MA140-01
2/11/09
Major Quiz 2
_________________, come on down!
Show all your work and explain your answers completely. I cannot give partial credit for answers that are both wrong and
unexplained. Even correct "bottom line" answers that are mysterious and unsupported will not be considered completely correct. Show
me what you are thinking. Try to keep your answers neat and organized so that I can follow them easily.
1.) Explain why f(x) has a root in the interval [0,1] where
5
( ) 2 3 1
f x x x
= + โ
.
๎ ๎๎๎๎๎
๎๎๎๎๎๎
๎
Since f(x) is a polynomial function, it is continuous everywhere, and thus, continuous on the closed
interval [0,1].
5 5
( ) 2 3 1 (0) 2(0) 3(0) 1 1
f x x x f
= + โ ๎= + โ = โ
and
5 5
( ) 2 3 1 (1) 2(1) 3(1) 1 4
f x x x f
= + โ ๎= + โ =
Since f is a continuous function on the closed interval [0,1] and the signs of f(0) and f(1) alternate, the
function must cross the x axis in the interval. Thus, there is a zero in the interval [0,1].
2.) Using the definition of the derivative, find
'( )
f x
where
( ) 2
f x x
= +
.
๎ ๎๎๎๎๎
๎๎๎๎๎๎
0
( ) ( )
lim
h
f x h f x
h
โ
+ โ
0
1
lim
2 2
h
x h x
โ
+ + + +
3.)
Find and equation of the line tangent to the graph of y = 4x
2
โ 3x + 1 when x = -1.
๎๎๎๎๎
๎๎๎๎๎๎
' 8 3
y x
= โ
If x = -1, y = 8.
1 1
( )
y y m x x
โ = โ
8 ( 11)( ( 1))
y x
โ = โ โ โ
11 3
y x
= โ โ
0
2 2
lim
h
x h x
h
โ
+ + โ +
0
2 2 2 2
lim
2 2
h
x h x x h x
hx h x
โ
+ + โ + + + + +
โ
+ + + +
( )
0
lim
2 2
h
h
h x h x
โ
+ + + +
1
2 2
x
+
'( 1) 8( 1) 3 11 Slope of tangent at x = -1
is -11
y
โ = โ โ = โ
