Chopter 3

1. Graph is correct.

2. Avs. ,u," = !€*i@

=

= 2.1 km/min

8. The line has slope 2 and point (5, 3)..'. y-3 =2(x- 5) + y =2x*7The diagram below shows this line and the graph of fwith a zoom by a factor of 10 in both the x- and y-directions. The graph looks more and more like a linethe closer you zoom. The phenomenon is known aslocal linearity.

I cu,u"/tin"

i

i-3 /i/i/i / 5

+. -' -'.' - -l'..'.r..............-.

9. The journal entry should include the insight that anexact derivative can be calculated algebraicallybywriting an equation for the difference quotient (i.e., theslope of the secant line) in terms of x, canceling to"remove" the factor in the denominator that causes thediscontinuity, and finding the limit of the simplifiedexpression by direct substitution. The journal entrycould also include the observation about local linearityfrom Problem 8.

3.21 - 30.1

x2 - 8x + 18 - 3 _x2 - 8x + 153. m(x)= x-5 x-5

7. The line is tangent to the graph.

= Jx-l)$--Q) - x-3, provided x * 5- x-54. Limit = 2. lt represents the instantaneous velocity of

the spaceship in km/min5. m(5) has the form 0/0, which is an indeterminate form.

It is undefined because of division by zero.6. Graph

il :

1'\'1'i\1r'\'i

.-l

1...

x

I l\ i.-+--j'tr'- !.

t.r.\,. r,- -- t--- r-.*-.

tttll

5

Q1. Derivative: lnstantaneous rate of changeQ2.x+9 Q3.1;1i1=18Q4.Graph. y=2x

S.a. t1x; = 0.6x2, f'(3) = lim "?=*

=l*sqf,9$d =s-o

b. Graph of the difference quotient m(x)

c and d. Graph. Tangent line: y - 3.6x - 5.4

Q5. 9x2 - 42x + 49Q7. Graph (example)

Q6. "-" signQ8. Graph (example)

Q9. Graph Q1 0. Newton and Leibniz

1. See text definition ol derivative.2. Physical; lnstantaneous rate of change of the

dependent variable with respect to the independentvariableGeometrical: Slope of the tangent line to the graph olthe function at that point

t(x)

26 Colculus: Concepts ond Applicotions Problem Set 3'2