Question

The Mercator map projection was proposed by the Flemish geographer Gerardus Mercator \((1512-1594) .\) The stretching of the Mercator map as a function of the latitude \(\theta\) is given by the function $$ G(\theta)=\int_{0}^{\theta} \sec x d x $$ Graph \(G,\) for \(0 \leq \theta<\pi / 2\)

Short answer

Answer: The graph of the stretching factor G(θ) represents how much stretching occurs at a particular latitude (θ) in the Mercator map projection. For angles between 0 and π/2, the graph starts at 0 and increases as the latitude increases, approaching infinity as θ approaches π/2. This indicates that the stretching factor increases with increasing latitude.

Step-by-step solution

Recall the definitions

Before we proceed to analyzing and graphing G(θ), we should remind ourselves of two key concepts we will be using: the integral and the secant function. The integral of a function represents the area under the curve of the function. The secant function, often abbreviated to sec, is the reciprocal of the cosine function, i.e., sec(x) = 1/cos(x). So in the context of the Mercator projection: $$G(\theta) = \int_{0}^{\theta} \sec x dx$$ Now that we have reviewed these concepts, let's proceed to calculate the integral.

Calculate the integral

The integral of sec(x) has a known result: $$\int \sec x dx = \ln \Bigg| \sec x + \tan x \Bigg| + C$$ Now, we will find G(θ) by setting limits as θ: $$G(\theta) = \int_{0}^{\theta} \sec x dx = \ln \Bigg| \sec x + \tan x \Bigg| \Bigg|^{\theta}_0$$ Evaluating it at the limits, we get: $$G(\theta) = \ln (\sec \theta + \tan \theta) - \ln(1)$$ Since, ln(1) is 0, we have: $$G(\theta) = \ln (\sec \theta + \tan \theta)$$ Now that we have the function G(θ), we can proceed to graph it.

Sketch the graph of G(θ)

To sketch the graph of G(θ) for θ ranging from 0 to just under π/2, we should consider some key points: - G(θ) is undefined at θ = π/2, as sec(π/2) is infinite. - G(θ) starts at 0 when θ = 0, as sec(0) = 1 and tan(0) = 0. Based on these insights, we can see that G(θ) will approximately start from 0 and increase over the interval 0 ≤ θ < π/2. Note that as θ approaches π/2, G(θ) will increase drastically and approach infinity. The graph shows how the stretching factor G(θ) increases with increasing latitude (θ). To draw the graph, plot the points on the x-axis using θ and the y-axis using the values of G(θ), observe the general trend of G(θ) over the interval, and create a smooth curve that reflects the behavior of the function. It might be helpful to use software or online graphing tools for plotting.

Key concepts

Integral Calculus

Integral calculus is a fundamental branch of mathematics focused on the concept of areas under curves. When we talk about an integral, we're referring to this integration process. It helps calculate the accumulation of quantities, which in this exercise, translates to understanding how the Mercator map's stretching changes with latitude.
The integral provided, \( \int_{0}^{\theta} \sec x \, dx \), represents the cumulative stretching from the equator (latitude \( \theta = 0 \)) to a specific latitude \( \theta \). By evaluating this integral, we determine the function \( G(\theta) \) that quantifies this stretching.
To solve integrals like the one given, mathematicians use known functions or rules. In this case, the integral of the secant function involves a logarithmic expression: \( \ln | \sec \theta + \tan \theta | \). This result highlights how integral calculus can turn complex problems into manageable expressions. It's a crucial tool for not only understanding theoretical concepts but also practical applications like map projections.

Secant Function

The secant function, abbreviated as \( \sec \), plays a key role in trigonometry and calculus. It is the reciprocal of the cosine function, meaning \( \sec(x) = \frac{1}{\cos(x)} \). This relationship is crucial as it highlights how closely \( \sec(x) \) and \( \cos(x) \) are related, yet different in properties, especially as angles approach \( \frac{\pi}{2} \), where cosine becomes zero and secant becomes undefined.
In the context of this exercise, the secant function defines how much each line of latitude stretches on a Mercator map. The integral of secant, which gives us \( G(\theta) \), uses this property to measure such stretching. When integrated, secant transforms into a logarithmic function: \( \ln | \sec \theta + \tan \theta | \). This transformation is a perfect example of how trigonometric functions, when used in calculus, reveal deeper insights.

Graphing Functions

Graphing a function involves plotting it on an axis to visually represent its behavior. In this exercise, we're graphing \( G(\theta) = \ln(\sec \theta + \tan \theta) \) to see how the Mercator map's stretching factor changes with latitude.
When we graph \( G(\theta) \), we start with the domain: \( 0 \leq \theta < \frac{\pi}{2} \). This is because the secant function becomes infinite at \( \theta = \frac{\pi}{2} \). As \( \theta \) increases from 0, \( G(\theta) \) starts at zero and rises as lines of latitude stretch more near the poles.
To construct the graph, plot different values of \( \theta \) on the x-axis and their corresponding \( G(\theta) \) values on the y-axis. As \( \theta \) nears \( \frac{\pi}{2} \), observe how \( G(\theta) \) climbs rapidly, indicating increasing map distortion. Sketching this graph provides a visual understanding of how integrals reflect changes over intervals, reinforcing the idea that mathematical functions are more than just equations; they are tools to visualize change.

Latitude

Latitude is a geographic coordinate that specifies the north-south position of a point on the Earth's surface. It's measured in degrees, with the equator at 0° and the poles at 90° north or south.
In relation to map projections like the Mercator, latitude is essential. The Mercator projection stretches the Earth's surface as latitude increases toward the poles, causing distortion. This is why \( G(\theta) = \int_{0}^{\theta} \sec x \, dx \), where the integral captures how each degree of latitude affects map distortion.
The function represents how much vertical stretching occurs based on latitude, with greater distortion nearer to the poles. This relationship demonstrates the importance of latitude in geographical mapping and navigation, as it ensures maps are both useful and accurate, despite the inevitable trade-offs in projecting a spherical surface onto a flat plane. Understanding latitude’s role is crucial when examining global geography or working with maps.