Problem 39 - Calculus: Graphical, Numerical, Algebraic

Question

Multiple Choice Find the average rate of change of \(f(x)=x^{2}+x\) over the interval \([1,3] .\) . \(\begin{array}{ll}{\text { (A) } y=-2 x} & {\text { (B) } y=2 x \text { (C) } y=-2 x+4} \\ {\text { (D) } y=-x+3} & {\text { (E) } y=x+3}\end{array}\)

Step-by-Step Solution

Answer

None of the provided options are correct. The average rate of change is 5.

Step 1: Identify the given interval

The given interval is from 1 to 3. Therefore, a = 1 and b = 3.

Step 2: Calculate f(a) and f(b)

Substitute a and b into the given function.\n For a=1, \( f(a) = (1)^2 + 1 = 2 \).\n For b=3, \( f(b) = (3)^2 + 3 = 12 \).

Step 3: Use the formula for average rate of change

The formula for average rate of change is \( \frac{f(b) - f(a)}{b - a} \). Substituting values we've calculated, we find \( \frac{12 - 2}{3 - 1} = 5 \).

Step 4: Match the result with the options given

The result 5 does not match with any of the options given, as they all represent equations, not constant values. Therefore, none of the provided options are correct.

Key Concepts

Calculus Problem-Solving

Calculus problem-solving often involves understanding and applying different formulas and methods to find solutions to mathematical problems. A common task is to determine rates of change, such as the average rate of change you might encounter when working with functions.

In the context of the provided exercise, calculating the average rate of change of a function over a certain interval is key. It requires a methodical approach: identifying the interval, evaluating the function at the endpoints, and applying the average rate of change formula. The solution steps indicate this process, which allows for determining how the function's output value changes, on average, between two input values. When solving calculus problems, it's crucial to pay attention to whether the question requires an incremental change like the average rate or a specific value like a function evaluation.

Quadratic Functions

Quadratic functions, which are of the form \( f(x) = ax^2 + bx + c \), are fundamental in calculus. They represent parabolas when graphed and are characterized by a variety of features—including a vertex, axis of symmetry, and potentially real or complex roots—depending on the values of \( a \) , \( b \) , and \( c \).

Properties of Quadratic Functions

These functions are unique in that they always have a U-shaped curve and they open upward if \( a > 0 \) and downward if \( a < 0 \). Additionally, understanding the behavior of quadratic functions over intervals is crucial for solving calculus problems. When you compute the average rate of change for a quadratic function, what you're essentially doing is finding the slope of the secant line that connects two points on its graph.

Interval Notation

Interval notation is a way of describing a set of numbers between two endpoints in mathematics. Recognizing and utilizing proper interval notation is crucial when working with functions and rates of change in calculus.

For example, in the exercise provided, the interval \[1, 3\] specifies the range from 1 to 3, inclusive of both endpoints. This is fundamental because the average rate of change is calculated over this specific interval. When an interval is given, it's essential to understand that it's laying out the domain over which we should analyze the function's behavior. Failing to apply the concept of interval notation correctly can lead to misunderstandings of the problem's requirements and therefore incorrect solutions or interpretations of a function's properties over the given range.