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Chapter 12: Problem 105

For the following exercises, find the average rate of change \(\frac{f(x+h)-f(x)}{h}\) $$ f(x)=\sqrt{x} $$

Short Answer

Expert verified
The average rate of change is \( \frac{1}{\sqrt{x+h} + \sqrt{x}} \).

Step by step solution

01

Substitute Function into Formula

The average rate of change formula given is \( \frac{f(x+h)-f(x)}{h} \). For the function \( f(x) = \sqrt{x} \), substitute \( x \) and \( x+h \) into the function. Thus, \( f(x+h) = \sqrt{x+h} \) and \( f(x) = \sqrt{x} \).
02

Substitute Values into the Formula

Replace \( f(x+h) \) and \( f(x) \) in the average rate of change formula: \( \frac{\sqrt{x+h} - \sqrt{x}}{h} \). This is our expression to simplify.
03

Rationalize the Numerator

To simplify \( \frac{\sqrt{x+h} - \sqrt{x}}{h} \), multiply the numerator and the denominator by the conjugate of the numerator, which is \( \sqrt{x+h} + \sqrt{x} \), to rationalize it: \( \frac{(\sqrt{x+h} - \sqrt{x})(\sqrt{x+h} + \sqrt{x})}{h(\sqrt{x+h} + \sqrt{x})} \).
04

Simplify the Expression

The numerator will be \((\sqrt{x+h})^2 - (\sqrt{x})^2 = x+h-x = h\). So, after rationalizing, we have \( \frac{h}{h(\sqrt{x+h} + \sqrt{x})} \).
05

Cancel Common Terms

Observe that \( h \) in the numerator and denominator cancels out: \( \frac{h}{h(\sqrt{x+h} + \sqrt{x})} = \frac{1}{\sqrt{x+h} + \sqrt{x}} \). This is the simplified average rate of change.

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Key Concepts

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

Rationalizing the Numerator
When working with expressions such as \( \frac{\sqrt{x+h} - \sqrt{x}}{h} \), rationalizing the numerator is a crucial step. This essentially means making the numerator a rational expression.To do this, we multiply both the numerator and the denominator by the conjugate of the numerator. The conjugate of the numerator \( \sqrt{x+h} - \sqrt{x} \) is \( \sqrt{x+h} + \sqrt{x} \). The purpose of this multiplication is to eliminate the radicals from the numerator.### How It Works:
  • Multiply the numerator by its conjugate: \((\sqrt{x+h} - \sqrt{x})(\sqrt{x+h} + \sqrt{x})\).
  • Use the difference of squares formula to simplify: \(a^2 - b^2 = (a-b)(a+b)\), where \(a = \sqrt{x+h}\) and \(b = \sqrt{x}\).
  • After applying the formula, the numerator becomes \((\sqrt{x+h})^2 - (\sqrt{x})^2 = x+h - x = h\).
The result effectively removes the radicals from the numerator, simplifying further manipulation of the expression.
Radical Functions
Radical functions include an expression with a variable inside a root, commonly a square root. The function \( f(x) = \sqrt{x} \) is a classic example of a radical function.Understanding radical functions helps in interpreting the behavior of their graphs, which are often curves that start at a point and gradually increase or decrease.### Key Properties of Radical Functions:
  • The domain generally includes only non-negative numbers since arithmetic with real numbers does not permit the square root of negatives.
  • The range consists of non-negative numbers as well, a reflection of the square root output always being non-negative.
Designated by a \( \sqrt{} \) symbol, radical functions also require unique techniques for manipulation and simplification, especially when working within calculus, where expressions need to be modified for differential and integral analysis.
Simplifying Expressions
Simplifying expressions is about reducing them to their simplest forms without changing their values. This often involves combining like terms, reducing fractions, and simplifying roots. In the step-by-step solution provided, simplifying took place after rationalizing the numerator.### Simplification Steps:
  • After rationalizing \( \frac{\sqrt{x+h} - \sqrt{x}}{h} \) to \( \frac{h}{h(\sqrt{x+h} + \sqrt{x})} \), we then simplify further by cancelling the common factor \( h \) from the numerator and denominator.
  • Thus, the expression simplifies to \( \frac{1}{\sqrt{x+h} + \sqrt{x}} \).
  • This reflects a transformation from a more complex to a simpler expression, facilitating easier application in calculus problems.
By practicing these techniques, you develop the skill to simplify complex algebraic and radical expressions, an essential skill for progressing in math, particularly calculus.

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Most popular questions from this chapter

For the following exercises, find the average rate of change \(\frac{f(x+h)-f(x)}{h}\) $$ f(x)=x^{2}+4 x-100 $$

For the following exercises, evaluate the limits algebraically. $$ \lim _{h \rightarrow 0}\left(\frac{(2-h)^{3}-8}{h}\right) $$

For the following exercises, determine whether or not the given function \(f\) is continuous. If it is continuous, show why. If it is not continuous, state which conditions fail. $$f(x)=\frac{x^{3}-4 x^{2}-9 x+36}{x^{3}-3 x^{2}+2 x-6}$$

For the following exercises, evaluate the limits algebraically. $$ \lim _{h \rightarrow 0}\left(\frac{\sqrt{5-h}-\sqrt{5}}{h}\right) $$

Find the equation of the tangent line to the graph of \(f(x)\) at the indicated \(x\) value. $$f(x)=-x^{3}+4 x ; x=2.$$
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