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Chapter 1: Problem 82

Complete the following for the given \(f(x)\) (a) Find \(f(x+h)\) (b) Find the difference quotient of \(f\) and simplify. $$ f(x)=1-x^{3} $$

Short Answer

Expert verified
(a) \( f(x+h) = 1 - x^3 - 3x^2h - 3xh^2 - h^3 \), (b) \( -3x^2 - 3xh - h^2 \)

Step by step solution

01

Substitute (x + h) into f(x)

To find \(f(x+h)\), substitute \(x+h\) into the function \(f(x) = 1 - x^3\).\[ f(x+h) = 1 - (x+h)^3 \]
02

Expand the Binomial (x + h)^3

Use the binomial theorem to expand \((x+h)^3\):\[(x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3\]Substitute back into the function:\[ f(x+h) = 1 - (x^3 + 3x^2h + 3xh^2 + h^3) \]
03

Simplify Expression for f(x + h)

Simplify the expression by distributing the negative sign:\[ f(x+h) = 1 - x^3 - 3x^2h - 3xh^2 - h^3 \]
04

Set Up the Difference Quotient

The difference quotient for \(f(x)\) is given by:\[ \frac{f(x+h) - f(x)}{h} \]
05

Substitute and Simplify Difference Quotient

Substitute \(f(x+h)\) and \(f(x) = 1 - x^3\) into the difference quotient:\[ \frac{1 - x^3 - 3x^2h - 3xh^2 - h^3 - (1 - x^3)}{h} \]Simplify:\[ \frac{-3x^2h - 3xh^2 - h^3}{h} \]
06

Simplify by Canceling h

Factor out \(h\) from the numerator and cancel it with the denominator:\[ \frac{h(-3x^2 - 3xh - h^2)}{h} = -3x^2 - 3xh - h^2 \]
07

Find the Simplified Difference Quotient

The simplified difference quotient is:\[ -3x^2 - 3xh - h^2 \]

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Key Concepts

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

Polynomial Functions
Polynomial functions are mathematical expressions that involve the sum of powers of a variable, each multiplied by a coefficient. In simple terms, they are functions that consist of terms involving variables raised to whole number exponents and their respective coefficients. Here are some key points to remember about polynomial functions:
  • The general form of a polynomial function is: \( f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \).
  • The highest exponent of the variable determines the degree of the polynomial.
  • Each term consists of a coefficient and a variable raised to an exponent, like \(a_nx^n\).
For instance, in the exercise, the function \(f(x) = 1-x^3\) is a simple polynomial function of degree 3. It consists of two terms, a constant (1) and a cubic term (-\(x^3\)). Understanding the structure of polynomial functions helps in solving various mathematical problems and expanding expressions, like computing \(f(x+h)\).
Binomial Expansion
Binomial expansion is an essential part of working with polynomial functions, especially when dealing with expressions of the form \((x+h)^n\). The binomial theorem gives us a way to expand these expressions efficiently:
  • It states that \((x + y)^n\) can be expanded into a series of terms in the form of \(a \cdot b\), where each term is a combination of the original variables raised to different powers.
  • For example, \((x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3\).
  • The coefficients of the expanded terms correspond to the binomial coefficients, found in Pascal's Triangle or by the formula for combinations \(\binom{n}{k}\).
In our exercise, expanding \((x+h)^3\) is crucial because it allows for further simplification of \(f(x+h)\). Knowing how to perform a binomial expansion helps simplify functions and is particularly useful in calculus when finding limits and derivatives.
Function Simplification
Simplifying functions is a crucial step in calculus and algebra that involves rewriting expressions in a simpler or reduced form without changing their value. This process is important as it makes equations easier to handle and interpret.Here are the steps typically involved in function simplification:
  • Substitution: Replace variables or constants as instructed, like finding \(f(x+h)\) by substituting \(x+h\) into \(f(x)\).
  • Expansion: Use techniques such as the binomial theorem to expand complex expressions into their expanded forms.
  • Distribution: Apply distribution laws to remove parentheses and combine like terms, simplifying the function further.
  • Reduction: Cancel common terms or factors across the numerator and denominator, reducing the expression to its simplest form.
In the given solution, the simplification step for the difference quotient of \(f(x)\) involves subtracting \(f(x)\) from \(f(x+h)\), and then canceling the \(h\) factor from both the numerator and denominator. This leaves a simpler expression that helps in evaluating limits or finding derivatives, which are key concepts in calculus.

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Most popular questions from this chapter

The table lists the marriages per 1000 residents in Kentucky for selected years. $$ \begin{array}{|c|c|c|c|}\hline\hline \text { Year } & 2001 & 2002 & 2003 & 2004 \\ \hline \text { Rate } & 9.0 & 9.0 & 9.1 & 8.9 \end{array} $$ (a) Could these data be modeled exactly by a constant function? (b) Determine a constant function \(f\) that models these data approximately. (c) Graph \(f\) and the data.

Analyzing Real Data For the given data set complete the following. (a) Make a line graph of the data. Let this graph represent a function \(f\) (b) Decide whether \(f\) is linear or nonlinear. Median incomes of full-time male workers $$ \begin{array}{|c|c|c|c|c|}\hline \hline \text { Year } & 1970 & 1980 & 1990 & 2000 \\ \hline \text { Income } & \$ 9,184 & \$ 19,173 & \$ 28,979 & \$ 37,435 \end{array} $$

Tuition Increases From 1976 to \(2004,\) average annual tuition and fees at public colleges and universities increased from \(\$ 433\) to \(\$ 5132 .\) Calculate the percent change over this time period.

Writing about Mathematics Suppose that a positive number \(a\) is written in scientific notation as \(a=b \times 10^{n}\), where \(n\) is an integer and \(1 \leq b \leq 10 .\) Explain what \(n\) indicates about the size of \(a\)

Suppose you are given a graphical representation of a function \(f .\) Explain how you would determine whether \(f\) is constant, linear, or nonlinear. How would you determine the type if you were given a numerical or symbolic representation? Give examples.
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