2.1.1Do I have enough gas?

Area Under a Curve Using Trapezoids

2-1.

Elena and J. T. are cruising at $65$ mph ( $95$ feet per second) on their way to Calculus Camp when their car runs out of gas. Elena knows that there is a gas station at the entrance to Calculus Camp in $1$ mile. She quickly decides to determine how far the car will travel by taking velocity measurements as the car decelerates. The velocity of the car at selected points in time is recorded in the table below.

Time (seconds)	$0$	$5$	$8$	$13$	$23$	$33$	$38$	$48$	$63$	$73$	$83$	$93$	$102$
Velocity (ft/seconds)	$95$	$85$	$81$	$70$	$62$	$48$	$44$	$35$	$25$	$19$	$12$	$4$	$0$

Sketch a velocity graph using the data shown in the table above.
Describe how the velocity is changing. When is the velocity changing the fastest? The slowest?
Approximately how far did the car travel in the first $5$ seconds? Use a trapezoid to approximate the distance traveled for $0 \le t \le 5$ .
With the data provided, use trapezoids to approximate the total distance traveled by the car after it ran out of gas. It is important to show your work in a systematic and organized way.
Did the car reach the gas station before stopping? If not, how far did Elena and J. T. have to walk?

2-2.

Sketch a scatterplot of the data below.

$x$	$0$	$3$	$6$	$9$	$12$	$15$
$f\left(x\right)$	$5$	$10$	$11$	$15$	$13$	$6$

“Connect” your scatterplot so that $f$ becomes a continuous function. Is there more than one way to do this? Explain.
Use five trapezoids to approximate the area under $f$ on $0 \le x \le 15$ . Organize your steps systematically.
Tristin organized his work like this:
$\frac { 3 } { 2 }(5 + 10) +\frac { 3 } { 2 }(10 + 11) +\frac { 3 } { 2 }(11 + 15) +\frac { 3 } { 2 }(15 + 13) +\frac { 3 } { 2 }(13 + 6) = A$
Why does the fraction $\frac { 3 } { 2 }$ keep recurring throughout this equation?
Simplify Tristin’s equation by “factoring out” the $\frac { 3 } { 2 }$ .
Describe any new patterns you see.
Use the Trapezoid Rule in the following Math Notes box to set up and compute an approximation for the area under $y = g(x) \text{ on } 2 \le x \le 10$ using four trapezoids of equal height.
$x$
$2$
$3$
$4$
$5$
$6$
$7$
$8$
$9$
$10$
$g(x)$
$1$
$1$
$2$
$4$
$5$
$4$
$4$
$4$
$3$

$x$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$g(x)$	$1$	$1$	$2$	$4$	$5$	$4$	$4$	$4$	$3$

2-3.

In previous problems, approximating the area under a curve required adding the areas of many rectangles or trapezoids together. Imagine the work required to write down the expression if $100$ rectangles were used! To reduce the amount of writing, mathematicians developed summation notation, which is explained in the following Math Notes box. Use the definition of summation notation to write out each sum in expanded form.

$\displaystyle\sum _ { j = 1 } ^ { 10 } j$

$\displaystyle\sum _ { m = 1 } ^ { 10 } m$

$\displaystyle\sum _ { n = 0 } ^ { 5 } ( 4 n - 3 )$

$\displaystyle\sum _ { k = 1 } ^ { 4 } k ( 3 k - 1 ) ^ { 2 }$

Math Notes

Summation Notation

The capital Greek letter sigma, $\sum$ , (equivalent to “S” in English) is used in mathematics as a compact way to indicate a sum. The notation is used as a shorter way to write a long list of numbers added together. For example:

$\displaystyle\sum _ { k = 1 } ^ { 3 } k ^ { 2 } = 1 ^ { 2 } + 2 ^ { 2 } + 3 ^ { 2 } = 14$

Translated, this expression means the sum from $k = 1$ to $k = 3$ of the expression $k^2$ equals $14$ .

Note: We call $k$ the index and $k^2$ the argument of the summation. The values of the index are consecutive integer values only, so the values of $k$ in this example are $1, 2,$ and $3$ .

2-4.

HELP!

Your teammate wrote the expansion for the sum below. Explain what was done incorrectly in this expansion.
$\displaystyle\sum _ { p = 3 } ^ { 5 } p ^ { 2 } = 3 ^ { 2 } + 3.5 ^ { 2 } + 4 ^ { 2 } + 4.5 ^ { 2 } + 5 ^ { 2 } = 82.5$
Realizing that their first expansion was incorrect, your teammate tried again. Explain what was done incorrectly this time.
$\displaystyle\sum _ { p = 3 } ^ { 5 } p ^ { 2 } = ( 3 + 4 + 5 ) ^ { 2 } = 144$
Demonstrate how to correctly expand and simplify this sum.

2-5.

Rewrite each of the following sums using summation notation.

$3 \cdot 4^1 + 3 \cdot 4^2 + 3 \cdot 4^3 + 3 \cdot 4^4$
$9 +16 + 25 + 36$
$\frac { 1 } { 2 } f ( 0 ) + \frac { 1 } { 2 } f ( 2 ) + \frac { 1 } { 2 } f ( 4 ) + \frac { 1 } { 2 } f ( 6 ) + \frac { 1 } { 2 } f ( 8 )$

2-6.

Hooree is learning to hula-hoop. Using a tachometer, her coach keeps a record of her hula-hooping rate (rotations per minute) at select times.

Time (minutes)	$2$	$4$	$9$	$15$	$19$	$26$	$30$
Rate (rotations per minute)	$14$	$18$	$22$	$16$	$10$	$8$	$6$

Sketch a graph of Hooree’s rate.
Find a way to approximate the number of rotations she makes during the $30$ -minute period. It is important to show your method in a systematic, organized way.
Do you think your approximation was over or under the actual number? Explain.
What appears to be happening to Hooree when $t > 9$ ?

2-7.

Consider the function $f(x) = \frac { 1 } { x - 4 }$ . 2-7 HW eToo. Homework Help ✎

Sketch a graph of the function. Label any holes or asymptotes.
Write an equation for the inverse of $f$ . Is the inverse a function? Why or why not?
Sketch a graph of the inverse and compare this sketch to the graph of $y = f(x)$ from part (a).

2-8.

The shaded region at right represents a flag (the upper boundary is a semi-circle). Calculate the volume of the solid formed when the flag is rotated about the pole. In a complete sentence, describe the rotated shape. To help you visualize this, use the 2-8 HW eTool. Homework Help ✎

Horizontal segment, with semicircle above, with it's diameter on the right 2 thirds of the segment, with segments connecting the left endpoint of diameter, the highest point of the semicircle, the right endpoint of diameter, & the center of the diameter, creating 2 right triangles, semicircle area outside the triangle is shaded, & vertical segment is labeled 3.

2-9.

For $f(x) = \sqrt {x-2}$ , he estimation of the area under the curve for $2 \le x \le 6$ is shown at right using four midpoint rectangles. Approximate the area using these rectangles. How reasonable is your result? Homework Help ✎

First quadrant grid on integer values from 0 to 7 on x axis, & from 0 to 4 on y axis, increasing curve opening down, starting at the point (2, comma 0), passing through (6, comma 2), & 4 shaded vertical bars, each with width of 1, starting at x = 2, bottom edges of bars on the x axis, midpoint of top edge of each bar, on the curve.

2-10.

Recall that the area between a function and the $x$ -axis is defined as negative if the region is below the $x$ -axis. Therefore, given $f(x) =\frac { 1 } { 2 }x − 6$ what is the area under the curve for $2 \le x \le 12$ ? 2-10 HW eTool. Homework Help ✎

2-11.

Expand and evaluate each of the following sums. Homework Help ✎

$\displaystyle\sum _ { n = - 4 } ^ { 4 } n ^ { 2 }$
$\displaystyle\sum _ { k = - 4 } ^ { 4 } k ^ { 3 }$
$\displaystyle\sum _ { j = - 3 } ^ { 3 } 2 ^ { j }$
$\displaystyle\sum _ { i = - 5 } ^ { 5 } \operatorname { sin } ( i )$

2-12.

Determine whether each function below is an even function, an odd function, or neither. Homework Help ✎

$f(x) = x^2$
$f(x) = x^3$
$f(x) = 2^x$
$f(x) = \sin(x)$

2-13.

A bug is walking on your graph paper along the $x$ -axis. The bug’s velocity (in feet per second) is shown on the graph at right. Homework Help ✎

When did the bug turn around?
When was the bug’s speed the greatest?
After $12$ seconds, how far is the bug from its starting position?
Remember that acceleration is the rate of change of velocity. Calculate the acceleration of the bug at the following times.
1. $1$ second
1. $5$ seconds
1. $10$ seconds

Continuous linear Piecewise, x axis labeled, t, y axis labeled, v of t, starting at the origin, turning down at (2, comma 4), turning right at (4, comma 3), turning down at (6, comma 3), ending at (12, comma negative 6).

2-14.

Let $f ( x ) = \left\{ \begin{array} { l l } { x ^ { 2 } - 1 } & { \text { for } x < 2 } \\ { 2 x + 3 } & { \text { for } x \geq 2 } \end{array} \right.$ . Homework Help ✎

What is $f(2)$ ?
As $x \rightarrow 2^+, y \rightarrow$ ?
As $x \rightarrow 2^−, y \rightarrow$ ?
What do the results from parts (b) and (c) indicate about the graph?

Calculus Third Edition

2.1.1Do I have enough gas?

The Trapezoid Rule

Summation Notation