Problem 57 - Calculus an Applied Approach

Question

(a) sketch the graphs of \(f\) and \(g,(b)\) find \(f^{\prime}(1)\) and \(g^{\prime}(1),(c)\) sketch the tangent line to each graph when \(x=1,\) and \((d)\) explain the relationship between \(f^{\prime}\) and \(g^{\prime}\). $$ \begin{array}{l}{f(x)=x^{3}} \\ {g(x)=x^{3}+3}\end{array} $$

Step-by-Step Solution

Answer

The graphs of the functions are cubic curves. Both \(f^{\prime}(1)\) and \(g^{\prime}(1)\) are equal to 3. The tangent line to each curve at \(x = 1\) are \(y = 3(x - 1) + 1\) for \(f(x)\) and \(y = 3(x - 1) + 4\) for \(g(x)\). The derivatives of \(f\) and \(g\) are identical, as the constant term in \(g(x)\) does not affect the slope.

Step 1: Graph Functions

Sketch the graphs of the two functions. The function \(f(x) = x^{3}\) is a basic cubic function, with a point at the origin (0,0), that increases to the right and decreases to the left along the x-axis. The graph of \(g(x) = x^{3} + 3\) is just a translation of \(f(x) = x^{3}\) upwards by 3 units.

Step 2: Calculate Derivatives at \(x = 1\)

Find \(f^{\prime}(1)\) and \(g^{\prime}(1)\) by first getting the derivatives of \(f(x)\) and \(g(x)\) which are \(f^{\prime}(x) = 3x^{2}\) and \(g^{\prime}(x) = 3x^{2}\), respectively. Then, substitute \(x = 1\) into each of the these expressions. Both \(f^{\prime}(1)\) and \(g^{\prime}(1)\) simplify to \(3\).

Step 3: Sketch Tangent Lines

Sketch the tangent line to each graph at \(x=1\). Remember that the tangent line at a point (p, q) on the graph of a function can be found using the equation \(y = f^{\prime}(p)(x-p) + f(p)\). Using this formula, you find that the tangent for \(f^{\prime}\) and \(g^{\prime}\) at \(x = 1\) are \(y = 3(x-1) + 1\) and \(y = 3(x-1) + 4\), respectively. Plot these lines on the same graph as \(f(x)\) and \(g(x)\).

Step 4: Explain Relationship

Explain the relationship between \(f^{\prime}\) and \(g^{\prime}\). Their derivatives are identical because the derivative of a constant term (like the +3 in g(x)) is zero, so it does not affect the slope of the function.

Key Concepts

Sketching Graphs

To understand the behavior of functions, particularly in calculus, it is often beneficial to start with graphical representations. Sketching graphs provides a visual comprehension of a function's growth or decay, roots, and symmetry.

When dealing with cubic functions like those in the exercise, their distinctive 'S' shaped curves reflect their changing slopes at different points. The function described by f(x) = x^3 is a fundamental cubic graph displaying these traits. Starting from the origin, the curve dips below the x-axis before curving back upwards, passing through the origin and continuing to rise indefinitely.

The graph of g(x) = x^3 + 3 is similar, but it is shifted higher on the y-axis by 3 units. Such translations in graphs do not alter the original function's shape; it merely moves it up or down the grid. Sketching both f(x) and g(x) helps us compare how parallel shifts affect the graph while maintaining the original function's characteristics.

Tangent Lines

In calculus, a tangent line is a straight line that touches a curve at precisely one point without intersecting it. The significance of a tangent line is that it represents the slope or derivative of a curve at that single point. This concept is fundamentally important for understanding rates of changes and dynamic behaviors in various fields such as physics and engineering.

For example, with our cubic functions, drawing a tangent line at x=1 provides insight into the curve's steepness at that point. To sketch tangent lines for f(x) and g(x), one uses the derivative of the function, which offers the slope of the tangent, and includes the point of tangency to create the line's equation. The similarity in tangency for f(x) and g(x) indicates analogous slope behaviors despite the vertical shift in the graph of g(x).

Derivative Calculations

The derivative of a function represents the rate at which the function's value is changing at any given point. Derivative calculations are at the heart of differential calculus and give us the ability to predict and understand the dynamism in models and real-world phenomena.

To find the derivative of a cubic function, we apply the power rule which states that the derivative of x^n is nx^(n-1). Thus, for f(x) = x^3, the derivative is f'(x) = 3x^2. At x=1, the value of the derivative, f'(1), represents the instantaneous rate of change or the slope of the tangent line at that point.

In our exercise, both f(x) and g(x) share the same derivative function f'(x) = g'(x) = 3x^2 which simplifies to 3 at x=1. Therefore, both functions grow at the same rate when x equals 1, which is crucial to understanding their behavior through the lens of calculus.

Cubic Functions

A cubic function is a polynomial of degree three, characterized by an equation of the form f(x) = ax^3 + bx^2 + cx + d where a, b, c, and d are constants and the highest power of x is 3. These functions are noteworthy due to their distinctive properties and graphs, which feature turns and inflection points.

The most basic cubic function is f(x) = x^3, which serves as the parent function to more complex cubic equations. Perturbations to this parent function, such as addition or subtraction of values, translate or stretch its graph, but do not alter its fundamental cubic nature.

In understanding the distinction between f(x) = x^3 and g(x) = x^3 + 3, we must recognize that while the additional constant in g(x) shifts its graph upwards, it does not affect the function's overall growth rate as x increases or decreases. This concept is reflected in the unaltered slopes of their tangents at any matching x-value and represents a foundational principle within the study of calculus.