Problem 24 - College Algebra and 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. $$ f(x)-\sqrt{x+1} ;(8,3) $$

Step-by-Step Solution

Answer

The slope of the tangent line to the graph of the function \( f(x) = \sqrt{x+1} \) at the point \((8,3)\) is \( \frac{1}{6} \).

Step 1: Express the derivative definition

The derivative of a function can be found using the limit definition of the derivative, which is given as \[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}. \]

Step 2: Substitute the function into the derivative definition

Substitute \( f(x) = \sqrt{x+1} \) into the derivative definition:\[f'(x) = \lim_{h \to 0} \frac{\sqrt{(x+h) + 1} - \sqrt{x + 1}}{h}\]

Step 3: Simplify the expression in the limit

Multiply the numerator and denominator of the fraction by the conjugate of the numerator to get rid of the square root: \[f'(x) = \lim_{h \to 0} \frac{((x+h) + 1) - (x + 1)}{h [ \sqrt{(x+h)+1} + \sqrt{x +1} ]}\]This simplifies to:\[f'(x) = \lim_{h \to 0} \frac{h}{h [ \sqrt{(x+h)+1} + \sqrt{x +1} ]}\]Simplify this further by cancelling out \(h\) to get \[f'(x) = \frac{1}{\sqrt{x+1} + \sqrt{(x+h)+1}}\]

Step 4: Evaluate the limit as h approaches 0

After simplifying the expression, take the limit as \(h\) approaches 0:\[f'(x) = \frac{1}{\sqrt{x+1} + \sqrt{x+1}} = \frac{1}{2\sqrt{x+1}}\]

Step 5: Calculate the derivative of the function at the given point

Lastly, you substitute \(x = 8\) (as this is the given x-coordinate of the tangent point) into the derivative. This will give you the slope of the tangent line at the given point.\[f'(8) = \frac{1}{2\sqrt{8+1}} = \frac{1}{6}\]

Key Concepts

Limit Definition of the Derivative

Understanding the limit definition of the derivative is crucial to grasp how we calculate the rate of change at a specific point on a function. It's based on the concept of finding the slope of the tangent line as it touches the curve at that point. In mathematical terms, if you have a function f(x), the derivative at x is defined as
\br> \br> $$ f'(x) =