Problem 5 - Calculus Early Transcendentals

Question

Describe the end behavior of \(f(x)=-2 x^{3}\)

Step-by-Step Solution

Answer

Answer: As \(x\) approaches positive infinity, \(f(x)\) approaches negative infinity. As \(x\) approaches negative infinity, \(f(x)\) approaches positive infinity.

Step 1: Identify the given function

The given function is \(f(x)=-2 x^{3}\), a cubic function with a negative coefficient.

Step 2: Analyzing behavior as x approaches positive infinity

To find the end behavior as x approaches positive infinity, we will examine the limit of the function as x approaches positive infinity: \(\lim_{x\to\infty}(-2 x^{3})\).

Step 3: Use properties of limits to find the limit as x approaches positive infinity

We know that as x gets larger, \(x^3\) also gets larger. Since the coefficient is -2, the entire expression will get more negative. Therefore, the limit for the function as x approaches positive infinity is: \(\lim_{x\to\infty}(-2 x^{3})=-\infty\).

Step 4: Analyzing behavior as x approaches negative infinity

To find the end behavior as x approaches negative infinity, we will examine the limit of the function as x approaches negative infinity: \(\lim_{x\to-\infty}(-2 x^{3})\).

Step 5: Use properties of limits to find the limit as x approaches negative infinity

We know that as x gets more negative, \(x^3\) also gets more negative, but it remains negative due to the cube. The coefficient (-2) multiplied by the negative value, as x approaches negative infinity, will result in a positive outcome. Therefore, the limit for the function as x approaches negative infinity is: \(\lim_{x\to-\infty}(-2 x^{3})=\infty\).

Step 6: Describing the end behavior of the function

With the two limits we've calculated, we can now describe the end behavior of \(f(x)=-2 x^{3}\): As \(x\) approaches positive infinity, \(f(x)\) approaches negative infinity. As \(x\) approaches negative infinity, \(f(x)\) approaches positive infinity.

Key Concepts

Cubic Functions

A cubic function is a type of polynomial function where the highest degree of the variable is three. Generally, it takes the form of \(f(x) = ax^3 + bx^2 + cx + d\). In this expression, \(a\), \(b\), \(c\), and \(d\) are constants, with \(a eq 0\) to ensure it's truly cubic. Cubic functions can exhibit a variety of behaviors depending on the values of these coefficients, particularly because they can have up to three roots and two turning points.

A key characteristic of cubic functions is their potential to change directions more than once. This attribute makes them useful in modeling real-world phenomena such as fluid dynamics or other physical processes where rates of change themselves change.

In the specific case of the function \(f(x) = -2x^3\), it's important to note that it lacks terms of degree two, one, and zero (\(bx^2 + cx + d = 0\)), making this a simplified or monomial cubic function. The absence of the quadratic and linear terms means that the function will not turn back on itself, following a more predictable upward or downward swoop based on the leading coefficient.

Limits in Calculus

Limits are a critical concept in calculus, particularly when analyzing the behavior of functions as they tend towards certain points or infinity. The limit of a function gives us a way to understand what value a function approaches as the input gets very large or very small.

Evaluating the limits of a function can teach us about its "end behavior"—how it behaves as \(x\) approaches positive or negative infinity. For large positive or negative values of \(x\), the dominant term in a polynomial (here, the cubic term \(-2x^3\)), has the most influence on the function's value.

When working with limits, especially in the context of functions like \(f(x) = -2x^3\), you'll notice the importance of considering the sign of the coefficient. As \(x\) becomes very large in the positive sense, the cubic term \(x^3\) dominates and its negative coefficient drives the overall expression further and further negative, tending to \(-\infty\). Conversely, as \(x\) becomes very large in the negative direction, \(x^3\) still dominates but with the negative sign flipping the direction of growth, driving it towards \(+\infty\).

Negative Coefficient

The coefficient in a polynomial function dictates the direction and orientation of the graph. Particularly in the cubic function \(f(x) = -2x^3\), the negative coefficient \(-2\) greatly influences the function's end behavior.

With polynomial functions, the largest power term (the leading term) determines how the function behaves as \(x\) approaches infinity or negative infinity. In \(f(x) = -2x^3\), the \(-2\) technically scales the values of \(x^3\), flipping the standard cubic graph upside down:

• As \(x\) approaches \(+fty\), instead of the function going to \(+\infty\) due to \(x^3\), the negative scaling factor drives it to \(-\infty\).
• As \(x\) goes to \(-\infty\), although \(x^3\) is negative, multiplying by \(-2\) makes the whole expression positive, hence it tends towards \(+\infty\).

Understanding coefficients is fundamental in knowing not just how steep or stagnant a curve may be, but also the direction it will face as it extends outwards. The negative sign plays a pivotal role in determining the ultimate direction of the sides of the graph, changing what one would otherwise expect with a positive-leading cubic term.