Problem 78 - Calculus and its applications

Question

For each function, find the points on the graph at which the tangent line is horizontal. If none exist, state that fact. $$ f(x)=\frac{1}{3} x^{3}+\frac{1}{2} x^{2}-2 $$

Step-by-Step Solution

Answer

Horizontal tangents are at points (0, -2) and (-1, -5/6).

Step 1: Differentiate the function

To find the horizontal tangent, we need to calculate the derivative of the function. The derivative of a function gives us the slope of the tangent at any point on the graph. For \( f(x) = \frac{1}{3}x^3 + \frac{1}{2}x^2 - 2 \), the derivative \( f'(x) \) is found by applying the power rule. Thus, \( f'(x) = x^2 + x \).

Step 2: Set the derivative to zero

A horizontal tangent line has a slope of 0. Therefore, we need to find the points where \( f'(x) = 0 \). Set the derivative equal to zero: \( x^2 + x = 0 \).

Step 3: Solve the equation

To solve \( x^2 + x = 0 \), factor the equation: \( x(x + 1) = 0 \). The solutions are \( x = 0 \) and \( x = -1 \).

Step 4: Verify the solution points

Verify that these points are on the graph by substituting back into the original function. \( f(0) = \frac{1}{3}(0)^3 + \frac{1}{2}(0)^2 - 2 = -2 \) and \( f(-1) = \frac{1}{3}(-1)^3 + \frac{1}{2}(-1)^2 - 2 = -\frac{5}{6} \). Hence, the points on the graph with horizontal tangents are \((0, -2)\) and \((-1, -\frac{5}{6})\).

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus that involves finding the derivative of a function. The derivative represents the rate at which a function changes and is crucial for understanding behaviors and trends in calculus. It’s essentially the formula we use to calculate the slope of a tangent line at any given point on a curve.

To differentiate a polynomial function like \( f(x) = \frac{1}{3}x^3 + \frac{1}{2}x^2 - 2 \), we apply the power rule. The power rule states that for any term \( ax^n \), the derivative is \( anx^{n-1} \). Thus, the derivative of our function is:

• For \( \frac{1}{3}x^3 \), derivative: \( x^2 \)
• For \( \frac{1}{2}x^2 \), derivative: \( x \)
• The constant \(-2\) disappears since constants have a derivative of 0.

By combining these, we find \( f'(x) = x^2 + x \). Differentiation allows us to explore how the function’s value changes instantaneously at points on its graph.

Tangent Lines

A tangent line to a curve at a given point is a straight line that just "touches" the curve at that point without crossing it. It mimics the local behavior of the curve near the point of tangency. The slope of the tangent line is precisely the value of the derivative of the function at that point.

In this exercise, we seek tangent lines that are horizontal. A horizontal tangent line indicates that the slope is zero—meaning no rise or fall. By setting the derivative equal to zero, \( f'(x) = 0 \), we find points on the curve where this occurs.

• Setting \( x^2 + x = 0 \) and solving gives points where the slope of the tangent line (the derivative) is zero.
• This results in the equation \( x(x + 1) = 0 \), which factors to yield the solutions \( x = 0 \) and \( x = -1 \).

These solutions identify where on the graph the tangent is horizontal; thus, these are the points where the curve levels out before moving up or down.

Polynomial Functions

Polynomial functions are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. They are a significant class of functions you will encounter in algebra and calculus, represented generally as \( a_nx^n + a_{n-1}x^{n-1} + \, ... \, + a_1x + a_0 \).

Our specific problem involves a cubic polynomial function: \( f(x) = \frac{1}{3}x^3 + \frac{1}{2}x^2 - 2 \). Each term's exponent provides information about the function's shape and behavior:

• The \( x^3 \) term contributes to the "cubic" nature, affecting the function's end behavior.
• The \( x^2 \) term shapes the mid-range curve of the polynomial.
• The constant \(-2\) shifts the entire graph vertically downward by two unit spaces.

Understanding how these components work together helps in predicting the function's characteristics such as where it may change direction or have a horizontal tangent. These insights are crucial when solving problems involving maximums, minimums, or points of inflection.