Linear Functions 1D  3D  Tutorial  Contents

Calculus is the study of functions.

Examples

The Zero Function

Constant Functions

Linear Functions



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When we describe a plane as the graph of a linear function f(x,y) = px  + qy + k, we are giving a special role to the origin.  Often it is more convenient to consider planes through a particular point (x0,y0,z0) in space, and we can describe such a plane with x-slope p and y-slope q by the condition z-z0 = p(x-x0) + q(y-y0).  Choosing different values of the slopes p and q, we obtain all non-vertical planes through (x0,y0,z0).

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Exercises

  • 1. The x-intercept of a linear function L(x) = px + qy + k is the point where the graph of L intersects the x-axis, i.e. the point (a,0,0) that lies on the graph of L, if this points exists.  Under what condition will the linear function have a unique x-intercept, and what will it be?  What about the y-intercept, defined similarly?
  • 2.  When p ≠ 0 and q ≠ 0 what is the volume of the tetrahedral pyramid with vertices given by the origin (0,0,0), the y-intercept, and the x-intercept?
  • 3. Show why the range of L(x) = px + qy + k is all real numbers if p ≠ 0 or q ≠ 0 i.e. show that for every z there is a point (x,y) such that L(x,y) = z.  What can be said about the collection of all points (x,y) for which L(x,y) = z for a given value z?
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    Domain, Range & Function Graphs  1D  3D   Polar Coordinates  Parametric Equations  Top of Page  Contents


    Two-Variable Calculus considers functions of two real variables.

    The domain of a function f is the set of points where the function is defined.

    When a function f is given by a formula, then the natural domain of f is the collection of (x,y) for which f(x,y) is defined. 

    Example:  The natural domain of f(x,y) = 1/x is the set of all (x,y) with x not equal to zero and the natural domain of f(x,y) = √(xy) is the set of all (x,y) such that xy ≥ 0, i.e. either x=0, or y=0, or x and y are both non-zero with the same algebraic sign. 

    The range of a real-valued function f is the collection of all real numbers f(x,y) where (x,y) is in the domain of f.


    The graph of a function of two variables is the collection of points (x,y,f(x,y)) in 3-space where (x,y) is in the domain of f.

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    Exercises

  • 1. If f(x,y) = x + y – 1, what are the natural domains of the functions f(x,y), 1/f(x,y), √f(x,y) and ln(f(x,y))?
  • 2. Same as Exercise 1, for the function f(x) = (1 – x2 – y2)?
  • 3. What is the range of the function f(x,y) = x2 + y2?  What about f(x,y) = x2 - y2? What about the range of f(x) = ax2 + cy2 where a and c are given constants? (The answer will depend on the constants.)
  • 4. What is the range of the function f(x) = -x4 + 2x2 -y2?  (This exercise can be done without using calculus techniques.  Make a table of values, form a conjecture about the maximum value taken on by the function for all real (x,y) and prove your conjecture algebraically.)
  • 5. What is the range of the function f(x) = x2 + 2bxy + y2, defined for all (x,y)?  (The answer will depend on b.  Try to cover all possible cases.)
  • 6. What is the range of the function f(x) = ax2 + 2bxy + cy2, defined for all (x,y)?  (The answer will depend on a, b, and c.  Try to cover all possible cases.)

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    Slice Curves  1D  3D  Polar Coordinates  Parametric Equations  Top of Page  Contents

    For every point (x0,y) in the domain of a function f, the intersection of the graph of f with the vertical plane x = x0 will be the x0-slice curve (x0,y,f(x0,y)).  The domain of the x0-slice curve is the set of y for which (x0,y) is in the domain of f.

    Similarly we define the y0-slice curve to be (x,y0,f(x,y0)) for all x such that (x,y0) is in the domain of f.

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    Exercises

  • Describe the x0-slice curves and the y0-slice curves for the function f(x,y) = ax2 + cy2.  More generally, what can be said about the x0-slice curves and the y0-slice curves for a function f(x,y) = r(x) + s(y) where r is a function of x defined for all x and s is a function of y defined for all y?
  • Describe the x0-slice curves and the y0-slice curves for the function f(x,y) = ax2 + 2bxy + cy2 where (x0,y0) = (0,0).
  • Describe the slice curves for the function f(x,y) = ax2 + 2bxy + cy2 through an arbitrary point (x0,y0)? 

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    Slice Curves Along an Arbitrary Line 3D  Top of Page  Contents

    We can generalize slice curves by considering the slice above any line (y - y0) = m(x - x0).  This gives a function of a single variable, g(x)=(x,f(x,m(x-x0)+y0).  Alternatively we may consider the slice curve over the line x(t)=x0+tcos(θ), y(t)=y0+tsin(θ).

    Figure9

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    Exercises at (x0,y0)=(0,0) for various values of m
  • 1. Analyze the slices of the general quadratic function f(x,y) = Ax2 + 2Bxy + Cy2 for various values of A, B, and C. For which A, B, and C will the range consist of all real numbers? All non-negative numbers? All non-positive numbers?
  • 2. Analyze the function f(x,y) = x2*y/(x4 + y2) for (x,y) ≠ (0,0) and f(0,0) = 0. What can be said about the restriction of this funciton to a line y = mx through the origin? What is the range of this function?



    • Level Sets and Contours 1D  3D  Polar Coordinates  Parametric Equations  Top of Page  Contents

    The collection of all points (x,y) in the domain of a function f for which f (x,y) = k is called the level set of f at level k.

    The collection of points (x,y,f(x,y)) in the graph of the function f such that f(x,y) = k is called the contour of f at height k.


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    Examples

    Contours of Linear Functions

    A Domain Color Graph in 2D 1D  3D  Top of Page  Contents
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    Exercises

  • 1. Describe the points in the level set for each of the following:
    f(x,y) = x + y -1, k = 0
    f(x,y) = x2 + y2 - 1, k = 0
    f(x,y) = sin(π*xy), k = √2/2
    f(x,y) = tan(y/x), k = 1
  • 2. For the four functions in Exercise 1, describe the level sets for arbitrary k.  In particular, for which k will the level set be empty?
  • 3. What is another name for the collection of values k for which the level set of the graph of f(x,y) is not empty?
  • 4. What is the level set of the function (fg)(x,y) = f(x,y)g(x,y) at k = 0, in terms of the level set of f(x,y) at k=0 and the level set of g(x,y) at k=0?

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    Continuity  1D  3D  Polar Coordinates  Parametric Equations  Top of Page  Contents

    One of the most important properties of functions of two real variables is continuity.  The basic intuition for continuity is that the range of a function f(x,y) will lie in an arbitrarily small interval centered at f(x0,y0) if (x,y) is restricted to lie in a sufficiently small disc centered at (x0,y0).  Geometrically, this means that the graph of f(x,y) will lie between a pair of parallel planes  z = f(x0,y0) + ε and
    z = f(x0,y0) – ε if (x,y) is required to lie in the disc of radius δ centered at (x0,y0), i.e. where √((x – x0)2 + (y – y0)2) < δ. 

    According to the epsilon-delta definition, a function f of two real variables is said to be continuous at (x0,y0) if for any ε > 0 there exists a δ such that | f(x,y) - f(x0,y0) | < ε whenever √((x – x0)2 + (y – y0)2) < δ.

    A function f of two real variables is said to be continuous if it is continuous at all points (x0,y0) in its domain.

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    Exercises

  • 1. Consider the function f(x) = (x - 1)2 + y2 for (x,y) ≠ (1,0) and f(1,0) = 2. Why is this function not continuous at (1,0)? Is there any constant k such that the function will be continuous at (1,0) if we define f(1,0) = k?
  • 2. Consider the function f(x,y) = 1/(x2 + y2) for (x,y)  ≠ (0,0) and f(0,0) = 0.  Why is this function not continuous at (0,0)? Is there any constant k such that the function will be continuous at (0,0) if we define f(0,0) = k?
  • 3. Consider the function f(x,y) = 2xy/(x2+y2) for (x,y) ≠ (0,0) and f(0,0) = 0. Why is this function not continuous at (0,0)? Is there any constant k such that the function will be continuous at (0,0) if we define f(0,0) = k?
  • 4. Show that the epsilon-delta definition can be expressed by using tolerances and accuracy in terms of number of decimal places, i.e. if a process requires us to estimate the value of f(x0,y0) up to three decimal place accuracy, can we achieve this by requiring that the distance from (x,y) to (x0,y0) is zero, up to a certain number of decimal places?
  • 5. Consider the function f(x,y) = x2y/(x4+y2) for (x,y) ≠ (0,0) and f(0,0) = 0.  Show that all of the slice functions through (0,0) for various choices of theta are all continuous, but the function itself is not continuous at (0,0).