Problem 13 - Calculus of a Single Variable

Question

Think About It In Exercises 13 and \(14,\) determine which value best approximates the area of the region bounded by the graphs of \(f\) and \(g .\) (Make your selection on the basis of a sketch of the region and not by performing any calculations.) $$ \begin{array}{l}{f(x)=x+1, \quad g(x)=(x-1)^{2}} \\\ {\begin{array}{llll}{\text { (a) }-2} & {\text { (b) } 2} & {\text { (c) } 10} & {\text { (d) } 4} & {\text { (e) } 8}\end{array}}\end{array} $$

Step-by-Step Solution

Answer

From visual observation of the overall image defined by the bounds, and approximating the area, it appears option (d) 4 would be the closest estimate for the area under the curves.

Step 1: Plot the graphs of given functions

Firstly, draw the graphs of functions \(f(x)=x+1\) and \(g(x)=(x-1)^2\). By plotting these graphs, one can identify where they intersect and what area they bound.

Step 2: Identify the points of Intersection

From observing the sketches, find the points where the two functions intersect. Solve the equation \(f(x) = g(x)\) to find the intersection points. This step is critical as it will define the boundaries of the area we are evaluating.

Step 3: Visualize the approximate area

After examining the plot, the next step is to visualize the area between the two functions. The bounded region needs to be estimated visually. Remember not to calculate exactly, but instead select the best approximate value.

Key Concepts

Sketching Graphs of Functions

When it comes to understanding the relationship between two functions, a visual representation can be incredibly powerful. This is where sketching graphs of functions comes into play. To sketch the graph of a function like f(x) = x + 1, start by plotting the y-intercept, which is the point where the function crosses the y-axis. In this case, that point is (0,1). Then, because this is a linear function with a slope of 1, you will draw a straight line with a 45-degree angle going through that point.

For the function g(x) = (x - 1)^2, noting that this is a parabola, the vertex can be identified at (1,0). Since it's an upward-opening parabola, sketch a u-shaped curve starting at the vertex, ensuring that the branches continue to rise as they move away from the vertex diagonally.

Once both graphs are plotted on the same coordinate system, the points where the two graphs intersect can be misleading from just looking at the equations alone. Thus, the visual comparison can guide us to a more accurate understanding of how these functions interact with each other over a range of values.

Points of Intersection

The points of intersection between two graphs represent the set of x and y values where the two functions have the same output. Determining these points is a critical step in finding the area between curves. In the case of f(x) = x + 1 and g(x) = (x - 1)^2, you can find the intersection points by setting the equations equal to each other and solving for x.

Solving for Intersection Points

To solve f(x) = g(x), you would write x + 1 = (x - 1)^2. This will typically result in a quadratic equation that you may need to simplify in order to find the values of x. It's these x-values that, when plugged back into either function, give you the corresponding y-values of the intersection points. Knowing where two graphs intersect helps to visualize and estimate the area between the curves because these points serve as boundaries for the region of interest.

Estimating Area Under Curve

The concept of estimating the area under a curve is fundamental in integral calculus, where you're often interested in finding the exact space between two curves. However, without performing any calculations, estimating can be done visually using the graphed functions. After sketching the functions and finding the points of intersection, you look for the region that is bounded by both curves.

In this exercise, the area we are examining is trapped between the linear function f(x) and the parabolic function g(x). Look at the space that's enclosed as the graphs travel from one point of intersection to the other. While it might be tempting to start calculating this space, the exercise asks for an approximation. You can achieve this by comparing the shape of the bounded area to known shapes, such as triangles or rectangles, and then make a judgment on its size relative to these shapes. This method is an effective way to visually estimate the area without the need for precise calculations, thus engaging your spatial reasoning skills.

Choosing the Best Approximation

Options are given for the estimated area, and it's up to the student to select the one that seems most reasonable based on the sketched graph. This process enhances your ability to analyze mathematical information visually, which is a key skill in many aspects of problem-solving.