Problem 19 - Calculus an Applied Approach

Question

Use the limit definition to find the slope of the tangent line to the graph of \(f\) at the given point. $$ f(x)=x^{2}-1 ;(2,3) $$

Step-by-Step Solution

Answer

The slope of the tangent line to the graph of \(f(x) = x^{2} - 1\) at the point (2,3) is 4.

Step 1: Identify \(a\), \( \) and \(f(x)\)

In this case, \(a\) is the x-coordinate of the given point, so \(a=2\). The function \(f(x)\) is given as \(x^{2} - 1\).

Step 2: Substitute \(a\), \(h\), and \(f(x)\) to the formula

The formula for the slope of the tangent line using limits is: \(m = \lim_{h->0}\frac{f(a+h) - f(a)}{h}\). If we substitute \(a=2\) and \(f(x) = x^{2} - 1\) into the formula, it becomes: \(m = \lim_{h->0}\frac{((2+h)^{2} - 1) - (2^{2} - 1)}{h}\). This simplifies to: \(m = \lim_{h->0}\frac{4+4h+h^{2} - 1 - 3}{h}\).

Step 3: Simplify the expression and compute the limit as \(h->0\)

Simplify the numerator to get: \(m = \lim_{h->0}\frac{4h + h^{2}}{h}\). Then factor \(h\) out of the numerator: \(m = \lim_{h->0} h(4 + h)\). As \(h\) is a factor of both the numerator and denominator, we can cancel \(h\) out to get: \(m = \lim_{h->0}(4 + h)\). Evaluating the limit as \(h\) approaches 0, we get: \(m = 4\).

Key Concepts

Limit Definition

The limit definition is a foundational concept in calculus that helps us understand the behavior of functions as they approach a particular point. When evaluating the slope of a tangent line, we use this definition to find the instantaneous rate of change at a specific point. This involves the expression:

• \( m = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h} \)

Here, \( m \) represents the slope of the tangent line, \( a \) is the specific x-value of the point of interest on the graph, and \( h \) is a small increment approaching zero.

To apply this definition, we evaluate how the function value at \( a + h \) differs from the function value at \( a \) as \( h \) shrinks towards zero. This gives us insight into how steep the graph of the function is at that point. The limit is essential because it allows us to calculate this exact slope without having a secant line's influence, which represents an average rate of change over an interval.

Slope of Tangent Line

The slope of the tangent line at a point on a curve is crucial for understanding the behavior of the function at that exact point. It effectively measures how the function is changing at that point, which can be thought of as its instantaneous rate of change.

For the function \( f(x) = x^2 - 1 \) at the point \( (2, 3) \), we calculate this slope using the limit definition discussed before. Substituting the point and the function into the limit expression:

• \( m = \lim_{h \to 0} \frac{(2+h)^2 - 1 - (2^2 - 1)}{h} \)
• This simplifies to \( m = \lim_{h \to 0} \frac{4h + h^2}{h} \).

Once simplified, you cancel \( h \) and evaluate the remaining part as \( h \) approaches 0, obtaining the slope \( m = 4 \).

This tells us that at the point \( (2, 3) \), the curve is rising steeply, with a slope of 4, meaning for every unit moved horizontally, the function moves up 4 units. Understanding this slope gives insights into the nature and behavior of the function at this specific point.

Tangent Line

The tangent line to a curve at a given point is a straight line that just "touches" the curve at that point. It doesn't intersect the curve anywhere near that vicinity (if the curve isn't sharply turning).

Having already found the slope of the tangent line at the point \( (2, 3) \), which is 4, we can determine the equation of this line using the point-slope form of a line:

• \( y - y_1 = m(x - x_1) \)

Substituting \( m = 4 \) and the point \( (2, 3) \), we have:

• \( y - 3 = 4(x - 2) \)

This simplifies to the line equation \( y = 4x - 5 \).

The tangent line equation is essential in calculus because it best approximates the function near that point. This concept can be expanded and used in many applications, such as finding the linear approximation of non-linear functions or solving optimization problems. It showcases the power of calculus in describing and predicting behavior at infinitesimally small scales.