Problem statement
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.
c) 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.
Again, a graph may help
it's just a step function, so just add up the areas of the rectangles with heights 0,1,2,...9 and width 0.1
To solve this problem, we need to define the functions f(x) and g(x), and then use a graphical interpretation of the definite integral to compute the given integrals.
a) The function f(x) is defined as the first digit in the decimal expansion for x. This means that we need to find the first digit of any given number between 0 and 1. To do this, we can convert the number into a decimal and then simply take the integer part. For example, for x = 0.719, the first digit would be found by converting it into a decimal as follows:
0.719 → 7.19 → 7
Now, we can sketch the graph of y = f(x) on the unit interval. Since f(x) gives us the first digit, we can plot points corresponding to the first digits of different numbers between 0 and 1. For example, for x = 0.5, f(0.5) = 5, so we plot the point (0.5, 5). Similarly, for x = 0.719, f(0.719) = 7, so we plot the point (0.719, 7). By plotting these points and connecting them, we can sketch the graph.
To compute the definite integral ∫_0^1▒f(x)dx using a graphical interpretation, we can calculate the area under the graph of y = f(x) between 0 and 1. In this case, since f(x) takes on integer values, the integral represents the sum of the function values over the interval [0, 1].
c) The function g(x) is defined as the second digit in the decimal expansion for x. Similar to f(x), we can convert the number x into a decimal and then consider the second digit. For example, for x = 0.437, the second digit would be found as follows:
0.437 → 4.37 → 3
Now, we need to compute the integral ∫_0^1▒g(x)dx using a graphical interpretation. The steps for this are the same as for part a) - we sketch the graph of y = g(x) on the unit interval, where g(x) represents the second digit of x, and then calculate the area under the graph between 0 and 1.
By following these steps, we can obtain the graphical solution and compute the desired integrals for both parts a) and c) of the problem.