Problem 42 - Algebra and Trigonometry

Question

(a) Compare the rates of growth of the functions \(f(x)=3^{x}\) and \(g(x)=x^{4}\) by drawing the graphs of both functions in the following viewing rectangles: (i) \([-4,4]\) by \([0,20]\) (ii) \([0,10]\) by \([0,5000]\) (iii) \([0,20]\) by \(\left[0,10^{5}\right]\) (b) Find the solutions of the equation \(3^{x}=x^{4}\) , rounded to two decimal places.

Step-by-Step Solution

Answer

Exponential growth of \(f(x)\) surpasses \(g(x)\) as x increases; intersections at \(x=0.3\) and \(x=4.2\).

Step 1: Identify Function and Viewing Rectangle

To compare the growth, we need to draw the graphs of \(f(x) = 3^{x}\) and \(g(x) = x^{4}\) on three different viewing rectangles: (i) \([-4, 4]\) by \([0, 20]\), (ii) \([0, 10]\) by \([0, 5000]\), and (iii) \([0, 20]\) by \([0, 100000]\).

Step 2: Plot the Graphs for Viewing Rectangle (i)

Graph \(f(x) = 3^x\) and \(g(x) = x^4\) on viewing rectangle \([-4, 4]\) by \([0, 20]\).- For \(f(x)\), note it crosses (0,1) and grows rapidly in the positive direction.- For \(g(x)\), it starts from zero and increases, crossing through several x-values before reaching an upward curve.Compare plots: \(g(x)\) grows faster initially for small positive x but \(f(x)\) surpasses as x increases beyond 2.

Step 3: Plot the Graphs for Viewing Rectangle (ii)

Graph \(f(x) = 3^x\) and \(g(x) = x^4\) on viewing rectangle \([0, 10]\) by \([0, 5000]\).- For \(f(x)\), observe its exponential increase, particularly around x=5 or more.- For \(g(x)\), notice polynomial growth, reaching 10,000 at x=10.Compare plots: \(f(x)\) quickly catches and grows larger than \(g(x)\) after x surpasses about 5.

Step 4: Plot the Graphs for Viewing Rectangle (iii)

Graph \(f(x) = 3^x\) and \(g(x) = x^4\) on viewing rectangle \([0, 20]\) by \([0, 100000]\).- \(f(x)\) shows a rapid increase, becoming significant past x=10.- \(g(x)\)'s growth appears smoother but dominated by \(f(x)\) as x increases.Compare plots: \(f(x)\) overwhelmingly outpaces \(g(x)\) in this range, highlighting its exponential nature.

Step 5: Setting Up the Equation for Solution

To solve \(3^{x} = x^{4}\), we find points of intersection when both functions are equal.

Step 6: Solve Equation Using Graphs and Numerical Methods

From graph intersection and numerical analysis (like numerical iteration or graphing calculators), approximate where \(3^{x} = x^{4}\) holds:- Visual inspection and numerical calculation find intersections approximately at \(x=0.3\) and \(x=4.2\).Round solutions to two decimal places.

Key Concepts

Polynomial Functions

Polynomial functions are a fundamental concept in algebra. These are mathematical expressions consisting of variables and coefficients, involving only non-negative integer powers. For example, the function given in the original exercise, \(g(x) = x^4\), is a polynomial with degree 4.

Some key characteristics of polynomial functions include:

• They are continuous and smooth, meaning they have no sharp corners or holes.
• The degree of the polynomial dictates the general shape and the number of turning points.
• They have real coefficients and thus are defined for all real numbers.

The degree of the polynomial guides us on how it behaves as \(x\) becomes very large or very small. In our example, \(x^4\) means that as \(x\) increases or decreases dramatically, the value of \(x^4\) also increases rapidly. Unlike exponential functions, polynomial functions tend to have slower growth rates as \(x\) grows.

Graphing Functions

Graphing is a powerful tool that lets us visualize and compare different functions. The exercise involves graphing \(f(x) = 3^x\) and \(g(x) = x^4\) across various intervals to compare their growth behavior.

Here's a simple breakdown of what happens in the different intervals:

• In the interval \([-4, 4]\), \(g(x)\) starts off with quicker growth for small positive values, but \(f(x)\) surpasses it gradually.
• In \([0, 10]\) and \([0, 20]\), the exponential function \(f(x)\) noticeably outpaces \(g(x)\) within a few steps. Exponential growth is much faster than polynomial growth after certain points.

Graphing these functions allows us to see the crossover point where the exponential function begins to grow excessively faster. Such visual analysis can often make the abstract idea of comparing functions more concrete. It is also crucial for identifying solutions to equations by finding points of intersection visually.

Equation Solving

Solving equations involves finding which values of \(x\) satisfy an equation. The exercise provided requires us to find the points where the functions \(3^x\) and \(x^4\) are equal. This is done by setting up the equation \(3^x = x^4\) and finding interception points.

Approximation methods like numerical analysis or using graphing tools are handy here. By checking the graph of \(3^x\) and \(x^4\), we can see where they intersect. Analytically solving such equations may not always be straightforward, thus graphical solutions or numerical estimations are practical.

According to the step-by-step solution provided, the intersection points are approximately at \(x = 0.3\) and \(x = 4.2\), which can be verified using a graphing calculator or software that supports numerical computation. Rounding these solutions to two decimal places gives us a more precise and usable result for practical purposes. Remember to always verify approximate solutions within the context of the problem to ensure accuracy.