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Math  /  Calculus

QuestionQuestion Pat 1 of 2 Completed: 6 of 11 My score: 6/116 / 11 pts (54.55\%) Save
Use the figures to calculate the left and right Riemann sums for f on the given interval and for the given value of n. f(x)=2x on [1,5];n=4f(x)=\frac{2}{x} \text { on }[1,5] ; n=4
The left Riemann sum for f is □\square . (Simplify your answer.)

Studdy Solution

STEP 1

1. We are given the function f(x)=2x f(x) = \frac{2}{x} .
2. The interval of interest is [1,5][1, 5].
3. We need to calculate the left Riemann sum with n=4 n = 4 .

STEP 2

1. Determine the width of each subinterval.
2. Identify the left endpoints of each subinterval.
3. Evaluate the function at each left endpoint.
4. Calculate the left Riemann sum by summing the products of function values and subinterval widths.

STEP 3

Determine the width of each subinterval:
The interval [1,5][1, 5] is divided into n=4 n = 4 subintervals. The width Δx\Delta x of each subinterval is calculated as:
Δx=5−14=1 \Delta x = \frac{5 - 1}{4} = 1

STEP 4

Identify the left endpoints of each subinterval:
Since Δx=1\Delta x = 1, the left endpoints are:
x0=1,x1=2,x2=3,x3=4 x_0 = 1, \quad x_1 = 2, \quad x_2 = 3, \quad x_3 = 4

STEP 5

Evaluate the function at each left endpoint:
f(x0)=f(1)=21=2 f(x_0) = f(1) = \frac{2}{1} = 2 f(x1)=f(2)=22=1 f(x_1) = f(2) = \frac{2}{2} = 1 f(x2)=f(3)=23 f(x_2) = f(3) = \frac{2}{3} f(x3)=f(4)=24=12 f(x_3) = f(4) = \frac{2}{4} = \frac{1}{2}

STEP 6

Calculate the left Riemann sum:
The left Riemann sum L L is given by:
L=Δx(f(x0)+f(x1)+f(x2)+f(x3)) L = \Delta x \left( f(x_0) + f(x_1) + f(x_2) + f(x_3) \right)
L=1(2+1+23+12) L = 1 \left( 2 + 1 + \frac{2}{3} + \frac{1}{2} \right)
L=2+1+23+12 L = 2 + 1 + \frac{2}{3} + \frac{1}{2}
Convert the fractions to a common denominator and simplify:
L=3+46+36 L = 3 + \frac{4}{6} + \frac{3}{6} L=3+76 L = 3 + \frac{7}{6} L=186+76 L = \frac{18}{6} + \frac{7}{6} L=256 L = \frac{25}{6}
The left Riemann sum for f f is:
256 \boxed{\frac{25}{6}}

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