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## Educators    + 6 more educators

### Problem 1

A tank holds 1000 gallons of water, which drains from the bottom of the tank in half an hour. The values in the table show the volume $V$ of water remaining in the tank (in gallons) after $t$ minutes.
$$\begin{array}{|c|c|c|c|c|c|c|} \hline t(\mathrm{~min}) & 5 & 10 & 15 & 20 & 25 & 30 \\ \hline V(\mathrm{gal}) & 694 & 444 & 250 & 111 & 28 & 0 \\ \hline \end{array}$$
(a) If $P$ is the point (15,250) on the graph of $V$, find the slopes of the secant lines $P Q$ when $Q$ is the point on the graph with $t=5,10,20,25,$ and 30
(b) Estimate the slope of the tangent line at $P$ by averaging the slopes of two secant lines.
(c) Use a graph of the function to estimate the slope of the tangent line at $P$. (This slope represents the rate at which the water is flowing from the tank after 15 minutes.) Evelyn C.

### Problem 2

A cardiac monitor is used to measure the heart rate of a patient after surgery. It compiles the number of heartbeats after $t$ minutes. When the data in the table are graphed, the slope of the tangent line represents the heart rate in beats per minute.
$$\begin{array}{|l|c|c|c|c|c|} \hline t \text { (min) } & 36 & 38 & 40 & 42 & 44 \\ \hline \text { Heartbeats } & 2530 & 2661 & 2806 & 2948 & 3080 \\ \hline \end{array}$$
The monitor estimates this value by calculating the slope of a secant line. Use the data to estimate the patient's heart rate after 42 minutes using the secant line between the points with the given values of $t$.
(a) $t = 36$ and $t = 42$
(b) $t = 38$ and $t = 42$
(c) $t = 40$ and $t = 42$
(d) $t = 42$ and $t = 44$ Daniel J.

### Problem 3

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

(a) If $Q$ is the point $(x, 1/(1-x))$, use your calculator to find the slope of the secant line $PQ$ (correct to six decimal places) for the following values of $x$:
(i) $1.5$ (ii) $1.9$ (iii) $1.99$ (iv) $1.999$
(v) $2.5$ (vi) $2.1$ (vii) $2.01$ (viii) $2.001$

(b) Using the results of part (a), guess the value of the slope of the tangent line to the curve at $P(2, -1)$.

(c) Using the slope from part (b), find an equation of the tangent line to the curve at $P(2, -1)$. Oswaldo J.

### Problem 4

The point $P(0.5, 0)$ lies on the curve $y = \cos \pi x$.

(a) If $Q$ is the point $(x, \cos \pi x)$, use your calculator to find the slope of the secant line $PQ$ (correct to six decimal places) for the following values of $x$:

(i) $0$ (ii) $0.4$ (iii) $0.49$
(iv) $0.499$ (v) $1$ (vi) $0.6$
(vii) $0. 51$ (viii) $0.501$

(b) Using the results of part (a), guess the value of the slope of the tangent line to the curve at $P(0.5, 0)$.

(c) Using the slope from part (b), find an equation of the tangent line to the curve at $P(0.5, 0)$.

(d) Sketch the curve, two of the secant lines, and the tangent line.

Anupa D.

### Problem 5

If a ball is thrown into the air with a velocity of $40 ft/s$, its height in feet $t$ seconds later is given by $y = 40t - 16t^2$.

(a) Find the average velocity for the time period beginning when $t = 2$ and lasting
(i) 0.5 seconds (ii) 0.1 seconds
(iii) 0.05 seconds (iv) 0.01 seconds
(b) Estimate the instantaneous velocity when $t = 2$. Numann M.

### Problem 6

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

(a) Find the average velocity over the given time intervals:
(i) $[1, 2]$ (ii) $[1, 1.5]$
(iii) $[1, 1.1]$ (iv) $[1, 1.01]$
(v) $[1, 1.001]$
(b) Estimate the instantaneous velocity when $t = 1$.

JB
Joshua B.

### Problem 7

The table shows the position of a motorcyclist after accelerating from rest.

(a) Find the average velocity for each time period:
(i) $[2, 4]$ (ii) $[3, 4]$ (iii) $[4, 5]$ (iv) $[4, 6]$
(b) Use the graph of $s$ as a function of $t$ to estimate the instantaneous velocity when $t = 3$.

DM
David M.

### Problem 8

The displacement (in centimeters) of a particle moving back and forth along a straight line is given by the equation of motion $s = 2 \sin \pi t + 3 \cos \pi t$, where $t$ is measured in seconds.

(a) Find the average velocity during each time period:
(i) $[1, 2]$ (ii) $[1, 1.1]$
(iii) $[1, 1.01]$ (iv) $[1, 1.001]$
(b) Estimate the instantaneous velocity of the particle when $t =1$. Carson M.
The point $P(1, 0)$ lies on the curve $y = \sin (10\pi /x)$.
(a) If $Q$ is the point $(x, \sin (10\pi /x))$, find the slope of the secant line $PQ$ (correct to four decimal places) for $x$ = 2, 1.5, 1.4, 1.3, 1.2, 1.1, 0.5, 0.6, 0.7, 0.8, and 0.9 .
(b) Use a graph of the curve to explain why the slopes of the secant lines in part (a) are not close to the slope of the tangent line at $P$.
(c) By choosing appropriate secant lines, estimate the slope of the tangent line at $P$. 