2.4.1How can I write sums?

Improving Approximation

2-132.

Sketch a sine wave on $0 < x < \pi$ . Determine the domain on which:

The left endpoint rectangles are an overestimate of the actual area.
The left endpoint rectangles are an underestimate of the actual area.
The right endpoint rectangles are an overestimate of the actual area.
The right endpoint rectangles are an underestimate of the actual area.
The left endpoint rectangles and right endpoint rectangles yield equal areas.

Explore using the Estimating Area Under a Curve eTool . Click in the lower right corner of the graph to view it in full-screen mode.

2-133.

Examine the graph at right. Use the graph to complete parts (a) and (b) below.

Assume that an area is being estimated using $n$ rectangles of equal width. Write an expression for $\Delta x$ using $a, b,$ and $n$ .
As the number of rectangles increases, what happens to the width of each rectangle?

Your teacher will provide you with a model.

2-134.

Rewrite the following Riemann sum by substituting your expression for $\Delta x$ from part (a) of problem 2-133 into the summation statement.

$A \approx \displaystyle\sum _ { i = 0 } ^ { n - 1 } [ \Delta x \cdot f ( a + \Delta x \cdot i ) ]$

Changing the value of $n$ varies the number of rectangles that are used in the approximation.

2-135.

OUT OF GAS, Part Two

After arriving at Calculus Camp, J.T. and Elena realized that they could have made a much better estimate of the distance traveled after running out of gas than they had in problem 2-1. Elena suggested using a function to approximate the data, thereby allowing them to use a Riemann sum.

Elena discovered that $y = 0.00538x^2 − 1.43x + 91.7$ represents a curve of best fit for the data in problem 2-1. Sketch this curve on your calculator, making sure you can see a complete picture of Elena and J.T.’s road trip.
Use sigma notation to write two Riemann sums that approximate the distance traveled. Write one expression for the left endpoint and one expression for the right endpoint rectangles. Assume that the area is divided into $n$ rectangles.
Consider the shape of the graph. Will left endpoint rectangles generate an over or and underestimate of the actual area? Justify your answer.
Have each member of your team choose a different number of rectangles ( $n = 10, n = 20,$ etc.) and calculate the corresponding distances using both right and left endpoint rectangles. Share this information with your team—keep a detailed record of everyone’s findings. Use your calculator so you have time to record many different values for $n$ . What happens as $n$ increases?
How can Elena get the best estimate? How many rectangles should she use? Should they be left or right endpoint rectangles?

2-136.

Sketch a graph of the region bounded by the functions $f(x)=x^2,g(x)=−2x+8,$ and the $x$ -axis. Homework Help ✎

How can you estimate the area in this region?
Using your method, estimate the area of the region.

2-137.

Write a single expression using the Trapezoid Rule, which will approximate the area under $f(x) = 2x^2 - 4x + 3$ over the interval $0\le x\le1$ using five trapezoids of equal height. Do not evaluate the expression. Homework Help ✎

2-138.

Examine the scenarios below, paying attention to the units. Homework Help ✎

While walking to school, Jaime’s distance from home (in miles) was $s(t)=3t^2$ , where $t$ is measured in hours. Sketch a graph of his distance. If it takes Jaime $30$ minutes to walk to school, what is his average velocity? Explain how you got your answer.
While walking home, Jaime walks so that his velocity (in miles per hour) is $v(t) = -2t$ , where $t$ is measured in hours. How long does it take him to get home?

2-139.

Write an expression that will calculate the slope between $(a, f(a))$ and $(b, f(b))$ . A sketch may help. Homework Help ✎

2-140.

WHICH IS BETTER? Part Three

Below is a comparison of using different numbers of rectangles to approximate the same area under a curve for $f$ . Decide which situation will best approximate the area under the curve for $a\le x\le b$ . Explain why. Homework Help ✎

If, in each situation, the rectangles all have equal widths, write expressions to approximate the areas under the curves.

$4$ rectangles

$10$ rectangles

$20$ rectangles

2-141.

A tangent line is drawn to the curve $y = -\frac { 1 } { 2 }(x – 3)^2 + 3$ at $x = 2$ . Trace the graph on your paper and add tangents at $x = 1, 3, 4,$ and $5$ . Visualize this using the Slope at a Point eTool. Homework Help ✎

Downward curve passing through the highlighted points (1 comma 1), (2, comma 2.5), (3, comma 3), which is the turning point, (4, comma 2.5), & (5, comma 1), & line passing through the point (2, comma 2.5) with slope of 1.

2-142.

Determine the values of the following limits. If the limit does not exist, indicate why not. Homework Help ✎

$\lim\limits _ { x \rightarrow 4 ^ { + } } ( \sqrt { x - 4 } - 5 )$

$\lim\limits _{ x \rightarrow - 2 } \large\frac { x ^ { 2 } - 4 x - 12 } { x + 2 }$

$\lim\limits _ { x \rightarrow 6 } \large\frac { x ^ { 2 } - 4 x - 12 } { x + 2 }$

$\lim\limits _ { x \rightarrow \infty } \large\frac { 1 } { x - 1 }$

$\lim\limits _ { x \rightarrow - \infty } \large\frac { x ^ { 2 } + 6 x - 7 } { x }$

$\lim\limits _ { x \rightarrow \infty } \large\frac { x ^ { 2 } - 7 x - 10 } { x ^ { 2 } }$

2-143.

Review the three conditions of continuity. Then, examine the graph below and determine at which values of $x$ the function is not continuous. Explain which condition the function fails at each discontinuity. Homework Help ✎

2-144.

Use the graph from problem 2-143 to complete the following table: Homework Help ✎

$a$	$\lim\limits _ { x \rightarrow a } f ( x )$	$f (a)$	Continuous at $x = a$ ?
$1$
$2$
$3$
$4$

2-145.

Given the function $f(x) = 2x^2 − x + 3$ , calculate the following values. Homework Help ✎

$\frac { f ( 3 ) - f ( 2 ) } { 1 }$

$\frac { f ( 2.1 ) - f ( 2 ) } { 0.1 }$

$\frac { f ( 2.01 ) - f ( 2 ) } { 0.01 }$

Estimate $\lim\limits _ { x \rightarrow 2 } \frac { f ( x ) - f ( 2 ) } { x - 2 }$ .

Calculus Third Edition