Question

Using the Intermediate Value Theorem In Exercises \(95-98\) , verify that the Intermediate Value Theorem applies to the indicated interval and find the value of \(c\) guaranteed by the theorem. $$ f(x)=x^{3}-x^{2}+x-2, \quad[0,3], \quad f(c)=4 $$

Short answer

By applying the Intermediate Value Theorem, it is verified that it can be applied to the function \(f(x) = x^3 - x^2 + x - 2\) over the interval [0,3] since 4 lies between the function’s values at the endpoints of the interval. By graphical method, the value of \(c\) such that \(f(c) = 4\) is approximately \(c = 1.650\).

Step-by-step solution

Verify The Function is Continuous on the Interval

The function \(f(x) = x^3 - x^2 + x - 2\) is a polynomial function and polynomial functions are continuous on all real numbers. Hence, the function is continuous on the interval \([0,3]\).

Verify the Value falls between the Function Values at the Endpoint

Calculate \(f(0)\) and \(f(3)\) and verify that \(4\) is in between these values. \(f(0) = (0)^3 - (0)^2 + 0 - 2 = -2\) and \(f(3) = (3)^3 - (3)^2 + 3 - 2 = 19\). Since \(4\) is between \(-2\) and \(19\), the IVT applies.

Find the c value

Use a method of approximation to find a \(c\) such that \(f(c) = 4\). This could involve graphical methods, numerical methods, or iterative methods. For simplicity, we'll identify \(c\) graphically by plotting the function and identifying approximately where it touches \(y = 4\). Note that this won't yield an exact value. For an exact value, more complex mathematical manipulation would be needed, such as applying the Newton-Raphson method to solve for \(c\). Graphing and inspecting visually, we find that \(f(c) = 4\) at around \(c = 1.650\).

Key concepts

Continuous Function

A continuous function is one that does not have any breaks, jumps, or holes at any point within its domain. Simply put, you can draw the graph of the function without lifting your pencil from the paper. This concept is crucial when we talk about the Intermediate Value Theorem (IVT), which requires the function to be continuous on a given interval.
Polynomials are a great example of continuous functions because they're smooth and connected throughout their entire domain, which includes all real numbers.

• For the function in our exercise, f(x) = x^3 - x^2 + x - 2, we are dealing with a polynomial function.
• This guarantees that the function will be continuous over any interval, including the interval [0, 3] given in the problem.

Understanding continuous functions helps us predict the behavior of the function over an interval, which is why they are essential for applying the IVT.

Polynomial Function

A polynomial function is a type of algebraic expression that involves terms in the form of \[ ax^n + bx^{n-1} + ... + k \] where each letter represents a constant and each exponent is a non-negative integer.

In our example, f(x) = x^3 - x^2 + x - 2 is a polynomial of degree 3. This is because the highest power of the variable the function reaches is 3.

• Polynomial functions are wonderful to work with because they are always continuous and differentiable, making them predictable and smooth.
• They provide a great context for understanding more advanced topics like the IVT as well as various approximation techniques.

Their continuity over all real numbers ensures that any polynomial function will meet the requirements for applying the Intermediate Value Theorem on any interval.

Numerical Approximation

Numerical approximation involves finding an estimate of the value or root of a function when an exact solution is difficult to obtain analytically. This concept often comes into play when we apply the Intermediate Value Theorem to non-trivial functions.
In the given exercise, we need to find a value of c for which f(c) = 4. To do so, graphical or numerical methods are used:

• Graphically, we plotted the function and visually inspected where it reaches y = 4.
• Numerically, methods like bisection or Newton-Raphson could offer more precise numerical approximations.

While visual inspection is helpful for an initial guess, numerical methods can provide a more exact value by iterating or "guessing and checking" until the error is negligible. Understanding numerical approximation helps in solving many real-world problems where exact methods are too complex or impossible to apply directly.

Graphical Methods

Graphical methods are useful tools for understanding the behavior of functions by visualizing their graph. This involves plotting the function on a coordinate plane and using it to identify key points, such as intercepts and intervals of increase or decrease.
In the context of the Intermediate Value Theorem, graphical methods can help us identify where a function takes on a specific value. In our exercise involving f(x) = x^3 - x^2 + x - 2, visualizing the graph lets us approximate where the function meets y = 4.

• By plotting the function, you gain insight into its slope, curvature, and other characteristics.
• This helps with making educated guesses for the numerical approximation process.

Graphical methods are especially beneficial for quickly estimating answers and understanding function behavior without rigorous computations.