Question

The following exercises are intended to derive the fundamental properties of the natural log starting from the definition ln(x)=?x1dtt, using properties of the definite integral and making no further assumptions. Compute the right endpoint estimates \(R_{50}\) and \(R_{100}\) of \(\int_{-3}^{5} \frac{1}{2 \sqrt{2 \pi}} e^{-(x-1)^{2} / 8}\).

Short answer

Calculate sums of evaluated function at right endpoints and multiply by their respective \\(\Delta x\\).

Step-by-step solution

Partition the Interval

To compute the right endpoint estimate, first partition the interval \([-3, 5]\) into equal subintervals, which will be dependent on whether we are computing \(R_{50}\) or \(R_{100}\). For \(R_{50}\), each subinterval length, or \(\Delta x\), will be \(\frac{5 - (-3)}{50} = \frac{8}{50} = 0.16\). For \(R_{100}\), each \Delta x\ becomes \(\frac{8}{100} = 0.08\).

Determine Right Endpoints

Next, identify the right endpoints of each subinterval for both \(R_{50}\) and \(R_{100}\). For \(R_{50}\), the right endpoints are \(-3+i\times0.16\) for \(i=1,2,\ldots,50\). For \(R_{100}\), they are \(-3+i\times0.08\) for \(i=1,2,\ldots,100\).

Apply Function to Right Endpoints

For each right endpoint, evaluate the function: \(\frac{1}{2\sqrt{2\pi}}e^{-((x-1)^2)/8}\). You will evaluate this function at \(-3+i\times0.16\) for \(R_{50}\) and \(-3+i\times0.08\) for \(R_{100}\).

Sum the Function Values

Calculate the sum of the function values for all right endpoints. For \(R_{50}\), this is \(\sum_{i=1}^{50} \frac{1}{2\sqrt{2\pi}}e^{-(((-3+i\times0.16)-1)^2)/8}\). For \(R_{100}\), it is \(\sum_{i=1}^{100} \frac{1}{2\sqrt{2\pi}}e^{-(((-3+i\times0.08)-1)^2)/8}\).

Multiply by \\Delta x to Get Estimate

Finally, multiply the sum obtained in Step 4 by \(\Delta x\) to estimate the integral. Thus, for \(R_{50}\), the estimate is \(\Delta x_{50}\sum_{i=1}^{50}\frac{1}{2\sqrt{2\pi}}e^{-(((-3+i\times0.16)-1)^2)/8}\), and for \(R_{100}\), it is \(\Delta x_{100}\sum_{i=1}^{100}\frac{1}{2\sqrt{2\pi}}e^{-(((-3+i\times0.08)-1)^2)/8}\).

Key concepts

Natural Logarithm

The natural logarithm, denoted as \(\ln(x)\), is a fundamental function in calculus and is the inverse of the exponential function \(e^x\). It has a unique base \(e\), approximately equal to 2.71828, making it vastly useful in various mathematical and scientific fields.

The natural logarithm of a positive number \(x\) can be understood through integration. Specifically, it's expressed as an integral from 1 to \(x\):\[\ln(x) = \int_{1}^{x} \frac{1}{t} \, dt\]This means that the natural log of \(x\) is the area under the curve \(y = \frac{1}{t}\) on the interval \([1, x]\).

• \(\ln(1) = 0\), since the area from 1 to 1 is zero.
• \(\ln(xy) = \ln(x) + \ln(y)\), making it useful for simplifying complex multiplications into additions.
• \(\frac{d}{dx} \ln(x) = \frac{1}{x}\), indicating how the function changes with respect to \(x\).

Understanding these properties helps in solving problems involving logarithms effectively.

Definite Integral

In calculus, a definite integral is a way to calculate the area under a curve over a specific interval \([a, b]\). It's written as:\[\int_{a}^{b} f(x) \, dx\]This notation indicates the sum of an infinite number of infinitesimally small areas. Each area is a product of a small distance \(dx\) and the function value \(f(x)\) at that point.

Key characteristics of a definite integral include:

• The integral calculates a number, representing the net signed area under the curve from \(a\) to \(b\).
• If \(f(x)\) is above the x-axis, it contributes positive area, while below contributes negative area.
• Changing the limits of integration affects both direction and value, i.e., \(\int_{b}^{a} f(x) \, dx = -\int_{a}^{b} f(x) \, dx\).

Definite integrals are widely used in computing solutions to problems in physics, engineering, and probability.

Riemann Sum

Riemann sums are the foundation of definite integrals. They provide a method of approximating the area under a curve by summing up multiple small rectangles.

Given a continuous function \(f(x)\) on an interval \([a, b]\), the interval is divided into \(n\) equal subintervals. The width of each subinterval is \(\Delta x = \frac{b-a}{n}\). For each subinterval, an endpoint—often the right endpoint—is chosen, and the function value at that point forms the height of the rectangle.

The sum is then expressed as:\[R_n = \sum_{i=1}^{n} f(x_i^*) \Delta x\]Where \(x_i^*\) is a representative sample point of each subinterval.

• In general, the more subintervals, the better the approximation of the integral.
• Both left and right endpoints, as well as midpoints, can be used for calculations, affecting the result differently.
• The concept guides understanding of how integrals are calculated and approximated.

The Riemann sum approach is essential for evaluating complex integrals numerically.

Gaussian Function

The Gaussian function is a significant mathematical function often appearing in statistics and probability. It is described by its bell-shaped curve called the Gaussian distribution or normal distribution.

A typical Gaussian function is given by:\[G(x) = \frac{1}{\sqrt{2\pi}\sigma} e^{-\frac{(x-\mu)^2}{2\sigma^2}}\]where \(\mu\) is the mean or expectation of the distribution, \(\sigma\) is the standard deviation, and \(e\) is Euler's number.

• This function is symmetric around the mean, making it pivotal in statistical analyses.
• The area under the entire Gaussian curve is exactly 1, indicating a full probability distribution.
• It describes data naturally clustering around a central value with no bias left or right.

The Gaussian function is essential in fields like data science and machine learning due to its significant role in normal distribution.