Problem 39 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find each limit by evaluating the derivative of a suitable function at an appropriate point. Hint: Look at the definition of the derivative. \(\lim _{h \rightarrow 0} \frac{3(2+h)^{2}-(2+h)-10}{h}\)

Step-by-Step Solution

Answer

The short answer is: For the given limit \(\lim _{h \rightarrow 0} \frac{3(2+h)^{2}-(2+h)-10}{h}\), the function is \(f(x) = 3x^2 - x - 10\), and its derivative is \(f'(x) = 6x-1\). Evaluating the derivative at the appropriate point \(x=2\), we get the limit value as \(f'(2) = 11\). Therefore, \(\lim_{h \rightarrow 0} \frac{3(2+h)^{2}-(2+h)-10}{h} = 11\).

Step 1: Identify the function and its derivative

Given the limit, we can say the difference quotient is \(\frac{3(2+h)^{2}-(2+h)-10}{h}\), which is for the function \(f(x) = 3x^2 - x - 10\). Now, let's find the derivative of \(f(x)\) using the formula \(f'(x)=\lim_{h \to 0}\frac{f(x+h)-f(x)}{h}\).

Step 2: Find the derivative of the function f(x)

To find the derivative, we will differentiate the function \(f(x) = 3x^2 - x - 10\) with respect to \(x\). Using the power rule, the derivative is \(f'(x) = 6x - 1\).

Step 3: Evaluate the derivative at the appropriate point

Now that we have the derivative, we need to find the value of \(h\) that corresponds to the derivative \(f'(x) = 6x - 1\). Notice that we are taking the limit as \(h\to 0\). So, we can replace \(h\) in the expression \((2 + h)\) with \(0\). Thus, we have a point \((2 + 0) = 2\). Now, we will evaluate the derivative at the point \(x=2\): \(f'(2) = 6(2) - 1 = 11\)

Step 4: Conclude the result

Now, given that we have found the value of the derivative \(f'(x)\) at the point \(x=2\), we can conclude that: \(\lim_{h \rightarrow 0} \frac{3(2+h)^{2}-(2+h)-10}{h} = 11\)

Key Concepts

Limits

Limits are a fundamental concept in calculus, acting as the foundation for derivatives and integrals. In calculus, a limit is the value a function approaches as the input approaches some value. To understand limits, think of them as the intended destination of a function's output. For instance, if we have a function and a particular point of interest, finding the limit helps us understand the behavior of the function as it nears that point.

• The notation \( \lim_{x \to a} f(x) = L \) implies that as \( x \) approaches \( a \), \( f(x) \) approaches \( L \).
• Limits are essential for defining the derivative, a core concept in calculus.
• Understanding limits aids in analyzing the continuity and growth behavior of functions.

By practicing with limits, you can better anticipate how functions behave near critical points or where they may not be explicitly defined. This is crucial for problem-solving in calculus.

Derivatives

Derivatives represent the rate at which a quantity changes. A derivative can be visualized as the slope of a function at a given point. In simple terms, if a function describes how a quantity changes over time, its derivative describes how fast it changes. Calculating derivatives involves using various rules and techniques to find these rates of change.

• The derivative of a function \( f(x) \) is often denoted \( f'(x) \) or \( \frac{d}{dx}f(x) \).
• It measures the instantaneous rate of change, meaning how the function is changing at just one specific point.
• Derivatives are used in numerous real-world applications, such as calculating velocity in physics.

Derivatives form the backbone of differential calculus, providing deeper insights into the behavior of functions and enabling more advanced analyses, such as identifying maxima and minima of curves.

Power Rule

The power rule is a quick and efficient way to find the derivative of polynomial functions. It simplifies the differentiation process by providing a straightforward formula. If \( f(x) = x^n \), where \( n \) is a constant, the derivative of \( f(x) \) is \( f'(x) = nx^{n-1} \). This rule is particularly useful for functions where variables are raised to a power.

• To apply the power rule: bring the exponent down as a coefficient, and reduce the original exponent by one to find the new exponent.
• For example, using the power rule, the derivative of \( 3x^2 \) is \( 6x \).
• The power rule helps quickly compute derivatives of functions involving powers, especially when dealing with more complex polynomial functions.

The power rule is a staple in calculus for efficiently finding derivatives, reducing the need for long calculations and enabling more focus on understanding the function's behavior.