Problem 59 - Calculus Early Transcendentals

Question

Fill in the blanks with right or midpoint, an interval, and a value of \(n\). In some cases, more than one answer may work.\(\sum_{k=1}^{4} f(1+k) \cdot 1\) is a ______ Riemann sum for \(f\) on the interval [ ____ ,_____ ]with \(n=\) ______ .

Step-by-Step Solution

Answer

Answer: The expression \(\sum_{k=1}^{4} f(1+k) \cdot 1\) is a right Riemann sum for \(f\) on the interval [2,5] with \(n=4\).

Step 1: Determine the type of Riemann sum

The type of Riemann sum depends upon the evaluation points within each subinterval. Since the summation has the form \(\sum_{k=1}^{4} f(1+k) \cdot 1\), and \(1+k\) produces an endpoint (right) value within each subinterval, it is a right Riemann sum.

Step 2: Determine the interval

To find the interval, we need to look at the range in the summation. The summation goes from \(k=1\) to \(k=4\). Since the function is \(f(1+k)\) at each step, the interval starts at \(x=1+1=2\) and ends at \(x=1+4=5\). Therefore, the interval is \([2,5]\).

Step 3: Determine the value of n

The value of \(n\) represents the number of subintervals in a Riemann sum. In this case, the summation has four terms (\(k=1,2,3,4\)), which means that there are 4 subintervals. Thus, \(n=4\). So, the given expression is a right Riemann sum for \(f\) on the interval \([2, 5]\) with \(n=4\). Answer: \(\sum_{k=1}^{4} f(1+k) \cdot 1\) is a right Riemann sum for \(f\) on the interval [2,5] with \(n=4\).

Key Concepts

Calculus

Calculus is a branch of mathematics that deals with continuous change. It's mainly divided into two areas: Differential Calculus and Integral Calculus. Riemann sums are part of Integral Calculus. They help us approximate the area under a curve, which is essentially finding an integral. By breaking down the area into small rectangles (subintervals), we evaluate the function at specific points—like the right endpoint—to approximate the total area. Understanding calculus requires grasping how these approximations lead to exact values as they become infinitely small.

Subintervals

Subintervals are smaller segments within a larger interval on which a function is evaluated in Riemann sums. If you have an interval \[a, b\], by dividing it into smaller parts, you create subintervals. For example, if you have a total interval from 2 to 5 and divide it into four parts, each with a width of 1, you create subintervals: [2,3], [3,4], [4,5], etc. The choice and number of subintervals (like four in our case with n=4) are crucial in determining the accuracy of the Riemann sum.

Function Evaluation

Function evaluation is about calculating the function's value at specific points within subintervals during a Riemann sum process. In the expression \(\sum_{k=1}^{4} f(1+k) \cdot 1\), the function \(f\) is evaluated at points like \(f(2), f(3), f(4),\) and \(f(5)\). This evaluation plays a key role in determining the height of the rectangles used in Riemann sum approximations for the integral. Here, since each evaluation shifts by a constant \(+1\), it's clear we're using the right endpoint method to determine each rectangle's height.

Interval Notation

Interval notation is a mathematical notation used to represent the set of numbers lying between two endpoints. For example, \[2, 5\] indicates the set of all real numbers \(x\) where \(2 \leq x \leq 5\). This includes both endpoints, hence it's a closed interval. In Riemann sums, this notation not only defines the domain over which the sum is calculated but also establishes the total span that is being divided into subintervals. Interval notation simplifies expressing these ranges clearly, making it a preferred choice in mathematical contexts for specifying the set of possible input values for a function.