Question

Using the Intermediate Value Theorem In Exercises \(91-94,\) use the Intermediate Value Theorem and a graphing utility to approximate the zero of the function in the interval \([0,1] .\) Repeatedly "zoom in" on the graph of the function to approximate the zero accurate to two decimal places. Use the zero or root feature of the graphing utility to approximate the zero accurate to four decimal places. $$ h(\theta)=\tan \theta+3 \theta-4 $$

Short answer

The zero of the function \(h(\theta)=\tan \theta+3\theta -4\) in the interval [0,1] is approximated to two decimal places and four decimal places respectively using the Intermediate Value Theorem and a graphing utility. The exact values would vary with different graphing utilities.

Step-by-step solution

Plot the function

Use a graphing utility to sketch the function \(h(\theta)=\tan \theta+3\theta -4\). Make sure to set your viewing window to include the interval [0,1] on the \(\theta\)-axis.

Apply Intermediate Value Theorem

The Intermediate Value Theorem states that if a function is continuous on a closed interval [a, b] and k is any number between \(f(a)\) and \(f(b)\), then there is at least one number c in the interval [a, b] such that \(f(c)=k\). Here, by looking at the graph, note that \(f(0)>0\) and \(f(1)<0\), therefore by IVT, there exists some c in [0,1] such that \(f(c) = 0\).

Approximate the zero to two decimal places

Now, 'zoom in' on the graph in the interval [0,1] to find approximately where the function crosses the x-axis. This will give an approximate zero of the function. Record this to two decimal places.

Approximate the zero to four decimal places

Further refine previous approximation by using the zero or root feature of the graphing utility. This should give a more accurate approximation, accurate to four decimal places.

Key concepts

Approximating Zeros

Approximating zeros is a critical skill in calculus and numerical analysis. It involves finding a number, or 'zero', where a function's output equals zero. In calculus exercises, this is often done within certain intervals if the exact solutions cannot be found analytically.

In the context of the Intermediate Value Theorem, this process becomes more intuitive. For the function given as \( h(\theta) = \tan \theta + 3\theta - 4 \), approximating zeros on the interval [0,1] starts with creating a visual representation of where the function crosses the x-axis. In step-by-step terms, this would mean plotting the function and narrowing down the interval where a sign change indicates the presence of a zero.

The art of zooming in on the graph allows students to iteratively get closer to the exact value where the function hits zero. Modern graphing utilities can aid in this process, giving us the facility to identify zeros to two decimal places initially and then refine to four decimal places for increased accuracy.

Continuity of Functions

The concept of the continuity of functions plays a central role in understanding the Intermediate Value Theorem. A function is considered continuous on an interval if there are no breaks, jumps, or holes in its graph over that interval.

The function \( h(\theta) = \tan \theta + 3\theta - 4 \) is continuous on the interval [0,1] as long there are no vertical asymptotes or discontinuities within that domain. This aspect ensures that values transition smoothly from one point to another across the interval.

Continuity is a precondition for applying the Intermediate Value Theorem. This theorem essentially tells us that for any continuous function defined over an interval [a, b], and for any value k between \( f(a) \) and \( f(b) \), there exists at least one point c in [a, b] where \( f(c) = k \). For example, when a function changes sign over an interval, a zero exists — a crossing point on the x-axis, which corresponds to 'k' being zero in this case.

Graphing Utility Usage

Leveraging a graphing utility is a practical approach to tackling calculus exercises involving the visualization of functions and the identification of important features such as zeros. A graphing utility can vary from simple online graphing calculators to advanced mathematical software packaging.

In our example, using a graphing utility helps us visualize the function \( h(\theta) \) over the selected interval and employ the zoom feature to 'hone in' on the zero. This interactive approach reduces the chance of overlooking asymptotic behavior or other anomalies that might not be evident through algebraic manipulation alone.

Students are advised to first familiarize themselves with the basic functions of a graphing utility, such as setting up viewing windows, using zoom functions, and employing the zero or root feature. Mastering these tools not only aids in exercises like the one discussed but also enriches understanding of function behavior and the practical application of theorems like IVT.

Calculus Exercises

Calculus exercises like the one presented are designed to enhance understanding of fundamental concepts like zeros, continuity, and the application of the Intermediate Value Theorem. These problems often combine analytical work with numerical methods and graphical analysis to provide a comprehensive understanding.

In the process of approximating zeros of \( h(\theta) \) within the interval [0,1], students engage in both visualization and the use of theorems to solidify their conceptual knowledge. It's crucial for students to work through such exercises regularly, as they form an integral part of the learning process in calculus. Each step, from plotting the function to fine-tuning the zero approximations, reinforces the connection between the abstract theory and its practical implications.

Variations of these exercises can also include exploring different intervals, functions, or even comparing graphical results with analytical solutions, providing a well-rounded approach to understanding continuous functions and the power of the Intermediate Value Theorem in calculus.