Problem 72 - Essential Calculus

Question

Sketch a graph of a curve defined by the parametric equations \(x=g(t)\) and \(y=f(t)\) such that \(d x / d t<0\) and \(d y / d t<0\) for all real numbers \(t\)

Step-by-Step Solution

Answer

The sketch should be a curve that starts from the upper right quadrant and continues downward and to the left, showing both \(x\) and \(y\) decrease with increasing \(t\).

Step 1: Understanding the Conditions

First, it should be understood that \(dx/dt<0\) and \(dy/dt<0\) mean both \(x\) and \(y\) are decreasing with time. As \(t\) increases, both \(x\) and \(y\) decrease.

Step 2: Draw the Axes

The plot will be in the cartesian coordinate system. So a vertical axis representing \(y\) and a horizontal axis representing \(x\) are drawn.

Step 3: Sketch the Graph

Since both \(x\) and \(y\) are decreasing functions of \(t\), start by plotting a point in the upper right quadrant. As \(t\) increases, \(x\) and \(y\) decrease, so draw a curve from the initial point downward and to the left. This curve represents the parameters \(x=g(t)\) and \(y=f(t)\).

Key Concepts

Parametric Functions

Parametric functions are a way to express a mathematical relationship by using multiple equations. Each equation, or parameter, defines one variable in terms of an independent variable, often denoted as t (for time). In the context of graphing, parametric equations, such as x=g(t) and y=f(t), create a set of points in the Cartesian coordinate system by taking the value of t and computing the corresponding x and y coordinates.

Imagine plotting a path of a moving object: the functions g(t) and f(t) describe the horizontal and vertical positions of the object at any given time. Unlike traditional functions, which relate y directly to x (i.e., y=f(x)), parametric functions can describe curves in the plane that may not represent functions in the traditional sense.

Moreover, the conditions given in the exercise, dx/dt<0 and dy/dt<0, imply that as the variable t increases, the values of both x and y decrease, indicating the moving object proceeds towards the bottom-left direction within the plane.

Calculus

Calculus is a branch of mathematics that studies continuous change. It is divided mainly into two fields: differential calculus and integral calculus. Differential calculus focuses on the concept of the derivative, which relates to the rate of change of a quantity. The derivative of a function at a certain point is a measure of how the function value changes as its input changes.

The process of finding a derivative is called differentiation. In the context of parametric equations, such as x=g(t) and y=f(t), you can take the derivative of each function with respect to the independent variable t to find how x and y change over time. The conditions dx/dt<0 and dy/dt<0 stem from this concept — indicating that as t progresses, the values of x and y are decreasing, which you would graphically represent with a curve moving downward and to the left, illustrating a declining motion.

Cartesian Coordinate System

The Cartesian coordinate system is a two-dimensional plane defined by two perpendicular axes: typically, the horizontal 'x-axis' and the vertical 'y-axis'. Used for plotting points, lines, and curves, this system is the foundation of algebraic geometry. Every point in the plane is defined by an ordered pair of numbers (x, y), which represent its coordinates.

When dealing with parametric equations, each value of the independent variable t provides a unique pair of (x, y) coordinates which can be plotted. Following the instructions from Step 2 in the textbook solution, you would draw the 'x' and 'y' axes with labeled arrows showing their positive direction. In adherence with the conditions provided in the exercise, the graph of the parametric functions will start in the upper right quadrant and, as time increases, will move down and left into the other quadrants, reflecting the decreasing nature of both x and y with respect to t.