4.a.f(x) =4.2x2, f'(6) = 1m t#{t

= ri, :8?G:i)$19 = -2.4x+6 x-bb. Graph of the difference quotient

x3-x2

15. Local Linearity Problem

a. First find f'(1), then plot a straight line through point(1, f(1)) using f'(1) as the slope.

b. Near the point (1, f(1)), which is (1 , 1), the tangentline and the curve appear coincidental.

c. The curve appears to get closer and closer to theline and finally touch it at the point (1, f(1)).

d. Near point (1 ,1 ) the curve looks linear.e. lf a graph has local linearity, the graph near that

point looks like the tangent line. Therefore, thederivative at that point cbuld be said to equal theslope of the graph at that point.

16. Local Nonlinearitv Problem

_ i-Za. f(x)= x2 + 0'1(!x - 1)-

f(1) = 12 + 0.1(1 - 1)2t3 - 1 + 0 = 1, o.E.D.b. Zoom by a factor of 10,000. Graph

c. and d. Graph. Tangent line: y = -2.4x + 7.2

:6

-7.2t.....-...

5.r(-2)= *gr-laf}|-= tim LL12)4i3)

= 1x+-2 X+Z

6.r(-4)= "go

tt;f4= ri, lLil)Gj2)

=_2x+-4 X+4

7.f'(1)=$,#= tim fU)Idi-Ql:2)=4

x+1 X-l

8. f(-1) = ,(E11

=*9'g*P=re.f(3)=, -ozlits

-0.7(x-3) _ __r\7x-5.. 1.3x-3-2.2rrm )(_+

= rim 1j0i0 = r.sxs4 X-4

11.f(-1)= "P,f# =o

-DrD12. f'(3) = lqt ta.= =O

13. The derivative of a linear function equals the slope.The tangent line coincides with the graph.

14. The derivative of a constant function is zero. Constantfunctions don't change! The tangent line coincideswith the graph.

:-4x+6-8x+1

c. The graph has a cusp at x = 1. lt changes directionabruptly, not smoothly.

d. lf you draw a secant line through (1 ,1) from a pointjust to the left of x = 1, it has a large negative slope.lf you draw one from a point just to the right, it has alarge positive slope. ln both cases, the secant linebecomes vertical as x approaches 1, and a verticalline has infinite slope. So there is no real numberequal to the derivative.

= limx+3

10. f'(4) =

Problem Sel3-2 Solufions Mqnuol 27