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Chapter 2: Problem 44

Difference Quotient Find \(f(a), f(a+h),\) and the difference quotient \(\frac{f(a+h)-f(a)}{h},\) where \(h \neq 0.\) $$f(x)=3 x^{2}+2$$

Short Answer

Expert verified
The difference quotient is \(6a + 3h\).

Step by step solution

01

Compute \(f(a)\)

To find \(f(a)\),\ we substitute \(a\) into the function \(f(x) = 3x^2 + 2\). Thus, \(f(a) = 3a^2 + 2\).
02

Compute \(f(a+h)\)

To find \(f(a+h)\),\ substitute \(a+h\) into the function \(f(x) = 3x^2 + 2\). That gives \(f(a+h) = 3(a+h)^2 + 2\). Expanding this, we have \(3(a^2 + 2ah + h^2) + 2 = 3a^2 + 6ah + 3h^2 + 2\).
03

Set Up the Difference Quotient

The difference quotient is \(\frac{f(a+h) - f(a)}{h}\). Substitute the expressions from Step 1 and Step 2: \[ \frac{3a^2 + 6ah + 3h^2 + 2 - (3a^2 + 2)}{h} \].
04

Simplify the Difference Quotient

Simplify the expression from Step 3: \[ \frac{3a^2 + 6ah + 3h^2 + 2 - 3a^2 - 2}{h} \]. Cancel terms to get \(\frac{6ah + 3h^2}{h}\) which reduces to \(6a + 3h\) after dividing by \(h\).

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Key Concepts

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

Function Evaluation
Function evaluation is a fundamental concept in algebra and calculus where you determine the value of a function at a specific input. To evaluate a function, you substitute the given input value into the function, replacing all instances of the variable with this input. Let's break this down with our example function:
  • Given the function: \( f(x) = 3x^2 + 2 \).
  • To find \( f(a) \), substitute \( a \) into \( f(x) \), resulting in \( f(a) = 3a^2 + 2 \).
Similarly, to find \( f(a+h) \), substitute \( a+h \) into the original function:
  • This results from replacing \( x \) with \( a+h \) in \( f(x) = 3x^2 + 2 \), leading to \( f(a+h) = 3(a+h)^2 + 2 \).
  • Expanding \( (a+h)^2 \) gives \( a^2 + 2ah + h^2 \), which makes \( f(a+h) = 3a^2 + 6ah + 3h^2 + 2 \).
Function evaluation serves as the starting point for various other calculations, such as determining difference quotients.
Polynomial Functions
Polynomial functions are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. The general form is \( ax^n + bx^{n-1} + ... + k \), where \( n \) is a non-negative integer.Our example, \( f(x) = 3x^2 + 2 \), is a simple polynomial where:
  • The highest power of \( x \) is 2, making it a quadratic polynomial.
  • Coefficients are 3 for \( x^2 \) and 2 for the constant term.
Polynomials can be evaluated at various inputs to analyze changes. They can also be expanded or factored to explore characteristics like roots and vertex behavior.In calculus, polynomial functions are particularly useful for understanding differences, changes, and growth through simplifications such as the difference quotient, leading to derivatives.
Simplifying Expressions
Simplifying expressions involves reducing a formula or equation to its simplest form. This often makes them easier to analyze or use in further computations.In our problem, after setting up the difference quotient, we have:\[\frac{3a^2 + 6ah + 3h^2 + 2 - (3a^2 + 2)}{h}\]The expression inside the numerator simplifies to:
  • Cancel \( 3a^2 \) and \( 2 \) terms, which leads to \( 6ah + 3h^2 \).
The next step is to divide each term in the new numerator by \( h \):
  • This yields \( \frac{6ah + 3h^2}{h} = 6a + 3h \).
This process highlights the importance of careful arithmetic and strategic thinking to make expressions clear and manageable. Simplified forms are crucial in uncovering underlying trends or rules, especially when studying limits, continuity, and derivatives in calculus.

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Most popular questions from this chapter

DISCUSS DISCOVER: Minimizing a Distance When we seek a minimum or maximum value of a function, it is sometimes easier to work with a simpler function instead. (a) Suppose $$g(x)=\sqrt{f(x)}$$ where \(f(x) \geq 0\) for all \(x .\) Explain why the local minima and maxima of \(f\) and \(g\) occur at the same values of \(x .\) (b) Let \(g(x)\) be the distance between the point \((3,0)\) and the point \(\left(x, x^{2}\right)\) on the graph of the parabola \(y=x^{2}\) Express \(g\) as a function of \(x\) (c) Find the minimum value of the function \(g\) that you found in part (b). Use the principle described in part (a) to simplify your work.

A family of functions is given. In parts (a) and (b) graph all the given members of the family in the viewing rectangle indicated. In part (c) state the conclusions that you can make from your graphs. \(f(x)=x^{2}+c\) (a) \(c=0,2,4,6 ; \quad[-5,5]\) by \([-10,10]\) (b) \(c=0,-2,-4,-6 ;[-5,5]\) by \([-10,10]\) (c) How does the value of \(c\) affect the graph?

DISCUSS: Obtaining Transformations Can the function \(g\) be obtained from \(f\) by transformations? If so, describe the transformations needed. The functions \(f\) and \(g\) are described algebraically as follows: $$ f(x)=(x+2)^{2} \quad g(x)=(x-2)^{2}+5 $$

The relative value of currencies fluctuates every day. When this problem was written, one Canadian dollar was worth 0.9766 U.S. dollars. (a) Find a function \(f\) that gives the U.S. dollar value \(f(x)\) of x Canadian dollars. (b) Find \(f^{-1}\). What does \(f^{-1}\) represent? (c) How much Canadian money would \(\$ 12,250\) in U.S. currency be worth?

Even and Odd Functions Determine whether the function \(f\) is even, odd, or neither. If \(f\) is even or odd, use symmetry to sketch its graph. $$f(x)=x+\frac{1}{x}$$
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