Problem 64 - Algebra and Trigonometry

Question

Sketch the function represented by the given parametric equations. Then use the graph to determine each of the following: a. intervals, if any, on which the function is increasing and intervals, if any, on which the function is decreasing. b. the number, if any, at which the function has a maximum and this maximum value, or the number, if any, at which the function has a minimum and this minimum value. \(x=e^{t}, y=t\)

Step-by-Step Solution

Answer

The function \(y=ln(x)\) is always increasing on its domain \(x > 0\). It doesn't have a maximum or minimum value.

Step 1: Converting to the Standard Function Form

To begin, rewrite the parametric equations as one function. As \(x=e^{t}\), replace \(t\) with \(ln(x)\) in the second equation, resulting in the function: \(y=ln(x)\)

Step 2: Sketching the function \(y=ln(x)\)

Using knowledge of the natural logarithm, sketch the graph, which starts at minus infinity when \(x=0\) and increases without bound as \(x\) increases.

Step 3: Determining where the function is increasing or decreasing

The derivative of \(y=ln(x)\) is \(1/x\), which is always greater than 0 (for x>0), indicating the function is always increasing.

Step 4: Determining the maximum or minimum value

The function \(y=ln(x)\) doesn't have any maxima or minima because it's always increasing on its domain \(x > 0\)

Key Concepts

Parametric Equations

Parametric equations represent a set of related quantities as explicit functions of an independent parameter, usually denoted by t. A pair of equations such as x=f(t) and y=g(t) defines a path in the Cartesian plane as the parameter t varies. Unlike standard function forms, parametric equations can describe more complex curves, including loops and intersections, which a single function y=f(x) cannot.

When dealing with parametric equations, we can sometimes eliminate the parameter to find a relationship directly between x and y. For instance, if we have the parametric equations x=e^t and y=t, we can eliminate t by expressing it in terms of x: t = ln(x), thus obtaining the standard form y = ln(x). This approach enables us to analyze the function using standard calculus techniques.

Natural Logarithm

The natural logarithm, denoted as ln(x), is the inverse function of the exponential function e^x. It is a logarithm to the base e, where e is an irrational and transcendental number approximately equal to 2.71828. The natural logarithm has several key properties that often come into play:

• It is defined for all positive real numbers x.
• It always passes through the point (1,0), since the logarithm of 1 is 0.
• The function approaches negative infinity as x approaches zero from the positive side.
• It increases without bound as x grows, though the rate of increase slows down.

Understanding the behavior of the natural logarithm is crucial when dealing with growth processes or when using certain compound interest formulas in finance.

Increasing and Decreasing Intervals

To determine intervals where the function increases or decreases, one typically examines the derivative of the function. If the derivative f'(x) is positive over an interval, the function is increasing on that interval. Conversely, if the derivative is negative, the function is decreasing. When the derivative equals zero, it may indicate a potential maximum or minimum, or a point of inflection.

For the natural logarithm function y = ln(x), the derivative is 1/x, which is positive for all positive values of x. Hence, the function is always increasing and does not decrease on its domain.

Maximum and Minimum Values of Functions

The maximum or minimum values of a function, often referred to as extrema, occur at points where the function’s derivative is zero (critical points) or where the derivative does not exist, provided these are also turning points. To determine whether a critical point is a maximum or a minimum, one can use the first or second derivative test. If a function f(x) has a maximum at a point x=c, then f(c) is the highest value f(x) attains on its domain. A similar interpretation holds for minimum values.

In the case of y = ln(x), since the function is always increasing and does not possess any critical points on its domain, it does not have any local extrema. Therefore, there are no maximum or minimum values within the function's domain.

Derivative of Natural Logarithm

The derivative of the natural logarithm function y = ln(x) gives us the slope of the tangent to the curve at any point (x, ln(x)). For y = ln(x), the derivative is 1/x. This derivative tells us that the slope of the tangent is inversely proportional to x. As x gets larger, the slope of the tangent gets smaller, but it remains positive. This is consistent with an increasing function that does not exhibit any maxima or minima but instead increases indefinitely with a decreasing rate.

The knowledge of this derivative is fundamental when solving various problems in calculus, particularly those involving rates of change and in understanding the behavior of logarithmic functions.