Problem 17 - The Calculus with Analytic Geometry

Question

Given \(f(x)=x^{3 / 2}\), prove that \(f_{+}^{\prime}(0)\) exists and find its value. Draw a sketch of the graph.

Step-by-Step Solution

Answer

The right-hand derivative \(f^{'}_{+}(0) = 0\).

Step 1: Understand the Function

Given function is \(f(x)=x^{3 / 2}\). This function needs to be analyzed to find the right-hand derivative at \(x = 0\).

Step 2: Right-Hand Derivative Definition

The right-hand derivative of \(f(x)\) at \(x = 0\) is given by: \[f^{'}_{+}(0) = \lim_{h \to 0^{+}} \frac{f(h) - f(0)}{h}\].

Step 3: Substitute the Function

Substitute \(f(x) = x^{3/2}\) into the right-hand derivative definition: \[f^{'}_{+}(0) = \lim_{h \to 0^{+}} \frac{(h)^{3/2} - (0)^{3/2}}{h} = \lim_{h \to 0^{+}} \frac{h^{3/2}}{h}\].

Step 4: Simplify the Expression

Simplify the expression inside the limit: \[\frac{h^{3/2}}{h} = h^{1/2}\]. Hence, \[f^{'}_{+}(0) = \lim_{h \to 0^{+}} h^{1/2}\].

Step 5: Evaluate the Limit

Evaluate \(\lim_{h \to 0^{+}} h^{1/2}\). As \(h\) approaches \(0\) from the right, \(h^{1/2}\) also approaches \(0\). Therefore, \[f^{'}_{+}(0) = 0\].

Step 6: Draw the Graph

To sketch the graph of \(f(x) = x^{3/2}\), note that it starts at \(f(0) = 0\) and increases as \(x\) increases. The graph will show a curve starting at the origin and curving upward to the right.

Key Concepts

Limit Definition

The concept of a limit is foundational in calculus and essential for understanding derivatives. When we talk about the right-hand derivative of a function at a point, we're actually discussing a specific type of limit. In this exercise, the right-hand derivative of a function at x = 0 is defined as follows:


< h h^{- a_{1} b_{1/2}} e f_{0,0.5,0} )

The limit here is capturing the behavior of the function as the independent variable 'h' approaches 0 from the positive side. Essentially, it allows us to analyze how the function changes infinitesimally close to the point of interest, but only from one direction, which is why it's called a right-hand derivative.

Derivative Calculation

Calculating the right-hand derivative involves using our limit definition of the derivative. Let's walk through each step to understand the process:

• First, identify the function. In this case, it's given by x^{3/2}.
• Next, substitute the function into the right-hand derivative formula:
  
  
  • Simplify the expression inside the limit: By substituting (h)^{3/2} and subtracting 0, we get:
    <
    
    As the steps show, the right-hand derivative evaluated to 0 because h^{1/2} approaches 0 as 'h' approaches 0.

These calculations reveal how the function behaves at an infinitesimally small interval from the right of 0, giving us valuable insight into its rate of change.

Graph Sketching

Drawing the graph of the function f(x) = x^{3/2} gives a visual representation of its behavior. Here are some key points to consider when sketching the graph:

• The function starts at (0, 0) because f(0) = 0.
• As x increases, the function x^{3/2} increases as well, but at a decreasing rate.
• Since the function is strictly positive for all positive x, the curve will always be above the x-axis to the right of the origin.

Sketching it, you will notice the curve trends upward and to the right, displaying an ever-decreasing slope as x increases. This slow growth is due to the x^{1/2} factor discussed in the derivative section.

Function Analysis

Analyzing f(x) = x^{3/2} provides deeper understanding on various properties:

• **Domain and Range**: The domain of function encompasses all non-negative real numbers. Its range also includes all non-negative real numbers because the function continuously increases.
• **Differentiability**: The function is differentiable for all x ≥ 0. The right-hand derivative at 0 exists and is equal to 0, showing a smooth transition.
• **Behavior**: As observed through the derivative calculations, the function's slope gradually decreases as we move rightward indicating it flattens out with increasing x.

Understanding these aspects ensures that we grasp how f(x) behaves and contributes to future problem-solving scenarios where these principles could be applied.