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.
It is not clear to me what the "first digit in the decimal expansion".
Is the first significant digit, the first digit after the decimal point, or otherwise?
What is f(0.004)? Is it 0, or 4?
Since 1.0 is in the domain, what about f(1.0), is it 1 or zero?
Note that 1 is not the first digit after the decimal point.
For b, the domain is not specified, is it also [0,1]?
I have the same exact workshop problem. Need help. It is written exactly as above.
Oh well, we'll make the best out of it.
I will assume the function is defined as f(x)=floor(x*10), so this will make a step function, something like:
f(x)=
0 0≤x<0.1
1 0.1≤x<0.2
2 0.2≤x<0.3
...
9 0.9≤x<1
10 x=1.0
Your graph should look like steps, but with a full circle on the left of the step (≤) and an empty circle on the right of each step (<). Finally, at x=1, there should be a single full cirlce at (1.0, 10).
To integrate the function, you only have to calculate the area under each step, i.e. height of step multiplied by its width (0.1), and sum them up.
(b)
The function is basically
f(x)=floor(x*100) mod 10
The graph consists of mini-steps from 0 to 9, repeated 10 times.
For the integral, you only have to integrate one set of steps and multiply the results by 10.
a) To sketch the graph of y = f(x), we need to identify the first digit in the decimal expansion of x for each value in the interval 0 ≤ x ≤ 1.
Start by partitioning the interval into subintervals. For example, you can divide the interval into 10 equal subintervals: [0, 0.1), [0.1, 0.2), [0.2, 0.3), ..., [0.9, 1.0].
For each subinterval, determine the first digit in the decimal expansion. For example, in the subinterval [0.2, 0.3), the first digit is 2.
Now, plot the points on a graph with x-coordinate corresponding to the midpoint of each subinterval and y-coordinate corresponding to the first digit in the decimal expansion.
The resulting graph of y = f(x) should consist of alternating horizontal line segments starting at each x-coordinate, such that the y-coordinate represents the first digit in the decimal expansion of that x-coordinate.
To compute ∫₀¹ f(x)dx, you can use the graphical interpretation of the definite integral. The integral represents the area under the curve of y = f(x) between x = 0 and x = 1.
Count the number of complete rectangles under the curve, each with width 0.1, and the height equal to the respective first digit. Add up the areas of each rectangle to find the total area under the curve. This sum will be the value of the definite integral ∫₀¹ f(x)dx.
b) To compute ∫₀¹ g(x)dx, we need to determine the second digit in the decimal expansion for each value in the interval 0 ≤ x ≤ 1 and use a graphical approach.
Similarly to part a), partition the interval [0, 1] into 10 equal subintervals: [0, 0.1), [0.1, 0.2), [0.2, 0.3), ..., [0.9, 1.0].
For each subinterval, determine the second digit in the decimal expansion. For example, in the subinterval [0.2, 0.3), the second digit is 0.
Plot the points on a graph with x-coordinate corresponding to the midpoint of each subinterval and y-coordinate corresponding to the second digit in the decimal expansion.
The resulting graph of y = g(x) should consist of alternating horizontal line segments starting at each x-coordinate, such that the y-coordinate represents the second digit in the decimal expansion of that x-coordinate.
To compute ∫₀¹ g(x)dx, use the same graphical interpretation as in part a). Count the number of complete rectangles under the curve, each with width 0.1, and the height equal to the respective second digit. Add up the areas of each rectangle to find the total area under the curve, which will be the value of the definite integral ∫₀¹ g(x)dx.