Question

Explain why the function has a zero in the given interval. $$ \begin{array}{lll} \text { Function } & \text { Interval } \\ g(t)=\left(t^{3}+2 t-2\right) \ln \left(t^{2}+4\right) & {[0,1]} \end{array} $$

Short answer

Following the analysis and the steps, we confirmed that the function \(g(t)=(t^{3}+2 t-2) \ln \left(t^{2}+4\right)\) indeed has at least one zero in the interval [0,1]. The solution primarily made use of the intermediate value theorem and found that the function evaluated at 0 is less than zero and at 1 is more than zero, which by the theorem confirms the existence of a root within this interval.

Step-by-step solution

Compute the value of the function at the endpoints of the given interval

We will evaluate \(g(t)\) at points \(t=0\) and \(t=1\) i.e. column values of \(f(0)\) and \(f(1)\).\n\n\[\begin{align*} g(0) &= (0^{3}+2*0-2) * \ln (0^{2}+4)\ &= -2\ln4 \\g(1) &= (1^{3}+2*1-2) * \ln (1^{2}+4)\ &= \ln5 \end{align*}\]

Applying the Intermediate Value Theorem

Now we have found that \(g(0)<0\) and \(g(1)>0\). Since \(g(t)\) is continuous for all real values of \(t\) (since \(t^{3}, t,\) and \(\ln(t^{2}+4)\) are all continuous and the sum, difference, or product of continuous functions is also continuous), we can apply the Intermediate Value Theorem on the interval [0,1]. According to this theorem, there must exist a number \(c\) in the interval [0,1] such that \(g(c)=0\), since \(g(0)<0<g(1)\). Thus, we can conclude that the function \(g(t)\) has at least one root between 0 and 1.

Key concepts

Zeroes of a Function

Understanding the zeroes of a function is crucial in the field of mathematics, particularly when dealing with polynomials, rational functions, and other algebraic equations. A 'zero' of a function is a value of the variable for which the function's output equals zero. In other words, if you have a function, say, f(x), then any value 'a' for which f(a) = 0 is considered a zero of the function.

These zeroes are significant because they represent the points where the graph of a function crosses the x-axis. The concept of zeroes is closely linked with solving equations; finding the zeroes can reveal the roots or solutions of the equation. In our example, we're trying to identify a zero of the function g(t) within the interval [0, 1]. Learning to find zeros is not only fundamentally important for one's conceptual understanding but also gives a tangible way to visualize the behavior of functions graphically.

Continuity of Functions

The continuity of a function is a foundational concept in calculus and analysis, with wide-ranging implications for both theoretical and practical problem-solving. A function is said to be continuous at a point if the following three conditions are met:

• The function is defined at the point.
• The limit of the function as it approaches the point exists.
• The limit of the function as it approaches the point is the same as the function's value at that point.

Continuity over an interval means the function is continuous at every point within that interval. In calculus, continuity is a necessary condition for the application of many important theorems, including the Intermediate Value Theorem, which is used in our example.

The benefit of continuity is that it guarantees no breaks, jumps, or holes in the interval we're examining. For g(t), since all the individual components t^3, t, and ln(t^2+4) are continuous, their combination, being a sum, difference, or product of continuous functions, is also continuous. This ensures that g(t) behaves predictably within the interval [0, 1].

Calculus Problem Solving

In calculus problem solving, approaching and interpreting problems methodically can make a significant difference. The application of theorems such as the Intermediate Value Theorem (IVT) allows us to draw conclusions about the behavior of functions without having to solve them algebraically or graph them precisely. The IVT, in particular, states that for a continuous function that changes signs over an interval, there must be at least one zero in that interval.

To apply this theorem effectively, as illustrated in our exercise, one must first ensure that the function is continuous over the considered interval and identify the change in sign by evaluating the function at the endpoints. Whenever you encounter functions in calculus, it's important to reflect on continuity and the existence of zeroes, as these will deeply impact your solutions. This step-by-step approach not only moves us towards the answer but reinforces the core concepts of calculus—continuity, zeroes, and the utility of theorems for efficient problem solving.