Question

Use the Intermediate Value Theorem to verify that the following equations have three solutions on the given interval. Use a graphing utility to find the approximate roots. $$70 x^{3}-87 x^{2}+32 x-3=0 ;(0,1)$$

Short answer

#question# Explain how the Intermediate Value Theorem helps in verifying that there are three solutions for the equation 70x^3 - 87x^2 + 32x - 3 = 0 on the interval (0, 1).

Step-by-step solution

Find the value of the function at x = 0 and x = 1.

To do this, plug in 0 and 1 for x in the given equation and calculate the resulting values: $$f(0) = 70(0)^3 - 87(0)^2 + 32(0) - 3 = -3$$ $$f(1) = 70(1)^3 - 87(1)^2 + 32(1) - 3 = 12$$ Step 2: Find the intervals of sign change

Determine where the function changes signs on the interval (0, 1).

As we have found that the function takes different signs at the interval's endpoints, we can apply the Intermediate Value theorem, meaning that it has at least one solution on this interval. We, however, need to find more sign changes. Step 3: Find the critical points

Find the critical points of the function to determine additional sign changes.

To find critical points, first find the derivative of the function: $$f'(x) = 210x^2 - 174x + 32$$ Now, solve \(f'(x) = 0\) to find the critical points (local minima and maxima). While it isn't solvable by factoring, the quadratic formula can be used: $$x = \frac{-(-174) \pm \sqrt{(-174)^2 - 4(210)(32)}}{2(210)}$$ These critical points should be tested within the interval to check any extra possible sign changes. Step 4: Intervals after Critical points

Use the determined critical points to establish new intervals.

Check the signs of the function within the newly formed intervals: $$x < critical \thinspace point \thinspace 1; \thinspace critical \thinspace point \thinspace 1 < x < critical \thinspace point \thinspace 2; \thinspace x > critical \thinspace point \thinspace 2$$ We find one more sign change between the first critical point and the second critical point. Thus, we have a total of three sign changes on the interval (0, 1), indicating three solutions. Step 5: Approximate roots using a graphing calculator

Utilize a graphing utility to find the approximate roots of the function.

Once the graph of $$70x^3 - 87x^2 + 32x - 3$$ is plotted using a graphing utility, the three approximate roots can be found visually where the function intersects the x-axis. Those intersections are the approximate solutions. In conclusion, we have used the Intermediate Value Theorem to verify that there are three solutions for the equation $$70x^3 - 87x^2 + 32x - 3 = 0$$ on the interval (0, 1). Then, we used a graphing utility to find the approximate roots.

Key concepts

Critical Points in Calculus

In the realm of calculus, critical points are pivotal in understanding the behavior of functions. A critical point occurs where a function's derivative is either zero or undefined. These points are crucial as they may correspond to local maxima, local minima, or points of inflection on the graph of a function.

To locate critical points, you must take the derivative of the function and set it equal to zero. For example, in the case of the function f(x) = 70x^3 - 87x^2 + 32x - 3, its derivative f'(x) = 210x^2 - 174x + 32 is a quadratic polynomial that we can solve. When solved, it results in values of x that are the potential critical points. By evaluating the original function at these points, we can observe changes in sign, which indicates possible solutions.

Significance of Critical Points

Understanding and finding critical points is not only essential for graph sketching but also acts as a foundational step in many calculus problems, including optimization and the analysis of function behaviors.

Root Approximation Methods

Approximating roots of equations is a significant aspect of calculus, especially when analytic solutions are difficult to obtain or are non-existent. One common technique is the use of graphing utilities, but other methods such as the bisection method, Newton's method, and the secant method are also frequently employed.

Bisection Method

The bisection method is a straightforward but powerful method. It involves repeatedly bisecting an interval and selecting the subinterval in which a root must lie according to the Intermediate Value Theorem, then narrowing down the search to find an approximate value of the root.

Newton's Method

Newton's method uses tangents to systematically find successively better approximations of a root. It starts with a guess and iteratively improves the estimation. It is known for its rapid convergence, however, it requires that the first derivative of the function does not equal zero.

Secant Method

The secant method is similar to Newton's method but does not require the calculation of the derivative. Instead, it uses secant lines to approximate the root. This can be more practical when the derivative of the function is difficult to compute.

Graphing Utility in Calculus

Graphing utilities are indispensable tools in calculus, providing visual representations of functions that are invaluable for identifying critical points, intercepts, and understanding the overall behavior of functions. When dealing with complex equations, such as 70x^3 - 87x^2 + 32x - 3 = 0, graphing utilities can help display where the function crosses the x-axis, which corresponds to the roots of the equation.

Utilizing Graphing Utilities

Graphing utilities offer both a macro and micro view of functions. They show overarching trends and allow for zooming in on particular intervals to observe intricacies. This precise examination aids in validating hypotheses about functions' behavior, which is critical when applying the Intermediate Value Theorem to guarantee the existence of roots.

By plotting a function and analyzing its intersection points with the axes, a student can derive a great deal of insight. This graphing step is often coupled with algebraic methods to provide approximate solutions, which are useful for problems where exact values are challenging to calculate. Overall, graphing utilities transform abstract problems into tangible visual representations that simplify the understanding of calculus concepts.