Problem 13 - Applied Calculus

Question

Find the average rate of change of \(f(x)=3 x^{2}+4\) between \(x=-2\) and \(x=1\). Illustrate your answer graphically.

Step-by-Step Solution

Answer

The average rate of change of the function is -3.

Step 1: Define the Formula for Average Rate of Change

The average rate of change of a function between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula \(\frac{f(x_2) - f(x_1)}{x_2 - x_1}\). In this case, \(x_1 = -2\) and \(x_2 = 1\).

Step 2: Calculate Function Values

Plug the values of \(x_1 = -2\) and \(x_2 = 1\) into the function \(f(x) = 3x^2 + 4\) to find \(f(-2)\) and \(f(1)\).- For \(x = -2\), \(f(-2) = 3(-2)^2 + 4 = 12 + 4 = 16\).- For \(x = 1\), \(f(1) = 3(1)^2 + 4 = 3 + 4 = 7\).

Step 3: Substitute in the Formula

Using the formula for average rate of change, substitute the function values obtained:\[\text{Average rate of change} = \frac{f(1) - f(-2)}{1 - (-2)} = \frac{7 - 16}{1 + 2}\].

Step 4: Calculate the Result

Simplify the expression:\[\frac{7 - 16}{3} = \frac{-9}{3} = -3\].Thus, the average rate of change of the function between \(x = -2\) and \(x = 1\) is \(-3\).

Step 5: Illustrate Graphically

Draw the graph of \(f(x) = 3x^2 + 4\). Identify points \((-2, 16)\) and \((1, 7)\) on the graph. The line connecting these two points has a slope of \(-3\), representing the average rate of change. This line is the secant line over this interval.

Key Concepts

Quadratic Functions

Quadratic functions are a type of polynomial function that feature prominently in many areas of mathematics and real-world applications. A standard form of a quadratic function is given by \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\).
In our example, the quadratic function is \(f(x) = 3x^2 + 4\). Here, the function involves the square of the variable \(x\), which creates a characteristic parabola shape when graphed.

• This parabolic graph opens upwards due to the positive coefficient of \(x^2\) (which is 3 in this case).
• Quadratic functions can model several physical phenomena like projectile motion, area calculations, and optimization problems.
• Understanding how the values of \(a\), \(b\), and \(c\) affect the graph’s shape and position is crucial when analyzing these functions.

Finding average rates of change in such functions, especially between specific points as seen in this exercise, helps in understanding how the function behaves over intervals.

Secant Line

The concept of the secant line is vital when looking into average rates of change of a function over an interval. A secant line intersects a curve at two distinct points and allows us to analyze how quickly a function is changing between those points.
In this exercise, you calculate the average rate of change of the function \(f(x) = 3x^2 + 4\) between \(x = -2\) and \(x = 1\).

• This is done by identifying the points \((-2, 16)\) and \((1, 7)\) on the graph of the function.
• The slope of the secant line, calculated as \(-3\) in this case, represents how steeply the function is changing over this interval.
• Connecting these two points with a straight line gives us the secant line, providing a clear graphical representation of the rate of change.

Understanding the secant line concept helps in linking graphical analysis with algebraic calculations, bridging the connection between the visual and numerical assessments of a function.

Graphical Illustration

Graphical representation is a powerful tool to visually understand and interpret mathematical concepts like functions and their rates of change.
Drawing the graph of the quadratic function \(f(x) = 3x^2 + 4\) provides a clear picture of how the function behaves across different \(x\) values.

• By plotting the points \((-2, 16)\) and \((1, 7)\), you can visually see the calculated change from one point to another.
• The secant line drawn between these points illustrates the average rate of change, showcasing it as a slope of the line.
• Visualizing this can help clarify why the average rate of change is \(-3\), as the secant line slopes downward.

Graphical illustrations help demystify abstract algebraic concepts, making learning more engaging and accessible. They allow students to observe and analyze patterns or behaviors of functions, reinforcing theoretical knowledge through tangible visual elements.