Assignment 1

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Assignment 1 - MAC2311-Fall2017

A tank holds 1000 gallons of water, which drains from the bottom of the tank in half an hour. The values in the table below show the volume $V$ of water remaining in the tank (in gallons) after $t$ minutes.


$t$ (min)	5	10	15	20	25	30

$V$ (gal)	694	444	250	111	28	0

(a) If $P$ is the point $(15,250)$ on the graph of $V$ , find the slopes of the secant lines $PQ$ when $Q$ is the point on the graph with the following values of $t$ :

$t$	slope

$5$	$\answer {-44.4}$
$10$	$\answer {-38.8}$
$20$	$\answer {-27.8}$
$25$	$\answer {-22.2}$
$30$	$\answer {-50/3}$

(b) Estimate the slope of the tangent line at $P$ by averaging the slopes of two secant lines.

$\mbox {Slope Estimate:}\; \answer {-30.5}$

The point $P(2,-1)$ lies on the curve $y=1/(1-x)$ .

(a) If $Q$ is the point $(x,1/(1-x))$ , use a calculator to find the slope of the secant line $PQ$ (correct to six decimal places) for the following values of $x$ : $\begin {array}{l|l} x & \mbox {slope} \\ \hline 1.5 & \answer {2}\\ 1.9 & \answer {10/9}\\ 1.99 & \answer {100/99}\\ 2.5 & \answer {2/3}\\ 2.1 & \answer {10/11}\\ 2.01 & \answer {100/101} \end {array}$

(b) Using the results of part (a), guess the value of the slope of the tangent line to the curve at $P(2,-1)$ . $\mbox {Slope Guess:}\;\answer {1}$

(c) Using the slope from part (b), find an equation of the tangent line to the curve at $P(2,-1)$ . $\mbox {Tangent line equation}:\; y=\answer {x-3}$

If a rock is thrown upward on the planet Mars with a velocity of $10$ m/s, its height in meters $t$ seconds later is given by $y=10t - 1.86t^2$ .

(a) Find the average velocity over the following time intervals: $[1,2]$ , $[1,1.5]$ , $[1,1.1]$ , $[1,1.01]$ .

$\begin {array}{l|l} \mbox {Interval} & \mbox {Avg. Velocity}\\ \hline [1,2] & \answer {4.42}\\ {[1,1.5]} & \answer {5.35}\\ {[1,1.1]} & \answer {6.094}\\ {[1,1.01]} & \answer {6.2614} \end {array}$

(b) Estimate the instantaneous velocity when $t=1$ .

$\mbox {Velocity Estimate}:\; \answer {6.28}$