Problem 72 - Calculus Single Variable

Question

Suppose that \(f\) is a function whose graph has a tangent line at each point. If \(g(x)=f(x)+\alpha\) for some constant \(\alpha\), show that the graph of \(g\) has a tangent line at each point and that the slope of the tangent line to the graph of \(g\) at \((c, g(c))\) is the same as the slope of the tangent line to the graph of \(f\) at \((c, f(c)) .\) Explain this geometrically.

Step-by-Step Solution

Answer

The slope of the tangent line to \( g(x) \) is the same as to \( f(x) \), as adding \( \alpha \) shifts the graph vertically but doesn't change slopes.

Step 1: Understand the Problem

We need to demonstrate that if the function \( g(x) = f(x) + \alpha \) is derived from \( f(x) \) by adding a constant \( \alpha \), the graph of \( g \) maintains the property of having a tangent line at every point. Additionally, the slope of this tangent line at any point \((c, g(c))\) should be equal to the slope of the tangent line of \( f \) at the same \( x \)-coordinate.

Step 2: Recognize the Tangent Line Formula

A tangent line at a point \((c, f(c))\) on the graph of a function \( f(x) \) has a slope given by the derivative \( f'(c) \). We will analyze how this derivative is affected by adding a constant \( \alpha \) to \( f(x) \) to form \( g(x) = f(x) + \alpha \).

Step 3: Differentiate \( g(x) \)

The derivative of a function gives the slope of the tangent line at any point. Since \( g(x) = f(x) + \alpha \), the derivative of \( g \) with respect to \( x \) is:\[g'(x) = \frac{d}{dx}[f(x) + \alpha] = f'(x) + 0 = f'(x).\]This shows that at every point, the slope of the tangent line of \( g \) is the same as the slope of the tangent line of \( f \).

Step 4: Explain Geometrically

Geometrically, the addition of \( \alpha \) to \( f(x) \) results in a vertical shift of the graph of \( f \) upwards by \( \alpha \) units. This shift does not affect the steepness or direction of the tangent lines, thus the slopes of the tangent lines remain unchanged across corresponding points on the graphs of \( f \) and \( g \).

Step 5: Conclude on Tangency

Since \( f(x) \) has a tangent line at every point, \( g(x) \), which is a vertically shifted form of \( f(x) \), will also have tangent lines at each point. The key point is that vertical shifts do not alter the horizontal behavior of the graph concerning tangents.

Key Concepts

Derivative

The derivative is a fundamental concept in calculus that describes how a function changes at any given point. It gives us the slope of the tangent line to the function's graph at a specific point. For a function \( f(x) \), its derivative, denoted as \( f'(x) \), tells us the rate at which \( f(x) \) changes as \( x \) changes.

Understanding derivatives is essential because:

• They help us find the slope of a curve at any given point, which is indispensable in analyzing the behavior of the function.
• They provide insights into the speed of change and can predict future behavior of a function.

Consider a scenario from the exercise where \( g(x) = f(x) + \alpha \). The derivative \( g'(x) \) becomes \( f'(x) \) because the constant term \( \alpha \) does not change as \( x \) changes; thus, its derivative is zero. This shows that adding a constant does not affect the rate of change of the function. Therefore, \( g'(x) \) and \( f'(x) \) are identical.

Function Transformation

A function transformation refers to modifications made to the graph of a function, such as translation, scaling, or reflection. A common transformation is to add or subtract a constant from a function. In this exercise, the transformation \( g(x) = f(x) + \alpha \) simply adds a constant \( \alpha \) to the function \( f(x) \).

• Addition of a constant \( \alpha \) to a function \( f(x) \) results in a **vertical shift** of the graph.
• The graph maintains its shape and slope, but each point on the graph moves vertically by \( \alpha \) units.

Function transformations are useful because they allow us to manipulate functions and study their behavior under different conditions without changing the essential characteristics of the function. Understanding such transformations helps in analyzing and solving problems that involve modified functions.

Graph Shifts

Graph shifts are movements of the entire graph of a function, either horizontally or vertically. They are among the simplest types of transformations and occur when constants are added or subtracted directly within the function's equation.

In the case of a vertical shift, seen in \( g(x) = f(x) + \alpha \), each point on the graph of \( f(x) \) is moved up by \( \alpha \) units if \( \alpha \) is positive, or down if \( \alpha \) is negative.
Here’s what happens:

• The shape and slope of the graph do not change. Only the vertical position changes.
• Tangent lines remain parallel before and after the shift, maintaining their original slopes.

This visualizes why the derivative remains unaffected by such a transformation. Understanding graph shifts is crucial as they allow you to predict how a function will behave when altered in straightforward ways such as moving the entire graph up or down.