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Chapter 7: Problem 24

Use the limit definition to find the slope of the tangent line to the graph of \(f\) at the given point. $$ f(x)-\sqrt{x+1} ;(8,3) $$

Short Answer

Expert verified
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 by step solution

01

Express the derivative definition

The derivative of a function can be found using the limit definition of the derivative, which is given as \[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}. \]
02

Substitute the function into the derivative definition

Substitute \( f(x) = \sqrt{x+1} \) into the derivative definition:\[f'(x) = \lim_{h \to 0} \frac{\sqrt{(x+h) + 1} - \sqrt{x + 1}}{h}\]
03

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 \[f'(x) = \frac{1}{\sqrt{x+1} + \sqrt{(x+h)+1}}\]
04

Evaluate the limit as h approaches 0

After simplifying the expression, take the limit as \(h\) approaches 0:\[f'(x) = \frac{1}{\sqrt{x+1} + \sqrt{x+1}} = \frac{1}{2\sqrt{x+1}}\]
05

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.\[f'(8) = \frac{1}{2\sqrt{8+1}} = \frac{1}{6}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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) =

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Most popular questions from this chapter

Find an equation of the tangent line to the graph of \(f\) at the point \((2, f(2)) .\) Use a graphing utility to check your result by graphing the original function and the tangent line in the same viewing window. $$ f(x)=x \sqrt{x^{2}+5} $$

Use the given information to find \(f^{\prime}(2)\) \(g(2)=3\) and \(g^{\prime}(2)=-2\) \(h(2)=-1 \quad\) and \(\quad h^{\prime}(2)=4\) $$ f(x)=3-g(x) $$

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