uaf-math251.github.io...Math F : Midterm Review November , d. Classify all critical points – using...
Transcript of uaf-math251.github.io...Math F : Midterm Review November , d. Classify all critical points – using...
Math F���: Midterm � Review November ��, ����
�. Consider the function f (x) and its derivatives:
f (x) = ex� + x
f ′(x) = xex� + x�
f ′′(x) = ex(x� + �)(� + x)� .
a. Find the critical numbers of f (x).
b. Find the open intervals on which the function is increasing or decreasing.
c. Find the open intervals on which the function is concave up or concave down.
( I
f' Ix ) -
- O : t = o only
( function not defined at x= - I )
f' G) -#i I
decreases lhcneasay
f''It ) -#
Math F���: Midterm � Review November ��, ����
d. Classify all critical points – using the �rst derivative test.
e. Classify all critical points – using the second derivative test.
f . Find the in�ection points.
g. Sketch the graph.
�
f'4)
O
✓ ← load min
f" lo ) = I s o
local m in
N one
!: .Ei Es
.
I = oo
¥ . ⇐x
.
.
= o.
.
Math F���: Midterm � Review November ��, ����
�. Find the linearization of f (x) =√x at a = � and use it to estimate√
�.�.
�. Show that the point (�, �) lies on the curve x� + xy − y� = �. �en �nd the slope of thetangent line to the curve at that point.
�
f'
Ix ) -
- ¥x
L ( x )= fla ) tf
'
la ) ( x - a )
= OFt try f x - t )
= 2 t ¥ ( * t )
of , an L ( Kl ) = 2 t to
22 +2.3 -
32 =t t 6 - 9=1 ✓
2x t y t try'
- Zy y'
= O
y' ( x - 4) =
- 2x - y
y ' -
-
-
=
-
= ¥ -
- I
Math F���: Midterm � Review November ��, ����
�. A ball of metal is being heated in an oven, and its radius is increasing at a rate of �.�cm/min. At what rate is the ball’s volume increasing when its radius is � cm?
�. Evaluate the following limits.
limx→�
� + x − exsin x limx→�+(� + �x)��x
�
V = 4g tr3
DVId
= 41T r> In
DE
did = HIT ' 9 . Yo = 3,6J IT Comynis
€÷i÷.
. .ioGii: 'Iii . .
!:* W¥¥ lay,
= 2
that -
-
I;m⇒e"' '
= e
'
Math F���: Midterm � Review November ��, ����
�. A landscape architect wishes to enclose a rectangular garden on one side by a brick wallcosting ��� per foot and on the other three sides with a metal fence costing ��� per foot.�e area of the garden is to be �����. What are the dimensions of the garden that mini-mize the cost of the fencing?
metal
metal
metalbrick ��� ��
�
¥←
Cost : C = Boyt
20×+104=40-11-20
,
Area : A = xy= 800
y = 801CEO : 160¥ =/
x
⇐ +008¥ " " t.ro= 40 .
C" =
-32000 +20
# 4--80×1=8+7%-2510
C"
=
2.32000-70on
+3( 0,007
Math F���: Midterm � Review November ��, ����
�.
a. State the Mean Value �eorem and draw a picture to illustrate it.
b. Suppose f (x) is continuous on [−�, �] and has a derivative at each x in (−�, �). Iff (−�) = � and f (�) = �, what does the Mean Value �eorem let you conclude?
�
If flx ) is continuous on Caio ] and differentiable
or ( a. b ),
then thee is a in ( a,b) where
f'(c) = fCb)
b- a
There is a car Gtl ) where
f-'
EH EET-
- I = - Ilie