Math  /  Calculus

QuestionGiven a graph of the rate of change of distance (in miles per hour) over time (in hours), you are asked to calculate the left and right hand estimates using Riemann sums.\text{Given a graph of the rate of change of distance (in miles per hour) over time (in hours), you are asked to calculate the left and right hand estimates using Riemann sums.}
The x-axis is divided into intervals corresponding to hours, and it is assumed that these intervals are equal in width.\text{The x-axis is divided into intervals corresponding to hours, and it is assumed that these intervals are equal in width.}
The rectangle is n=1.\text{The rectangle is } n=1.
Determine the width of each interval and calculate the left and right hand estimates.\text{Determine the width of each interval and calculate the left and right hand estimates.}
If the intervals aren’t uniform or if there are specific points given, provide that information as well.\text{If the intervals aren't uniform or if there are specific points given, provide that information as well.}

Studdy Solution

STEP 1

What is this asking? We're looking at a graph showing how speed changes over time, and we need to estimate the total distance traveled using left and right Riemann sums with a single rectangle (n=1n=1). Watch out! Remember, the *area* under the speed-time graph gives us the distance traveled.
Don't mix up speed and distance!

STEP 2

1. Find Interval Width
2. Calculate Left Riemann Sum
3. Calculate Right Riemann Sum

STEP 3

We're given that the x-axis (time) goes from 00 to 44 hours, and we're using only one rectangle (n=1n=1).
This means our rectangle spans the entire interval.

STEP 4

To find the width of our rectangle, we subtract the **starting time** from the **ending time**: 40=44 - 0 = 4.
So, our interval width is **4 hours**.

STEP 5

The left Riemann sum uses the function's value at the *left* endpoint of the interval.
Our interval starts at 00 hours.

STEP 6

Looking at the graph, at t=0t = 0, the speed is f(0)=0f(0) = 0 miles per hour.

STEP 7

Now, we multiply the **height** (speed) by the **width** (time) to get the area of our rectangle, which represents the distance.
So, 04=00 \cdot 4 = 0.
Our **left Riemann sum is 0 miles**.

STEP 8

The right Riemann sum uses the function's value at the *right* endpoint of the interval.
Our interval ends at 44 hours.

STEP 9

Looking at the graph, at t=4t = 4, the speed is f(4)=40f(4) = 40 miles per hour.

STEP 10

We multiply the **height** (speed) by the **width** (time) to get the area of the rectangle: 404=16040 \cdot 4 = 160.
Our **right Riemann sum is 160 miles**.

STEP 11

The left Riemann sum estimate is **0 miles**, and the right Riemann sum estimate is **160 miles**.

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