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Chapter 5: Problem 56

Find the average rate of change of \(f(x)=x^{2}+4 x-3\) from -2 to \(1 .\)

Short Answer

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Step by step solution

01

- Understand the Average Rate of Change

The average rate of change of a function over an interval \text{[a, b]} is given by the formula \ \ \[ \frac{f(b) - f(a)}{b - a} \] \ where in this case, \ \ \( a = -2 \) \ and \ \( b = 1 \).
02

- Calculate f(a)

Evaluate the function \( f(x) = x^2 + 4x - 3 \) at \( x = -2 \). \ \ \[ f(-2) = (-2)^2 + 4(-2) - 3 \] \ \ This simplifies to \[ 4 - 8 - 3 = -7 \].
03

- Calculate f(b)

Evaluate the function \( f(x) = x^2 + 4x - 3 \) at \( x = 1 \). \ \ \[ f(1) = 1^2 + 4(1) - 3 \] \ This simplifies to \[ 1 + 4 - 3 = 2 \].
04

- Apply the Average Rate of Change Formula

Substitute \( a = -2 \), \( b = 1 \), \( f(a) = -7 \), and \( f(b) = 2 \) into the average rate of change formula: \ \ \[ \frac{f(1) - f(-2)}{1 - (-2)} \] \ \ This simplifies to \[ \frac{2 - (-7)}{1 - (-2)} = \frac{2 + 7}{1 + 2} = \frac{9}{3} = 3 \].

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Key Concepts

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

Function Evaluation
Before solving any mathematical problem involving a function, it's crucial to understand how to evaluate a function. Function evaluation merely means calculating the value of a function at a specific point. For example, given a function \( f(x) = x^2 + 4x - 3 \) and a point \( x = -2 \), we substitute -2 for \( x \) in the function.
Here鈥檚 the step-by-step:
  • Replace all instances of \(x\) in the function with \(-2\)
  • Perform arithmetic operations (squaring, multiplication, addition, and subtraction)
So:\[ f(-2) = (-2)^2 + 4(-2) - 3 \= 4 - 8 - 3 \= -7 \]
Similarly, to find \( f(1) \), replace \( x \) with 1:
  • Replace all instances of \(x\) in the function with 1
  • Perform arithmetic operations
So:\[ f(1) = 1^2 + 4(1) - 3 \= 1 + 4 - 3 \= 2 \]
Algebra
Algebra plays a significant role in solving problems involving functions. The fundamental operations involve simplifying expressions and solving equations. For the function \( f(x) = x^2 + 4x - 3 \), understanding how to expand, simplify, and manipulate algebraic expressions is essential.
When we evaluated \( f(-2) \) and \( f(1) \), we performed several algebraic steps, like:
  • Substituting the value of \( x \)
  • Calculating the square \( x^2 \)
  • Performing multiplications and additions/subtractions
Combining all these steps correctly is how we use algebra to find specific values of a function.
Quadratic Function
A quadratic function is one of the simplest types of polynomial functions and has the general form \( ax^2 + bx + c \). In our case, \( f(x) = x^2 + 4x - 3 \) is a quadratic function where:
  • \( a = 1 \)
  • \( b = 4 \)
  • \( c = -3 \)
Quadratic functions create parabolic graphs that open either up or down. Determining values at specific points helps us understand the behavior of the quadratic function in a specified interval. These functions are immensely valuable in various fields, such as physics and economics.
Interval
An interval defines the boundary within which we evaluate a function or study its behavior. For example, the problem specifies the interval from \(-2\) to \(1\).
Using intervals ensures we are calculating the rate of change or other properties of the function within a specific range.
When looking at the average rate of change in this interval, we use:
  • \(a = -2\)
  • \(b = 1\)
Next, we find \( f(a) \) and \( f(b) \). Finally, we apply the average rate of change formula:
The formula is:\[ \frac{f(b) - f(a)}{b - a} \]
Substituting the values, we get:\[ \frac{2 - (-7)}{1 - (-2)} = \frac{2 + 7}{1 + 2} = \frac{9}{3} = 3 \]

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Most popular questions from this chapter

Find bounds on the real zeros of each polynomial function. $$ f(x)=x^{4}+x^{3}-x-1 $$

Solve each equation in the real number system. $$ x^{3}-\frac{2}{3} x^{2}+\frac{8}{3} x+1=0 $$

Use the Rational Zeros Theorem to find all the real zeros of each polynomial function. Use the zeros to factor \(f\) over the real numbers. $$ f(x)=2 x^{3}+x^{2}+2 x+1 $$

Find \(k\) so that \(x-2\) is a factor of $$ f(x)=x^{3}-k x^{2}+k x+2 $$

Solve each equation in the real number system. $$ 2 x^{3}-11 x^{2}+10 x+8=0 $$
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