Problem 68 - Single Variable Calculus

Question

Let \(f(x)=x^{5}-3 x^{2}+2 x+5\) a. Show that there is at least one number \(c\) in the interval \([0,2]\) such that \(f(c)=12\). b. Use a graphing utility to find all values of \(c\) accurate to five decimal places. Hint: Find the point(s) of intersection of the graphs of \(f\) and \(g(x)=12\)

Step-by-Step Solution

Answer

We used the Intermediate Value Theorem (IVT) to show that there exists at least one \(c\) in the interval \([0,2]\) such that \(f(c)=12\), where \(f(x)=x^5-3x^2+2x+5\). By finding the intersection points between \(f(x)\) and \(g(x)=12\) using a graphing utility, we can determine the value(s) of \(c\) accurate to five decimal places.

Step 1: Check if the function is continuous in the given interval

Since \(f(x)\) is a polynomial function, it is continuous in its domain which includes our given interval \([0, 2]\).

Step 2: Calculate f(a) and f(b) using the endpoints of the interval

We will use the given interval [0, 2] and compute the values of the function at the endpoints: \(f(a)=f(0)\) and \(f(b)=f(2)\). \(f(0) = 0^5 - 3 *0^2 + 2*0 + 5 = 5\) \(f(2) = 2^5 - 3 * 2^2 + 2*2 + 5 = 32 -12 +4 + 5= 29\)

Step 3: Check if there exists a value between f(a) and f(b) equal to 12

We see that \(f(a) = 5\) and \(f(b) = 29\) which confirms that \(12\) is within the range of values between \(f(a)\) and \(f(b)\). Since the function is continuous and \(12\) is between \(f(a)\) and \(f(b)\), according to the IVT, there exists at least one \(c\) in the interval \([0, 2]\) such that \(f(c) = 12\). b. Finding the values of c using a graphing utility

Step 1: Graph the functions f(x) and g(x)

Using a graphing utility or software, graph the functions \(f(x) = x^5 - 3x^2 + 2x + 5\) and \(g(x) = 12\) on the same graph.

Step 2: Locate the point(s) of intersection

Observe the graph and identify the points where the functions intersect on the interval \([0,2]\). These intersections correspond to the values of \(x\) for which \(f(x)=12\).

Step 3: Use the graphing utility to find the intersection point(s) accurate to five decimal places

Use the software or tool's built-in functions to find the exact coordinates of intersection(s) within the interval \([0,2]\) accurate to five decimal places. Note down the \(x\)-coordinate(s) of the intersection points as this represents the value(s) of \(c\).

Key Concepts

Polynomial Functions

Polynomial functions are one of the cornerstones of algebra and calculus. A polynomial function is a mathematical expression consisting of variables (also known as indeterminates) and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables. An example of a polynomial function is the one given in our exercise, where we have an expression like \(f(x) = x^5 - 3x^2 + 2x + 5\).

The significance of polynomial functions extends to their predictability and smoothness. Each term in a polynomial has a degree which is based on the exponent of the variable. The degree of the polynomial is the highest degree of any single term. This particular function \(f(x)\) is a quintic polynomial because the highest exponent is 5. One important characteristic of polynomial functions is that they are continuous and differentiable everywhere, which makes them quite manageable and easy to work with in calculus.

Continuity in Calculus

Continuity is a fundamental concept in calculus that deals with the behavior of functions. Informally, a function is continuous at a point if there is no 'break' or 'jump' in the graph of the function at that point. More formally, a function is continuous at a point \(x = a\) if the following three conditions are met:

• The function \(f(a)\) is defined.
• The limit of \(f(x)\) as \(x\) approaches \(a\) exists.
• The limit of \(f(x)\) as \(x\) approaches \(a\) is equal to \(f(a)\).

For the polynomial function given in the exercise, the concept of continuity assures us that we can graph this function without lifting our pencil, thus it has no breaks or holes. The Intermediate Value Theorem (IVT) applies to continuous functions over a closed interval. It states that if a function is continuous on a closed interval \[a, b\], then it takes on every value between \(f(a)\) and \(f(b)\) at least once. This theorem plays a crucial role in solving many calculus problems, as it allows us to confirm the existence of roots or solutions within a certain interval without necessarily identifying the exact values.

Graphing Utility

A graphing utility is a tool, often software-based, used for plotting the graphs of functions to visually interpret their behavior. With advances in technology, such utilities have become indispensable in teaching and learning mathematics, providing immediate visual feedback that aids in understanding complex functions and their interactions.

When dealing with the graph of the polynomial function and the constant function \(g(x) = 12\) as given in the exercise, a graphing utility allows for a precise and detailed visualization of these functions and their points of intersection. The ability to zoom in and out, as well as calculate exact values of intersections, bring an interactive element to learning and problem-solving. Indeed, using a graphing utility not only complements analytical methods but can also enhance the user's intuition with respect to the behavior of functions over an interval.

Points of Intersection

Points of intersection are the coordinates where two or more graphs meet or cross each other. In calculus, finding these points is essential in solving a plethora of problems, such as optimization, area computations, and solving equations graphically.

The exercise provided demonstrates finding the points of intersection between the graph of the polynomial function \(f(x)\) and the constant function \(g(x) \). The x-coordinates of these points are significant because they represent the values where \(f(x) = g(x)\), which translates to the solution of the equation \(f(x) = 12\) in the context of our problem. Graphing utilities help us to not only vizualize where these points lie but also to calculate their exact values to a required precision. This visualization and calculation are key in understanding how functions behave and interact with one another.