Problem 21 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

For Problems 21 through 24 use the limit definition of \(f^{\prime}(a)\) to find the derivative of \(f\) at the point indicated. $$ f(x)=\frac{3}{2-x} \text { at } x=1 $$

Step-by-Step Solution

Answer

The derivative of the function \( f(x) = \frac{3}{2 - x} \) at the point \( x = 1 \) is 3. Therefore, \( f'(1) = 3 \).

Step 1: Substitute \( f(x) \) into the Formula

Substitute \( f(x) \) into the limit definition of the derivative formula. So, we have \( \lim_{h \to 0} \frac{f(1 + h) - f(1)}{h} \) which transforms to \( \lim_{h \to 0} \frac{\frac{3}{2 - (1 + h)} - \frac{3}{2 - 1}}{h} \). Now, simplify fractions inside the limit operation: \( \lim_{h \to 0} \frac{\frac{3}{2 - (1 + h)} - 3}{h} \).

Step 2: Simplify Further

Next, simplify the fractions further, to look like this: \( \lim_{h \to 0} \frac{\frac{3}{1 - h} - 3}{h} \). To get a common denominator for the fractions in the numerator, multiply the second term by \( \frac{1 - h}{1 - h} \). Now, the expression is \( \lim_{h \to 0} \frac{\frac{3 - 3(1 - h)}{1 - h}}{h} \). We simplify the numerator in the fraction to obtain \( \lim_{h \to 0} \frac{\frac{3h}{1 - h}}{h} \).

Step 3: Cancel Out the Common Factor

We now cancel out the common factor of \( h \) in the numerator and the denominator to get \( \lim_{h \to 0} \frac{3}{1 - h} \).

Step 4: Compute the Limit

Finally, we compute the limit as \( h \) approaches \( 0 \) which gives \( \frac{3}{1 - 0} \) and hence, \( f'(1) = 3 \).

Key Concepts

Derivative Calculation

To find the derivative at a specific point, we use the limit definition of the derivative. The formula we start with is \[ f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h} \]This expression captures how the function's value changes as we move from point \(a\) to \(a + h\).

• Start by substituting \(a\) with your given point, in this case, \(x = 1\) into the formula.
• Then plug in the function \(f(x)\) into the formula.
• The result is an expression involving \(h\) and our function evaluated at \(x = 1 + h\) and at \(x = 1\).

By calculating this limit, we derive the rate of change of our function at the specific point, revealing vital information about the function's behavior at that location.

Simplifying Fractions

Simplifying fractions is key to handling complex expressions, especially when computing derivatives. In our context, we initially encounter a fraction\[ \frac{\frac{3}{1-h} - 3}{h} \]This fraction's complexity lies within nested operations. To simplify:

• Find a common denominator for terms in the numerator. This involves multiplying each term by an expression that will equalize their denominators.
• After establishing a common denominator, combine the fractions into a single one. This simplifies the problem greatly and often reveals cancellations.
• Upon simplifying the numerator fully, ensure any like terms or potential factors with the denominator are canceled out.

This process transforms a seemingly complex problem into a clearer, more manageable expression.

Computing Limits

Limits are foundational to calculus, helping us understand how functions behave around specific points. In our exercise, we compute the limit\[ \lim_{h \to 0} \frac{3}{1-h} \]Here's how we handle such limits:

• Evaluate the simplified expression as \(h\) approaches \(0\). Our task is to substitute \(0\) into the expression where \(h\) appears.
• Verify that the expression becomes defined or well-behaved as \(h\) approaches the specified limit. If it might cause undefined conditions (like division by zero), see if any adjustments or simplifications beforehand may help.
• After substituting, assess the final expression to ensure it logically complies with mathematical principles, confirming it as the derivative.

By computing this correctly, you determine the exact slope of the function at the point in question.

Differentiation Techniques

Differentiation techniques refer to various strategies used to take derivatives. Typically, you will use the limit definition and special rules to ease this task. Here's a breakdown of approaches:

• **Using Limit Definitions**: As shown, compute derivatives directly using limits for a stronger fundamental understanding. This method is crucial for more complex functions.
• **Utilizing Simplification**: Simplify expressions before differentiating to reduce unnecessary complexity.
• **Employ Derivative Rules**: Once familiar with the basic concept, rules such as the power rule, product rule, and quotient rule are invaluable for quickly finding derivatives.
• **Practicing Problem Solving**: Regular practice solidifies these concepts, equipping students with the confidence to tackle more difficult derivative problems.

By employing these techniques, differentiation becomes an approachable skill, essential for exploring relationships within mathematics and science.