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17Calculus Precalculus - Difference Quotient

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Difference Quotient

An important expression in calculus is a difference quotient. When you have a function, for example \(f(x)=x^2+3x\), we can evaluate this expression at a point, say \(x=5\) by replacing all \(x\)'s in the function with the number \(5\). This gives us \( f(5) = 5^2+3(5) = 25+15 = 40 \).

But we can also evaluate a function with something other than a constant. So \( f(a) = a^2+3(a) = a^2+3a = a(a+3)\). We can also replace \(x\) with \(x+h\) to get \(f(x+h)\). For this example, this looks like \(f(x+h) = (x+h)^2 + 3(x+h)\). We use this in the following difference quotient.

Difference Quotient of \(f(x)\)

\[ \frac{f(x+h) - f(x)}{h} \]

This is a strange looking rational expression but it is very important in calculus. So it is important for you to understand how to calculate it. Let's calculate the difference quotient for the example above, \(f(x)=x^2+3x\).

\(\displaystyle{ \frac{f(x+h) - f(x)}{h} }\)

\(\displaystyle{ \frac{[(x+h)^2+3(x+h)] - [x^2+3x]}{h} }\)

\(\displaystyle{ \frac{[x^2+2xh+h^2+3x+h] - [x^2+3x]}{h} }\)

\(\displaystyle{ \frac{x^2+2xh+h^2+3x+h - x^2-3x}{h} }\)

\(\displaystyle{ \frac{2xh+h^2+h}{h} }\)

\(\displaystyle{ \frac{h(2x+h+1)}{h} }\)

\( 2x+h+1 \)

So the difference quotient for \(f(x)=x^2+3x\) is \(2x+h+1\).

Okay, let's work the practice problems.

Practice

Unless otherwise instructed, calculate the difference quotient of these functions.

\( f(x) = 7x \)

Problem Statement

Calculate the difference quotient of \( f(x) = 7x \)

Final Answer

\( 7 \)

Problem Statement

Calculate the difference quotient of \( f(x) = 7x \)

Solution

The Organic Chemistry Tutor - 4173 video solution

Final Answer

\( 7 \)

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\( f(x) = 3x-5 \)

Problem Statement

Calculate the difference quotient of \( f(x) = 3x-5 \)

Final Answer

\( 3 \)

Problem Statement

Calculate the difference quotient of \( f(x) = 3x-5 \)

Solution

Thinkwell - 4183 video solution

video by Thinkwell

Final Answer

\( 3 \)

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\( f(x) = 5x+4 \)

Problem Statement

Calculate the difference quotient of \( f(x) = 5x+4 \)

Final Answer

\( 5 \)

Problem Statement

Calculate the difference quotient of \( f(x) = 5x+4 \)

Solution

The Organic Chemistry Tutor - 4174 video solution

Final Answer

\( 5 \)

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\( f(x) = 2x-5 \)

Problem Statement

Calculate the difference quotient of \( f(x) = 2x-5 \)

Final Answer

\( 2 \)

Problem Statement

Calculate the difference quotient of \( f(x) = 2x-5 \)

Solution

MIP4U - 4179 video solution

video by MIP4U

Final Answer

\( 2 \)

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\( f(x) = x^2 \)

Problem Statement

Calculate the difference quotient of \( f(x) = x^2 \)

Final Answer

\( 2x+h \)

Problem Statement

Calculate the difference quotient of \( f(x) = x^2 \)

Solution

The Organic Chemistry Tutor - 4175 video solution

Final Answer

\( 2x+h \)

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\( f(x) = 3x^2+4x-5 \)

Problem Statement

Calculate the difference quotient of \( f(x) = 3x^2+4x-5 \)

Final Answer

\( 6x+4+3h \)

Problem Statement

Calculate the difference quotient of \( f(x) = 3x^2+4x-5 \)

Solution

The Organic Chemistry Tutor - 4178 video solution

Final Answer

\( 6x+4+3h \)

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\( f(x) = x^2-3x+4 \)

Problem Statement

Calculate the difference quotient of \( f(x) = x^2-3x+4 \)

Final Answer

\( 2x+h-3 \)

Problem Statement

Calculate the difference quotient of \( f(x) = x^2-3x+4 \)

Solution

MIP4U - 4180 video solution

video by MIP4U

Final Answer

\( 2x+h-3 \)

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\( f(x) = 3x^2-5x+2 \)

Problem Statement

Calculate the difference quotient of \( f(x) = 3x^2-5x+2 \)

Final Answer

\( 6x+3h-5 \)

Problem Statement

Calculate the difference quotient of \( f(x) = 3x^2-5x+2 \)

Solution

MIP4U - 4181 video solution

video by MIP4U

Final Answer

\( 6x+3h-5 \)

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\( f(x) = 1/x \)

Problem Statement

Calculate the difference quotient of \( f(x) = 1/x \)

Final Answer

\(\displaystyle{ \frac{-1}{x(x+h)} }\)

Problem Statement

Calculate the difference quotient of \( f(x) = 1/x \)

Solution

The Organic Chemistry Tutor - 4177 video solution

Final Answer

\(\displaystyle{ \frac{-1}{x(x+h)} }\)

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\( f(x) = \sqrt{x} \)

Problem Statement

Calculate the difference quotient of \( f(x) = \sqrt{x} \)

Hint

Once you form the difference quotient, rationalize the square root expression.

Problem Statement

Calculate the difference quotient of \( f(x) = \sqrt{x} \)

Final Answer

\(\displaystyle{ \frac{1}{\sqrt{x+h}+\sqrt{x}} }\)

Problem Statement

Calculate the difference quotient of \( f(x) = \sqrt{x} \)

Hint

Once you form the difference quotient, rationalize the square root expression.

Solution

The Organic Chemistry Tutor - 4176 video solution

Final Answer

\(\displaystyle{ \frac{1}{\sqrt{x+h}+\sqrt{x}} }\)

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\( f(x) = \sqrt{x-1} \)

Problem Statement

Calculate the difference quotient of \( f(x) = \sqrt{x-1} \)

Final Answer

\(\displaystyle{ \frac{1}{\sqrt{x+h-1} + \sqrt{x-1}} }\)

Problem Statement

Calculate the difference quotient of \( f(x) = \sqrt{x-1} \)

Solution

The Math Jim - 4182 video solution

video by The Math Jim

Final Answer

\(\displaystyle{ \frac{1}{\sqrt{x+h-1} + \sqrt{x-1}} }\)

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Practice Instructions

Unless otherwise instructed, calculate the difference quotient of these functions.

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