Problem 81 - Calculus Volume 1

Question

Use \(f^{\prime \prime}(x)=\lim _{h \rightarrow 0} \frac{f^{\prime}(x+h)-f^{\prime}(x)}{h}\) to find \(f^{\prime \prime}(x)\). $$ f(x)=2-3 x $$

Step-by-Step Solution

Answer

The second derivative \( f''(x) \) is 0.

Step 1: Find the First Derivative

To find the second derivative, we need the first derivative first. The given function is \( f(x) = 2 - 3x \). The first derivative of \( f(x) \), denoted as \( f'(x) \), is found by differentiating \( f(x) \) with respect to \( x \). The derivative of a constant is 0, and the derivative of \(-3x\) with respect to \( x \) is \(-3\). Thus, \( f'(x) = 0 - 3 = -3 \).

Step 2: Apply the Formula for the Second Derivative

Given \( f'(x) = -3 \), we use the formula for the second derivative: \( f''(x) = \lim_{h \to 0} \frac{f'(x+h) - f'(x)}{h} \). Substitute \( f'(x) = -3 \): \( f''(x) = \lim_{h \to 0} \frac{-3 - (-3)}{h} = \lim_{h \to 0} \frac{0}{h} \).

Step 3: Evaluate the Limit

Evaluate \( \lim_{h \to 0} \frac{0}{h} \). Since the numerator is 0, the limit is 0. Thus, \( f''(x) = 0 \).

Key Concepts

First Derivative

The first derivative, represented by \( f'(x) \), signifies the rate at which a function is changing at any point \( x \). It essentially tells us the slope of the tangent line to the function at a particular point. In simpler terms, if your function is a curve on a graph, the first derivative helps you understand how steep the curve is and in which direction it's going. To find \( f'(x) \), we differentiate the given function. For example, given \( f(x) = 2 - 3x \), the derivative is calculated by performing differentiation on each term:

• the derivative of a constant (like 2) is 0, because constants do not change,
• the derivative of \(-3x\) is \(-3\), since the power of \( x \) is 1 and decreasing by 1 gives 0.

So, \( f'(x) = -3 \). This result aligns with what we expect: the function \( f(x) = 2 - 3x \) is a straight line with a constant slope, meaning it does not curve, and hence, the derivative is a constant \(-3\). This constant indicates that for every increase by 1 unit in \( x \), the function decreases by 3 units.

Limit Definition

The limit definition is a fundamental concept in calculus. It forms the basis for finding derivatives and integrals. When we use the limit definition for derivatives, we are essentially finding the exact rate of change of a function at a particular point.For derivatives, the limit definition is generally expressed as: \[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\]This formula allows us to calculate the derivative from first principles by considering the change in the function \( f(x) \) as \( x \) incrementally increases by a tiny amount \( h \). This idea of making \( h \) approach zero is crucial as it captures the instantaneous rate of change, rather than an average over a larger interval.When finding the second derivative, we apply a similar concept. For instance, \[ f''(x) = \lim_{h \to 0} \frac{f'(x+h) - f'(x)}{h}\]Here, we use the first derivative \( f'(x) \) as our new function to find how its rate of change is varying. If \( f'(x) \) is constant, as in our example, the process yields \( f''(x) = 0 \) because there is no additional change happening.

Differentiation

Differentiation is the process of finding a derivative. It allows us to determine how a function behaves, especially how it increases or decreases. This is helpful in many real-world applications, like physics to determine speed or acceleration, economics to find cost functions, and much more.The rules of differentiation, like the power rule, product rule, and chain rule, help simplify the process. For functions like polynomials, where each term is differentiated separately, finding derivatives becomes routine:

• Constants disappear because their rate of change is zero,
• For any term \( ax^n \), the derivative is \( nax^{n-1} \).

In the case of the function \( f(x) = 2 - 3x \), we used basic differentiation rules to find \( f'(x) = -3 \). This tells us the function decreases linearly, which is further confirmed as we proceed to find the second derivative and discover \( f''(x) = 0 \), indicating the function has no curvature, maintaining a constant linear decrease.