homework homework assignment #1 review section 2.1/read section 2.2 page 66, exercises: 1 – 9...

Homework

Homework Assignment #1 Review Section 2.1/Read Section 2.2 Page 66, Exercises: 1 – 9 (Odd), 13 – 29

(EOO)

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

Homework, Page 661. A ball is dropped from a state of rest at t = 0. The distance traveled after t sec is s(t) = 16t2.

(a) How far does the ball travel during the time interval [2, 2.5]?

(b) Compute the average velocity over [2, 2.5].


2

2

2.5 16 2.5 16 6.25 100

2 16 2 64 2.5 2 36

s ft

s ft s s ft

36avg. velocity 72

0.5

ftft s

s

Homework, Page 661. (c) Compute the average velocity over the time intervals [2, 2.01], [2, 2.005], [2, 2.00001]. Use this to estimate the object’s instantaneous velocity at t = 2.


2

2

2

2.01 2 64.6416 6464.16

2.01 2 0.01

2.005 2 64.6416 6464.08

2.005 2 0.005

2.00001 2 64.00064 6464.0

2.00001 2 0.00001

Instantaneous velocity at 2 is 64

s sROC

s sROC

s sROC

t ft s

Homework, Page 663. Let . Estimate the instantaneous ROC of v with respect to T when T = 300.


20 300.01 300 0.005770.577

300.01 300 0.01

20 300.0001 300 0.00005770.577

300.0001 300 0.0001

20 300.00001 300 0.000005770.577

300.00001 300 0.00001

Instantaneous ROC at 300 is 0.577

mROC s

mROC s

mROC s

mT s

20v T

Homework, Page 66A stone is tossed in the air from ground level with an initial velocity of 15 m/s. Its height at time t is h(t) = 15t – 4.9t2 m.

5. Compute the stones average velocity over the time interval [0.5, 2.5] and indicate corresponding the corresponding secant line on a sketch of the graph of h(t).


2.5 0.5

2.5 0.50.6

0.3 /2

h hROC

m s

x

y

(2.5, 6.875)(0.5, 6.275)

Homework, 67. With an initial deposit of $100, the balance in a bank account after t years is f(t) = 100(1.08)t dollars.

(a) What are the units of the ROC of f(t)?

The units of the ROC of f (t) are dollars per year.

(b) Find the average ROC over [0, 0.5] and [0, 1].


0.5 0 103.92 1007.85$ /

0.5 0 0.51 0 108 100

8$ /1 0 1

f fROC yr

f fROC yr

Homework, Page 66(c) Estimate the instantaneous ROC at t = 0.5 by computing the average ROC over intervals to the right and left of t = 0.5.


0.5 0.4999 103.9230 103.9222

0.5 0.4999 0.00018.00$ /

0.5001 0.5 103.9238 1009230

0.5 0.0001 18.00$ /

Instantaneous ROC 0.5 0.08

f fROC

yr

f fROC

yr

t

Homework, Page 66


29. 4 3; 2P x x t

2 1.9999 13 12.998415.9996

2 1.9999 0.00012.0001 2 13.0016 13

16.00040.5 0.0001 0.0001

Instantaneous ROC 2 16

P PROC

P PROC

x

Estimate the instantaneous ROC at the indicated point.

Homework, Page 66


13. ; 0xf x e x

0.0001 0 1.0001 11

0.0001 0 0.00010 0.0001 1 0.9999

10 0.0001 0.0001

Instantaneous ROC 0 1

f fROC

f fROC

x

Estimate the instantaneous ROC at the indicated point.

Homework, Page 6617. The atmospheric temperature T (in ºF) above a certain point is T = 59 – 0.00356h, where h is the altitude in feet (h ≤ 37,000 ft). What are the average and instantaneous ROC of T with respect to h? Why are they the same? Sketch the graph of T.

The secant line and

the graph of T are the

same line, so the average

and instantaneous ROC

are both equal to – 0.00356. Rogawski Calculus

Copyright © 2008 W. H. Freeman and Company

x

y

50

40,000 ft

Tem

pera

ture

-50

Homework, Page 6621. Assume that the period T (in sec) of a pendulum is

where L is the length (in m).

(a) What are the units for ROC of the T with respect to L? Explain what this rate measures.

The units of the ROC of T are sec/m, giving the rate at which the period changes when length changes.

(b) What quantities are represented by the slopes of lines A and B in Figure 10?

Lines A and B represent instantaneous and average ROC.


3

2T L

Homework, Page 6625. The fraction of a city’s population infected by a flu virus is plotted as a function of time in weeks in Figure 14.

(a) Which quantities are represented by the slopes of lines A and B? Estimate these slopes.

The slope of line A represents the average ROC over weeks 4 through 6. The slope of line B represents the instantaneous ROC in week 6. The slopes appear to be about , respectively.


0.09 0.120.045 and 0.02

2 6

Homework, Page 6625. (b) Is the flu spreading most rapidly at t = 1, 2, or 3?

The flu is spreading most rapidly at t = 3 , as the slope is greatest at that point.

(c) Is the flu spreading most rapidly t = 4, 5, or 6?

The flu is spreading most rapidly at t = 4, as the slope is greatest at that point.


Homework, Page 6629. Sketch the graph of f (x) = x (1 – x) over [0, 1]. Refer to the graph and find:

(a) The average ROC over [0, 1].

average ROC = 0

(b) The instantaneous ROC at x = 0.5.

instantaneous ROC = 0

(c) The values of x for which ROC is positive.

ROC is positive for x < 0.5.



Chapter 2: LimitsSection 2.2: Limits: A Numerical and

Graphical Approach

Jon Rogawski

Calculus, ET

First Edition


In previous courses, we have considered limits when looking at the end behavior of a function. For instance, the function y = 2 + e–x asymptotically approaches y = 2 as x approaches infinity.

lim 2 2x

xe

lim 2 x

xe

Mathematically, we could write this as:

Similarly:

In this section we will look at the behavior of the values of f (x) as x approaches some number c, whether or not f (c) is defined.


Consider the function as x approaches 0. The function is not defined at x = 0, but it is defined for values of x very close to 0. By examining these values in the table below, we see that f (x) approaches 1 as x approaches 0 or

sin xf x

x

0

sinlim 1x

x

x

Example, Page 76Fill in the table and guess the value of the limit.


30

sin4. lim ,where f f

θ ±0.002 ±0.0001 ±0.00005 ±0.00001

f (+θ)

f (–θ)


Recalling that the distance between points a and b is |b – a|,

lim (DNE)x c

f x does not exist

If the values of f (x) do not converge to any finite value as x approaches c, we say that

Example, Page 76Verify each limit using the limit definition.


3

14. lim 5 7 8x

x


Theorem 1 states two simple, yet important concepts, which are illustrated in the graph below.

x

y

y = k

c

The value of constant k is independent of the value of x, as is the value of c.

Graphical InvestigationUse the graphing calculator to graph the function in the vicinity of the value of x in question.


If the graph appears to approach the same point from either side of the value in question, then we may say that the limit exists, and the limit may frequently be estimated directly from the graph.

Example, Page 76Verify each limit using the limit definition.


3

14. lim 5 7 8x

x

Numerical Investigation1. Construct a table of values of f (x) for x close to, but

less than c.

2. Construct a second table of values of f (x) for x close to, but greater than c.

3. If both tables indicate convergence to the same number L, we take L to be an estimate for the limit.


limx c

f x L


The graph and table below illustrate graphical and numerical investigation of:

9

9lim

3x

x

x



2

4limx

x



0

1lim

h

h

e

h


The graph below doesn’t really answer the question of the existence of a limit, but the table shows that it does not exist as the values of sin (1/x) have opposite signs when the same distance from zero in the positive and negative directions is considered. Thus:

0

1limsin D.N.E.x x


In Figure 6, we see that does

not approach the same value of

as approaches 0 from the negative

and positive directions.

f x

y

x

0

In this case, lim D.N.E.,x

f x

0 0

but we may state: lim 1 and lim 1.x x

f x f x

0 0

lim and lim are referred to as one-sided

limits and read as the limit as approaches 0 from the

positive and negative directions, respectively.

x xf x f x

x


Figure 7 shows the graph of a piecewise function. The limitsas x approaches 0 and 2 do notexist, but the limit as x approaches 4 appears to exist,as the graph on both sides of x = 4 seems to converge on the same value of y. The following one-sided limits also appear to exist:

0 2 2

lim , lim , and lim , x x x

f x f x f x

0

limx

f x

but DNE as the values of f (x) continue to

oscillate between ±1 as x gets ever closer to 0.

Definition

The limit of a function at a given point exists if, and only if, the one-sided limits at the point are equal. Mathematically, we state:


lim exists lim limx c x c x c

f x f x f x

or

lim lim limx c x c x c

f x f x f x

Example, Page 7638. Determine the one-sided limits at c = 1, 2, 4 of the function g(t) shown in the figure and state whether the limit exists at these points.


x

y


Graphs (A) and (B) in Figure 8 illustrate functions with infinite discontinuities. Graph (C) illustrates a function

with the interesting property: 0

lim andx

f x

lim .

xf x

Example, Page 76Draw the graph of a function with the given limits.


1 3 3

48. lim , lim 0, limx x x

f x f x f x

x

y

Homework

Homework Assignment #2 Read Section 2.3 Page 76, Exercises: 1 – 53 (EOO) Quiz next time


homework homework assignment #1 review section 2.1/read section 2.2 page 66, exercises: 1 – 9...

Documents

instantaneous roc of

time t

roc of f t

graph of t

period t

t dollars

t years

average roc