Problem 77 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

In Exercises 77 and 78 , determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(f\) and \(g\) are differentiable, then $$ \frac{d}{d x}[2 f(x)-5 g(x)]=2 f^{\prime}(x)-5 g^{\prime}(x) $$

Step-by-Step Solution

Answer

The statement is true because the constant rule and the sum rule apply to differentiable functions. Applying these rules, we find that the derivative of \( 2f(x) - 5g(x) \) is indeed equal to \( 2f'(x) - 5g'(x) \).

Step 1: Understand the Differentiation Rules

Differentiation rules we'll use are: 1. \( (c f(x))' = c f'(x) \) - where c is a constant, also known as the constant rule. 2. \( (f(x) + g(x))' = f'(x) + g'(x) \) - known as the sum rule. 3. \( (f(x) - g(x))' = f'(x) - g'(x) \) - Known as the difference rule.

Step 2: Analyze the given statement

The given statement says that if f and g are differentiable functions, the derivative of \( 2f(x) - 5g(x) \) is equal to \( 2f'(x) - 5g'(x) \).

Step 3: Apply Differentiation Rules to the Given Functions

First, let's apply the constant rule to the functions f(x) and g(x), multiplied by the constants 2 and 5 respectively: 1. The derivative of 2f(x) is \( 2f'(x) \) 2. The derivative of -5g(x) is \( -5g'(x) \) Now we will apply the sum rule (which also include differences as sums) to determine the derivative of \( 2f(x) - 5g(x) \): 3. The derivative of \( 2f(x) - 5g(x) \) = derivative of (2f(x) + (-5g(x))) = \( 2f'(x) - 5g'(x) \)

Step 4: Compare the Results

Now, let's compare our results with the given statement: $$ \frac{d}{d x}[2 f(x)-5 g(x)]=2 f^{\prime}(x)-5 g^{\prime}(x) $$ We found that the derivative of \( 2f(x) - 5g(x) \) is indeed equal to \( 2f'(x) - 5g'(x) \) using differentiability and differentiation rules. So, the statement is

Step 5: True

and the reason is because the constant rule and the sum rule apply to differentiable functions.

Key Concepts

Constant Rule of Differentiation

The constant rule of differentiation is a simplifying mantra in calculus that deals with derivatives of functions multiplied by a constant. Imagine that we have a function denoted by f(x) and a constant denoted by c. According to this rule, when we want to find the derivative of the function multiplied by the constant, \( c \cdot f(x) \), we can simply take the derivative of f(x) alone and then multiply the result by c. So, the rule is expressed mathematically as
\[ ( c \cdot f(x) )' = c \cdot f'(x) \].
Intuitively, this rule indicates that the slope of the line affected by a constant factor doesn't change in its steepness; the constant just scales it. This rule is fundamental when dealing with linear combinations of functions, and it plays nicely with the sum rule, as we will see next.

Sum Rule of Differentiation

When calculus students first encounter the sum rule of differentiation, they often breathe a sigh of relief. This rule states that the derivative of a sum of functions is simply the sum of their derivatives. In a formula, it's elegantly expressed as
\[ (f(x) + g(x))' = f'(x) + g'(x) \].
But wait, it gets even better! The sum rule also covers the subtraction of functions, making it even more versatile. So, when faced with a function like \( f(x) - g(x) \) the derivative is just as friendly:
\[ (f(x) - g(x))' = f'(x) - g'(x) \].
So, if you get a function like \(2f(x) - 5g(x)\), according to the sum rule, you break it down and differentiate piece by piece, just like solving a puzzle. This rule is a cornerstone of differentiation, making it approachable and systematic.

Differentiable Functions

A concept that is pivotal to making sense of the previous rules is that of differentiable functions. These are the functions we can actually apply differentiation rules to. But what exactly makes a function differentiable? In a nutshell, if you can draw the function without lifting your pen off the paper, and at every point along the curve you can neatly place a tangent line that touches the curve at just that point, then you've got yourself a differentiable function.
Mathematically, for a function to be differentiable at a point, the function must be continuous, and its derivativemust exist at that point. These twin conditions ensure that, just like an artist with a brush, the calculus student can smoothly apply rules like the constant and sum rule to paint a picture of the function's rate of change at any point. When functions tick these boxes of differentiability, they open up to a world of calculus possibilities.