Problem 69 - Calculus

Question

Are the statements true or false? Give an explanation for your answer. If \(f(a) \neq g(a),\) then \(f^{\prime}(a) \neq g^{\prime}(a)\).

Step-by-Step Solution

Answer

False, because functions can have the same derivative even if their values differ at a given point.

Step 1: Understanding the Functions

We have two functions, \(f(x)\) and \(g(x)\). We know that at a point \(a\), these functions have different values, meaning \(f(a) eq g(a)\). This means their graphs do not intersect at this point.

Step 2: Analyzing Derivatives

The problem asks if this inequality at a point implies that their derivatives are not equal at that point, that is, whether \(f^{\prime}(a) eq g^{\prime}(a)\). The derivative represents the slope of the tangent line to the function at a given point.

Step 3: Considering the Possibility

It is possible for two different functions to have the same derivative at a point without having the same value. For example, the functions \(f(x) = x^2\) and \(g(x) = x^2 + 1\) have different values at any point but the same derivative \(f'(x) = 2x\) for any \(x\).

Step 4: Conclusion

Since it's possible for \(f(a) eq g(a)\) and yet \(f^{\prime}(a) = g^{\prime}(a)\), the original statement is false. The lack of equality between the functions' values does not necessarily lead to a difference in their derivatives.

Key Concepts

Derivative

Derivatives are at the heart of calculus and refer to the rate at which a function changes. Imagine a car driving along a road. The speedometer reading at any instant shows the car's instantaneous velocity, akin to a derivative at a point for a function. Mathematically, the derivative of a function \(f(x)\) at a point \(x = a\) is represented as \(f'(a)\). It signifies the slope of the tangent line to the graph of the function at that point.
The derivative helps determine how fast or slow a function is increasing or decreasing. A large positive derivative means the function is increasing rapidly, while a large negative derivative indicates a rapid decrease.
It provides insights into the behavior of functions, making derivatives an indispensable tool in calculus.

Functions

Functions are mathematical relationships between inputs and outputs. They are like machines where you feed in a number (input), and out comes another number (output). For example, the function \(f(x) = 2x\) doubles the input value.

In the context of this exercise, we consider functions \(f(x)\) and \(g(x)\). These functions can look different at specific points, indicated by different output values, such as \(f(a) eq g(a)\). It's important to realize that functions can differ at certain points and yet behave similarly in terms of their derivatives.
This means that even if their values at a point are not equal, their rate of change can still match, illustrating the diverse nature of functions and their behaviors.

Graph Analysis

Graph analysis involves interpreting the visual representation of functions, particularly focusing on slopes and intersections. In this exercise, analyzing the graphs of \(f(x)\) and \(g(x)\) at point \(a\) can reveal important information.

Even if the two functions do not intersect at this point, meaning their values differ (as in \(f(a) eq g(a)\)), their graphs might still have parallel tangent lines, leading to equal derivatives \(f'(a) = g'(a)\).

• Parallel tangents indicate equal slopes, irrespective of where the functions themselves stand vertically.
• Therefore, different graph positions don't necessarily imply different derivatives.

This thorough graph analysis verifies that differences in function values do not imply differences in their derivatives, though it might intuitively seem so.