a) Suppose f(x) is defined on 0 ≤x≤1 by the following rule: f(x) is the first digit in the decimal expansion for x. For example, f(1/2) = 5 and f(0.719) = 7. Sketch the graph of y = f(x) on the unit interval with appropriate scales for x and for y. Use a graphical interpretation of the definite integral to compute ∫_0^1▒f(x)dx.
b) Suppose the function g(x) is defined as follows: g(x) is the second digit in the decimal expansion for x. For example, g(0.437) = 3. Compute ∫_0^1▒g(x)dx. Use graph.
I don't know what you need. We cant do the graph for you.
a) To sketch the graph of y = f(x), we need to determine the value of f(x) for each x in the interval [0, 1]. We know that f(x) is the first digit in the decimal expansion of x.
Let's consider a few values of x to find their corresponding first digits:
For x = 0.1, the first digit in its decimal expansion is 1.
For x = 0.2, the first digit in its decimal expansion is 2.
For x = 0.3, the first digit in its decimal expansion is 3.
...
For x = 0.9, the first digit in its decimal expansion is 9.
Using these values, we can plot the graph of y = f(x). The graph will consist of horizontal lines at y = 1, y = 2, y = 3, ..., y = 9. Each line segment will span a width of 0.1 (since the first digit only changes at intervals of 0.1).
Now, to compute the definite integral ∫ from 0 to 1 of f(x)dx, we can interpret it graphically. The definite integral represents the area under the graph of f(x) between x = 0 and x = 1.
Since the graph of f(x) consists of horizontal lines, the area under each line segment is simply the width of the segment multiplied by the height of the segment. In this case, the width is 0.1, and the height is the first digit of each line segment.
We can calculate the area under each line segment and sum them up to find the total area. Here's the calculation:
Area under y = 1: 0.1 * 1 = 0.1
Area under y = 2: 0.1 * 2 = 0.2
...
Area under y = 9: 0.1 * 9 = 0.9
Summing up these areas gives us the total area under the graph of f(x) between x = 0 and x = 1:
∫ from 0 to 1 of f(x)dx = 0.1 + 0.2 + ... + 0.9 = 4.5
Therefore, the definite integral ∫ from 0 to 1 of f(x)dx is equal to 4.5.
b) Similarly, to compute the definite integral ∫ from 0 to 1 of g(x)dx, we need to consider the function g(x), which is defined as the second digit in the decimal expansion of x.
Let's determine the second digit for a few values of x:
For x = 0.01, the second digit in its decimal expansion is 0.
For x = 0.02, the second digit in its decimal expansion is 2.
For x = 0.12, the second digit in its decimal expansion is 1.
...
For x = 0.99, the second digit in its decimal expansion is 9.
Based on these values, we can sketch the graph of y = g(x). Similar to the graph of f(x), the graph of g(x) will consist of horizontal lines at y = 0, y = 1, ..., y = 9. Again, each line segment will span a width of 0.1.
To compute the definite integral ∫ from 0 to 1 of g(x)dx, we need to calculate the area under each line segment and sum them up. Each line segment's width is 0.1, and the height is the second digit of each segment.
Here's the calculation:
Area under y = 0: 0.1 * 0 = 0
Area under y = 1: 0.1 * 1 = 0.1
...
Area under y = 9: 0.1 * 9 = 0.9
Summing up these areas gives us the total area under the graph of g(x) between x = 0 and x = 1:
∫ from 0 to 1 of g(x)dx = 0 + 0.1 + ... + 0.9 = 4.5
Therefore, the definite integral ∫ from 0 to 1 of g(x)dx is also equal to 4.5.