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Chapter 3: Problem 88

Find the difference quotient of \(f\); that is, find \(\frac{f(x+h)-f(x)}{h}, h \neq 0,\) for each function. Be sure to simplify. \(f(x)=3 x^{2}-2 x+6\)

Short Answer

Expert verified
The difference quotient is \(6x + 3h - 2.\)

Step by step solution

01

Substitute into the Difference Quotient Formula

Start by substituting the given function, \( f(x) = 3x^2 - 2x + 6 \), into the difference quotient formula: \[ \frac{f(x+h) - f(x)}{h} \]. This gives: \[ \frac{(3(x+h)^2 - 2(x+h) + 6) - (3x^2 - 2x + 6)}{h} \]
02

Expand and Simplify the Expression

Expand the polynomial \(3(x+h)^2 - 2(x+h) + 6 \) in the numerator: \[ 3(x+h)(x+h) - 2(x+h) + 6 = 3(x^2 + 2xh + h^2) - 2x - 2h + 6 = 3x^2 + 6xh + 3h^2 - 2x - 2h + 6. \]
03

Subtract the Original Function

Subtract the function \( f(x) = 3x^2 - 2x + 6 \) from the expanded form: \[ 3x^2 + 6xh + 3h^2 - 2x - 2h + 6 - (3x^2 - 2x + 6) = 3x^2 + 6xh + 3h^2 - 2x - 2h + 6 - 3x^2 + 2x - 6. \]
04

Simplify the Expression

Combine like terms in the numerator: \[ 3x^2 + 6xh + 3h^2 - 2x - 2h + 6 - 3x^2 + 2x - 6 = 6xh + 3h^2 - 2h. \]
05

Divide by h

Divide each term in the numerator by \( h \): \[ \frac{6xh + 3h^2 - 2h}{h} = \frac{6xh}{h} + \frac{3h^2}{h} - \frac{2h}{h} = 6x + 3h - 2. \]
06

State the Difference Quotient

The difference quotient simplified is: \[ 6x + 3h - 2. \]

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Key Concepts

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

polynomials
Polynomials are algebraic expressions that consist of variables and coefficients. They take the form of sums of monomials, such as \( ax^n \), where \( a \) is a coefficient and \( n \) is a non-negative integer exponent. In our problem, the function \( f(x) = 3x^2 - 2x + 6 \) is a second-degree polynomial. Here are some key features of polynomials:
  • Degree: The highest power of the variable. In \( f(x) = 3x^2 - 2x + 6 \), the degree is 2.
  • Coefficients: The numerical factors multiplying the variables. For \( f(x) \), these are 3, -2, and 6.
  • Terms: Each monomial in the polynomial. The terms here are \( 3x^2 \), \( -2x \), and 6.
Polynomials can be added, subtracted, multiplied, and divided (except by zero) to form new polynomials. They are fundamental in calculus as they are easy to differentiate and integrate compared to other types of functions.
function composition
Function composition involves applying one function to the results of another. If you have two functions \( f \) and \( g \), the composition \((f \circ g)(x) \) means \( f(g(x)) \). In the context of the difference quotient, we are effectively composing \( f \) with the function that shifts \( x \) by \( h \) (i.e., \( x+h \)).
  • Applying \( f \) to \( x + h \): Here, we substitute \( x + h \) into the polynomial, resulting in \( f(x+h) = 3(x+h)^2 - 2(x+h) + 6 \).
  • Distributing and combining like terms: Expanding \( f(x+h) \) involves distributing and simplifying each part of the polynomial to find the new expression.
Function composition is helpful in identifying changes and shifts in functions, playing an essential role in differentiation and integration.
limits
Limits are a core concept in calculus used to define the behavior of a function as it approaches a specific point. For the difference quotient, we are interested in the limit as \( h \) approaches 0. This limit gives us the derivative of the function.
  • Understanding the limit: The limit evaluates what the value of \( \frac{f(x+h) - f(x)}{h} \) approaches as \( h \) gets infinitely small. This signifies the instantaneous rate of change of the function at a point \( x \).
  • Simplifying the quotient: To find the limit, we simplify the difference quotient: \( \frac{6xh + 3h^2 - 2h}{h} = 6x + 3h - 2 \).
  • Taking the limit: As \( h \) approaches 0, the term \( 3h \) vanishes, simplifying our expression to \( 6x - 2 \), representing the derivative of the polynomial at any point \( x \).
Limits provide a foundational framework for calculus, underpinning the concepts of continuity, derivatives, and integrals.

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Most popular questions from this chapter

Find the sum function \((f+g)(x)\) if $$f(x)=\left\\{\begin{array}{ll}2 x+3 & \text { if } x<2 \\\x^{2}+5 x & \text { if } x \geq 2\end{array}\right.$$ and $$g(x)=\left\\{\begin{array}{ll}-4 x+1 & \text { if } x \leq 0 \\\x-7 & \text { if } x>0\end{array}\right.$$

The slope of the secant line containing the two points \((x, f(x))\) and \((x+h, f(x+h))\) on the graph of a function \(y=f(x)\) may be given as \(m_{\mathrm{sec}}=\frac{f(x+h)-f(x)}{(x+h)-x}=\frac{f(x+h)-f(x)}{h} \quad h \neq 0\) (a) Express the slope of the secant line of each function in terms of \(x\) and \(h\). Be sure to simplify your answer. (b) Find \(m_{\text {sec }}\) for \(h=0.5,0.1\), and 0.01 at \(x=1 .\) What value does \(m_{\text {sec }}\) approach as h approaches \(0 ?\) (c) Find an equation for the secant line at \(x=1\) with \(h=0.01\). (d) Use a graphing utility to graph fand the secant line found in part ( \(c\) ) in the same viewing window. Problems \(85-92\) require the following discussion of a secant line. The slope of the secant line containing the two points \((x, f(x))\) and \((x+h, f(x+h))\) on the graph of a function \(y=f(x)\) may be given as \(m_{\mathrm{sec}}=\frac{f(x+h)-f(x)}{(x+h)-x}=\frac{f(x+h)-f(x)}{h} \quad h \neq 0\) (a) Express the slope of the secant line of each function in terms of \(x\) and \(h\). Be sure to simplify your answer. (b) Find \(m_{\text {sec }}\) for \(h=0.5,0.1\), and 0.01 at \(x=1 .\) What value does \(m_{\text {sec }}\) approach as h approaches \(0 ?\) (c) Find an equation for the secant line at \(x=1\) with \(h=0.01\). (d) Use a graphing utility to graph fand the secant line found in part ( \(c\) ) in the same viewing window. \(f(x)=-3 x+2\)

Find the quotient and remainder when \(x^{3}+3 x^{2}-6\) is divided by \(x+2\)

The slope of the secant line containing the two points \((x, f(x))\) and \((x+h, f(x+h))\) on the graph of a function \(y=f(x)\) may be given as \(m_{\mathrm{sec}}=\frac{f(x+h)-f(x)}{(x+h)-x}=\frac{f(x+h)-f(x)}{h} \quad h \neq 0\) (a) Express the slope of the secant line of each function in terms of \(x\) and \(h\). Be sure to simplify your answer. (b) Find \(m_{\text {sec }}\) for \(h=0.5,0.1\), and 0.01 at \(x=1 .\) What value does \(m_{\text {sec }}\) approach as h approaches \(0 ?\) (c) Find an equation for the secant line at \(x=1\) with \(h=0.01\). (d) Use a graphing utility to graph fand the secant line found in part ( \(c\) ) in the same viewing window. Problems \(85-92\) require the following discussion of a secant line. The slope of the secant line containing the two points \((x, f(x))\) and \((x+h, f(x+h))\) on the graph of a function \(y=f(x)\) may be given as \(m_{\mathrm{sec}}=\frac{f(x+h)-f(x)}{(x+h)-x}=\frac{f(x+h)-f(x)}{h} \quad h \neq 0\) (a) Express the slope of the secant line of each function in terms of \(x\) and \(h\). Be sure to simplify your answer. (b) Find \(m_{\text {sec }}\) for \(h=0.5,0.1\), and 0.01 at \(x=1 .\) What value does \(m_{\text {sec }}\) approach as h approaches \(0 ?\) (c) Find an equation for the secant line at \(x=1\) with \(h=0.01\). (d) Use a graphing utility to graph fand the secant line found in part ( \(c\) ) in the same viewing window. \(f(x)=2 x^{2}+x\)

Medicine Concentration The concentration \(C\) of a medication in the bloodstream \(t\) hours after being administered is modeled by the function $$ C(t)=-0.002 t^{4}+0.039 t^{3}-0.285 t^{2}+0.766 t+0.085 $$ (a) After how many hours will the concentration be highest? (b) A woman nursing a child must wait until the concentration is below 0.5 before she can feed him. After taking the medication, how long must she wait before feeding her child?
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