## Educators

DJ

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.

(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 $t$ = 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.)

DJ
Daniel J.

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.

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 $t = 44$

DJ
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 $P(2, -1)$.

DJ
Daniel 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.

DJ
Daniel J.

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 $t = 2$.

DJ
Daniel J.

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 $t = 1$.

DJ
Daniel J.

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 $t = 3$.

DJ
Daniel J.

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 $t =1$.

DJ
Daniel J.
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 $P$.