Calculus Applications

Understanding Turning Points

A turning point (or stationary point) on the graph of a function $y=f(x)$ is a point where the derivative $f'(x)$ is zero or undefined. At these points, the tangent to the curve is horizontal (if $f'(x)=0$).

Turning points can be classified as:

• Local Maximum: The function value is greater than or equal to the values at nearby points. The curve changes from increasing to decreasing.
• Local Minimum: The function value is less than or equal to the values at nearby points. The curve changes from decreasing to increasing.
• Stationary Point of Inflexion: A point where $f'(x)=0$ but it's neither a local maximum nor a local minimum (e.g., $y=x^3$ at $x=0$).

Tests for Nature of Turning Points (where $f'(c)=0$):

First Derivative Test:

• If $f'(x)$ changes from positive to negative at $x=c$, then $f(c)$ is a local maximum.
• If $f'(x)$ changes from negative to positive at $x=c$, then $f(c)$ is a local minimum.
• If $f'(x)$ does not change sign at $x=c$, then $f(c)$ is a stationary point of inflexion.

Second Derivative Test:

• If $f''(c) < 0$, then $f(c)$ is a local maximum.
• If $f''(c) > 0$, then $f(c)$ is a local minimum.
• If $f''(c) = 0$, the test is inconclusive; use the first derivative test.

Functions of Two Independent Variables

A function of two independent variables, typically denoted as $z = f(x,y)$, assigns a unique output value $z$ to each pair of input values $(x,y)$ from its domain.

Graphically, such a function represents a surface in three-dimensional space. The $x$ and $y$ axes form a horizontal plane (the domain), and the $z$-axis represents the output value (height) of the function at each point $(x,y)$.

Understanding the shape of this surface, including its peaks (local maxima), valleys (local minima), and other interesting features like saddle points, is a key aspect of multivariable calculus.

Explaining Saddle Points

A saddle point of a function $z=f(x,y)$ is a critical point (where $f_x=0$ and $f_y=0$) that is neither a local maximum nor a local minimum.

Imagine the shape of a horse's saddle. If you move along the saddle in one direction (e.g., from front to back), the point is a local minimum. However, if you move in another direction (e.g., side to side), the same point is a local maximum.

At a saddle point, the surface curves up in one direction and curves down in another. This means that in some paths leading to the point, the function values increase, while in other paths, they decrease. The tangent plane at a saddle point is horizontal, just like at a local maximum or minimum.

Example: The function $f(x,y) = x^2 - y^2$ has a saddle point at $(0,0)$. Along the x-axis ($y=0$), $f(x,0)=x^2$ which has a minimum at $x=0$. Along the y-axis ($x=0$), $f(0,y)=-y^2$ which has a maximum at $y=0$.

Determining Maxima, Minima, and Saddle Points (f(x,y))

To find local maxima, local minima, and saddle points of a function $z=f(x,y)$:

1. Find Critical Points: Calculate the first-order partial derivatives $f_x = \frac{\partial f}{\partial x}$ and $f_y = \frac{\partial f}{\partial y}$. Solve the system of equations $f_x(x,y) = 0$ and $f_y(x,y) = 0$. The solutions $(a,b)$ are the critical points. At these points, the tangent plane to the surface is horizontal.
2. Apply the Second Derivative Test: Calculate all second-order partial derivatives: $f_{xx}$, $f_{yy}$, and $f_{xy}$. Evaluate these at each critical point $(a,b)$. Compute the discriminant (or Hessian determinant) $D$ at $(a,b)$: $$ D(a,b) = f_{xx}(a,b) \cdot f_{yy}(a,b) - [f_{xy}(a,b)]^2 $$
3. Classify Critical Points based on $D$ and $f_{xx}$:
   • If $D(a,b) > 0$ and $f_{xx}(a,b) > 0$: $f$ has a local minimum at $(a,b)$.
   • If $D(a,b) > 0$ and $f_{xx}(a,b) < 0$: $f$ has a local maximum at $(a,b)$.
   • If $D(a,b) < 0$: $f$ has a saddle point at $(a,b)$.
   • If $D(a,b) = 0$: The test is inconclusive. Further investigation is needed (e.g., examining the function's behavior near the point).

Sketching Contour Maps

A contour map (or level set plot) is a way to visualize a function of two variables $z=f(x,y)$ in two dimensions. It consists of curves in the $xy$-plane along which the function $f(x,y)$ has a constant value.

Each curve is called a level curve or contour line, defined by $f(x,y) = c$ for some constant $c$. Different values of $c$ give different level curves.

Properties of contour maps:

• Closely spaced contour lines indicate a steep slope of the surface.
• Widely spaced contour lines indicate a gentle slope.
• Closed, concentric contour lines often suggest a local maximum or minimum (a "hill" or "depression").
• Contour lines that cross or form an "X" shape can indicate a saddle point.
• The gradient vector $\nabla f = \langle f_x, f_y \rangle$ at a point $(x,y)$ is always perpendicular to the level curve passing through that point and points in the direction of the steepest ascent.

Contour maps are widely used in various fields, such as cartography (topographic maps showing elevation), meteorology (isobars for pressure, isotherms for temperature), and engineering (visualizing potential fields or stress distributions).