91Ó°ÊÓ

\(A\) and \(B\) are points on the graph of \(k(x)\). The \(x\) -coordinate of point \(A\) is 6 and the \(x\) -coordinate of point \(B\) is \((6+h)\). Write mathematical expressions, using functional notation, for each of the following. (a) The change in value of the function from point \(A\) to point \(B\) (b) The average rate of change of the function \(k\) over the interval \([6,6+h]\) (c) Suppose that the average rate of change of the function \(k\) over the interval \([6,6+h]\) is \(-5\). The functions \(f, g\), and \(h\) are de ned as follows: $$ f(x)=k(x)+2, \quad g(x)=k(x+2), \quad h(x)=2 k(x). $$ i. Which of the following must also be equal to \(-5\) ? A. The average rate of change of the function \(f\) over the interval \([6,6+h]\) B. The average rate of change of the function \(g\) over the interval \([6,6+h]\) C. The average rate of change of the function \(h\) over the interval \([6,6+h]\) ii. One of the functions \(f, g\), and \(h\) has an average value of \(-10\) on the interval \([6,6+h]\). Which is it? Explain brie y.

Step by step solution

01

Find the change in value of the function

To find the change in value of the function from point \(A\) to point \(B\), subtract the value of the function at point \(A\) from the value of the function at point \(B\). In terms of the function \(k\), this is \(k(6+h) - k(6)\). This value represents how much the function value increases or decreases as \(x\) changes from 6 to \(6+h\).

02

Find the average rate of change

The average rate of change of the function over the interval is the change in the function value divided by the change in \(x\). In terms of the function \(k\) and the given interval, the average rate of change is \(\frac{k(6+h) - k(6)}{h}\).

03

Determine the average rate of change of the functions \(f, g, h\)

When the average rate of change of function \(k\) is \(-5\), the average rates of change of the functions \(f, g, h\) can be calculated with similar methods as in step 2. For function \(f(x)=k(x)+2\), the constant 2 does not affect the rate of change, so its average rate of change is also \(-5\). For function \(g(x)=k(x+2)\), the +2 instead shifts the graph of \(k\) to the left, but it does not change the slope, so its average rate of change is also \(-5\). For function \(h(x)=2k(x)\), the coefficient 2 scales the output of \(k\) by 2, so its average rate of change is \(-5 * 2 = -10\).

04

Determine the function with an average value of \(-10\)

The average value of a function on an interval \([a,b]\) is given by \(\frac{1}{b-a}\int_a^b f(x) dx\), which in this case simplifies to \(k(x)\) since \(a=6\) and \(b=6+h\). In order for the average value of the function to be -10, the value of the function must be -10 at all points in the interval. This rule excludes options \(f\) and \(g\), leaving \(h(x)=2k(x)\) as the only option, since it multiplies the output of \(k\) by 2.

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Key Concepts

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

Calculus

Calculus is a rich field of mathematics that explores continuous change. One of its fundamental concepts is the rate of change. Understanding the average rate of change between two points on a graph is crucial. It provides insights into how a function behaves over an interval, similar to finding the slope of a line segment joining two points.

The average rate of change of a function over an interval \[a, b\] is determined by assessing the change in the function's output values relative to the change in its input values. For a function like \(k(x)\), the formula is given by:

• Average rate of change = \(\frac{k(b) - k(a)}{b-a}\)

This formula helps us understand how the function \(k\) increases or decreases as \(x\) changes from \(a\) to \(b\). In our exercise, focusing on the interval \[6, 6+h\], allows us to analyze the behavior of this function between the x-values 6 and \(6+h\). This approach encapsulates the essence of calculus and its applications in real-world scenarios.

Functional Notation

Functional notation is a compact and powerful way to describe functions. It simplifies the expression of how inputs relate to outputs. When working with functions, it is important to use the correct notation to avoid confusion.

In our problem, we deal with the function \(k(x)\). By using functional notation, we express the values at specific points efficiently:

• \(k(a)\) indicates the value of the function at an input \(a\).
• Similarly, \(k(6)\) and \(k(6+h)\) are used to denote the function values at 6 and \(6+h\) respectively.

Functional notation helps in breaking down complex problems into manageable parts by relating changes in input to changes in output simply. This method enables clearer communication about how specific values affect the overall function, particularly when calculating the average change between different inputs, as shown in our exercise.

Graph Analysis

Graph analysis involves examining and interpreting graphs to understand the behavior of functions. Graphs offer a visual representation of a function's operational rule over a domain.

In analyzing graphs for the average rate of change, we focus on points \(A\) and \(B\) on the graph of \(k(x)\). This means observing how the graph moves between the x-values of 6 and \(6+h\). The points on the graph can provide clues about the function's nature, such as identifying linear or non-linear trends.

Through graph analysis:

• We can verify the computations of average rate by visualizing the slope between points \(A\) and \(B\).
• A steeper slope indicates a higher rate of change.
• Understand transformations (e.g., shifts or scaling) in functions \(f(x)\), \(g(x)\), and \(h(x)\) as related to \(k(x)\).

This comprehensive analysis can reveal deeper insights into problem solving, like why certain operations result in specific rates of change, as illustrated by the different average rates of change for the given functions.