Problem 17 - Calculus Early Transcendentals

Question

Let \(f(x)=x^{2}.\) (a) Estimate the values of \(f^{\prime}(0), f^{\prime}\left(\frac{1}{2}\right), f^{\prime}(1),\) and \(f^{\prime} 2)\) by using a graphing device to zoom in on the graph of f. (b) Use symmetry to deduce the values of \(f^{\prime}\left(-\frac{1}{2}\right), f^{\prime}(-1)\) and \(f^{\prime}(-2).\) (c) Use the results from parts (a) and (b) to guess a formula for \(f^{\prime}(x)\). (d) Use the definition of a derivative to prove that your guess in part (c) is correct.

Step-by-Step Solution

Answer

The derivative \(f'(x) = 2x\).

Step 1: Understanding the Derivative Concept

The derivative of a function at a point provides the slope of the tangent line to the function's graph at that point. It reflects how the function value changes as the input changes.

Step 2: Using Graphing Device for Estimation (Part a)

Use a graphing device to zoom in on the graph of \(f(x) = x^2\) at points 0, \(\frac{1}{2}\), 1, and 2. You will observe that as you zoom in, the curve appears more linear, and you can estimate the slope of the tangent lines at these points. The estimated slopes will be 0 at \(x=0\), 1 at \(x=\frac{1}{2}\), 2 at \(x=1\), and 4 at \(x=2\). Therefore, \(f'(0)=0\), \(f'\left(\frac{1}{2}\right)=1\), \(f'(1)=2\), and \(f'(2)=4\).

Step 3: Utilizing Symmetry (Part b)

The function \(f(x) = x^2\) is symmetric about the y-axis, which means the derivative of the function at positive x is equal in magnitude but opposite in direction to its derivative at negative x. Therefore, using symmetry: \(f'\left(-\frac{1}{2}\right)=-1\), \(f'(-1)=-2\), and \(f'(-2)=-4\).

Step 4: Guessing the Derivative Formula (Part c)

From the results of parts (a) and (b), it's evident that the derivative is twice the input value. This suggests the formula for \(f'(x)\) is \(2x\).

Step 5: Using the Definition of a Derivative (Part d)

Let's rigorously derive \(f'(x)\) using the definition. The definition of a derivative is \[f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\]For \(f(x)=x^2\),\[f(x+h) = (x+h)^2 = x^2 + 2xh + h^2\]Substituting back into the derivative definition gives:\[f'(x) = \lim_{h \to 0} \frac{(x^2 + 2xh + h^2) - x^2}{h} = \lim_{h \to 0} \frac{2xh + h^2}{h}\]Simplifying the expression:\[f'(x) = \lim_{h \to 0} (2x + h) = 2x\]The derivative of \(f(x) = x^2\) is indeed \(f'(x) = 2x\).

Key Concepts

Understanding Tangent Lines

A tangent line provides a unique snapshot of the behavior of a function at a particular point. It's the straight line that just touches a curve at a single point, matching the curve's slope at that point. Imagine a rocket launching: the tangent line tells us the rocket's instantaneous direction and speed at any given time.
For the function given, a tangent line at any point on the parabola can be approximated by visually inspecting or calculating the derivative at that point. The closer you zoom in on the graph of a smooth curve, the more the curve resembles its tangent line. This idea is pivotal in understanding not just where a function goes, but how it moves.

Exploring Symmetry in Derivatives

The symmetry in functions can greatly simplify our calculations, especially in derivatives. For even functions like the parabola given by \(f(x) = x^2\), symmetry around the y-axis ensures that the slope of the tangent line on one side of the y-axis is the same magnitude but opposite in sign on the other side.
This symmetry implies that the change in direction is natural and expected as we move across the y-axis. If you find that \(f'(1) = 2\), then \(f'(-1)\) naturally equals \(-2\), showing a mirror effect. This gives us a deeper insight into understanding the patterns and behaviors of graphs without recalculating every point from scratch.

Utility of Graphing Calculators in Calculus

Graphing calculators are powerful allies in learning calculus. They allow you to visualize complex functions, test hypotheses, and verify calculations from your manual work. With a graphing calculator, you can zoom into a graph at specific points, helping you to approximate the derivative through tangent lines visually.
In this exercise, a graphing calculator becomes extremely useful for approximating derivatives at specific points before you derive the precise calculations. This practical approach helps build intuition about the relationship between a function and its tangent lines before diving deeper into algebraic methods.

The Definition of a Derivative

A derivative represents an essential concept in calculus, reflecting the rate of change of a function with respect to its variable. It signifies how a function changes at a particular instant, like the speed of a car on a winding road. The formal definition states \(f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\), offering a precise formula to calculate this rate of change.
For \(f(x) = x^2\), the derivative tells us how steeply the curve rises or falls as \(x\) changes. Using this definition, you can systematically derive that the slope of the tangent at any point on this parabola is \(2x\), confirming your intuitive guess from visual approximations. This rigorous approach bridges the gap between intuitive understanding and mathematical proof.