as x varies, if y is constant.
  • Similarly, the slope with respect to the y-axis indicates how the function changes as y varies, if x is constant.
Both are negative here, meaning the surface decreases as we move along increasing x or y.
Understanding these slopes allows us to finesse our sketch of the surface as it gives directions of change from any point. Unlike in single-variable calculus, you're looking at how the surface tilts or bends instead of just upward or downward sloping lines.
3D Visualization in Calculus
Visualizing mathematics in three dimensions might seem challenging, but it's rewarding as it provides a more comprehensive understanding of surfaces like planes or curves. Here, you're taking what you might have sketched on paper in 2D and building it out in 3D. Here's how to approach it:
  • Start by plotting the intercepts, which provide foundational points in your space.
  • Next, incorporate slope information to understand how the surface behaves between these intercepts.
  • The directional arrows from the slopes help predict where and how the surface shifts, gives a sense of the "rise and run" in all directions.
  • Click and drag with digital tools, or rotate your paper drawings, to see the complete picture from various angles.
Visualization tools or software can help when manual drawing becomes difficult. Remember, practicing 3D visualization hones your ability to apply calculus concepts effectively in engineering, physics, or computer science fields.
It鈥檚 like sculpting your understanding into a comprehensive whole and seeing the beauty and utility of calculus in bringing theories to life.

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Most popular questions from this chapter

Differentiate implicitly to find the first partial derivatives of \(z\) \(z=e^{x} \sin (y+z)\)

Describe the relationship of the gradient to the level curves of a surface given by \(z=f(x, y)\).

Use the gradient to find a unit normal vector to the graph of the equation at the given point. Sketch your results $$ 3 x^{2}-2 y^{2}=1,(1,1) $$

Find \(\partial w / \partial s\) and \(\partial w / \partial t\) using the appropriate Chain Rule, and evaluate each partial derivative at the given values of \(s\) and \(t\) $$ \begin{array}{l} \text { Function } \\ \hline w=\sin (2 x+3 y) \\ x=s+t, \quad y=s-t \end{array} $$ $$ \frac{\text { Point }}{s=0, \quad t=\frac{\pi}{2}} $$

Differentiate implicitly to find the first partial derivatives of \(w\). \(w-\sqrt{x-y}-\sqrt{y-z}=0\)
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