Math 220 March 7 I. If 1200 cm2 of material is available to make a ...
Math 220 March 7 I. If 1200 cm2 of material is available to make a ...
Math 220 March 7 I. If 1200 cm2 of material is available to make a ...
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<strong>Math</strong> <strong>220</strong> <strong>March</strong> 7<br />
I. <strong>If</strong> <strong>1200</strong> cm 2 <strong>of</strong> <strong>material</strong> <strong>is</strong> <strong>available</strong> <strong>to</strong> <strong>make</strong> a box with a square base<br />
and an open <strong>to</strong>p, find the largest possible volume <strong>of</strong> the box.<br />
II. Find the point on the curve y = √ x that <strong>is</strong> closest <strong>to</strong> point (4,0).<br />
III. At which points on the curve y = 1 + x + 60x 3 − x 5 does the tangent<br />
line have the largest slope?<br />
IV. A farmer whats <strong>to</strong> fence in area <strong>of</strong> 400 square feet in a rectangular<br />
field and then divide it 3 sections with a two fence parallel <strong>to</strong> one <strong>of</strong> the sides<br />
<strong>of</strong> the rectangle. How can he do th<strong>is</strong> so as <strong>to</strong> minimize the cost <strong>of</strong> the fence?<br />
V. A Fr<strong>is</strong>bee has landed in the ocean 150 from the shore. A dog <strong>is</strong> standing<br />
200 feet down the shore from where the Fr<strong>is</strong>bee landed. The dog can run<br />
5 feet per second and swim 2 feet per second. How far should the dog run<br />
down the shore before he get jumps in the water <strong>to</strong> get the Fr<strong>is</strong>bee in the<br />
least amount <strong>of</strong> time?<br />
VI. Find the limit.<br />
1.<br />
2.<br />
3.<br />
4.<br />
5.<br />
e<br />
lim<br />
x→∞<br />
x<br />
x3 sin(3x)<br />
lim<br />
x→0 cos(4x)<br />
lim<br />
x→0 +<br />
ln( √ x)<br />
x2 5<br />
lim<br />
x→0<br />
x − 7x x<br />
sin(x) − x<br />
lim<br />
x→0 x<br />
1
6.<br />
7.<br />
8.<br />
9.<br />
10.<br />
11.<br />
12.<br />
13.<br />
14.<br />
15.<br />
lim<br />
x→0<br />
sin(x<br />
lim<br />
x→0<br />
2 )<br />
x<br />
lim<br />
x→0 +<br />
cos(x)<br />
x<br />
√ 1 + 2x − √ 1 − 4x<br />
x<br />
2x − sin(x)<br />
lim<br />
x→0 2x − tan(x)<br />
lim<br />
x→0 +<br />
<br />
1 1<br />
−<br />
x ex <br />
− 1<br />
lim (sin(x) ln(x))<br />
x→0<br />
lim<br />
x→0 +<br />
2<br />
x − ln(x )<br />
lim (tan(3x))6x<br />
x→0 +<br />
lim (x)2/(1−x)<br />
x→1 +<br />
x 2<br />
lim e + x<br />
x→∞<br />
1/(1−x) 2
I. <strong>If</strong> <strong>1200</strong> cm 2 <strong>of</strong> <strong>material</strong> <strong>is</strong> <strong>available</strong> <strong>to</strong> <strong>make</strong> a box with a square base<br />
and an open <strong>to</strong>p, find the largest possible volume <strong>of</strong> the box.<br />
Answer:<br />
V olume = x 2 y<br />
SurfaceArea = <strong>1200</strong> = 4x 2 + xy =⇒ y =<br />
v = x 2 y<br />
v(x) = x 2 <strong>1200</strong> − 4x2<br />
( )<br />
x<br />
v(x) = x(<strong>1200</strong> − 4x 2 )<br />
v ′ (x) = (<strong>1200</strong> − 4x 2 ) + x(−8x)<br />
0 = <strong>1200</strong> − 12x 2<br />
0 = 100 − x 2<br />
x = ±10<br />
v ′′ (x) = −24x<br />
v ′′ (10) < 0<br />
<strong>1200</strong> − 4x2<br />
x<br />
There <strong>is</strong> a maximum when x = 10 and y = 12<br />
The largest possible volume <strong>of</strong> the box <strong>is</strong> <strong>1200</strong> cm 3 .<br />
II. Find the point on the curve y = √ x that <strong>is</strong> closest <strong>to</strong> point (4,0).<br />
Answer:<br />
D<strong>is</strong>tance = (x − 4) 2 + (y − 0) 2<br />
3
D = (x − 4) 2 + (y − 0) 2<br />
<br />
D(x) = (x − 4) 2 + ( √ x) 2<br />
D(x) = √ x 2 − 8x + 16 + x<br />
D(x) = √ x 2 − 7x + 16<br />
D ′ 2x − 7<br />
(x) =<br />
2 √ x2 − 7x + 16<br />
2x − 7<br />
0 =<br />
2 √ x2 − 7x + 16<br />
0 = 2x − 7<br />
x = 7<br />
2<br />
D ′ (x) < 0 when x < 7/2<br />
D ′ (x) > 0 when x > 7/2<br />
Thus there <strong>is</strong> a minimum at x = 7/2<br />
The point (7/2, 7/2) <strong>is</strong> closest.<br />
III. At which points on the curve y = 1 + x + 60x 3 − x 5 does the tangent<br />
line have the largest slope?<br />
Answer:<br />
slope = y ′ = 180x 2 − 5x 4<br />
4
m ′′ (x) = 360 − 60x 2<br />
m = 180x 2 − 5x 4<br />
m ′ (x) = 360x − 20x 3<br />
m ′ (x) = 20x(18 − x 2 )<br />
m ′′ (0) > 0<br />
m ′′ ( √ 18) < 0<br />
m ′′ ( √ 18) < 0<br />
0 = 20x(18 − x 2 )<br />
x = 0, ± √ 18<br />
There <strong>is</strong> a maximum at x = ± √ 18<br />
So the tangent line will have the largest slope at the points<br />
( √ 18, 1+ √ 18+60(18) 3/2 −(18) 5/2 ) and (− √ 18, 1+ √ 18+60(18) 3/2 −(18) 5/2 )<br />
IV. A farmer whats <strong>to</strong> fence in area <strong>of</strong> 400 square feet in a rectangular<br />
field and then divide it 3 sections with a two fence parallel <strong>to</strong> one <strong>of</strong> the sides<br />
<strong>of</strong> the rectangle. How can he do th<strong>is</strong> so as <strong>to</strong> minimize the cost <strong>of</strong> the fence?<br />
Answer:<br />
400 = xy<br />
Cost = 4x + 2y<br />
5
A ′′ (x) = 1600x −3 > 0<br />
C = 4x + 2y<br />
C(x) = 4x + 800<br />
x<br />
C ′ (x) = 4 − 800x −2<br />
0 = 4 − 800x −2<br />
x = 4x 2 − 800<br />
200 = x 2<br />
x = ± √ 200<br />
There <strong>is</strong> maximum at x = √ 200<br />
Cost <strong>is</strong> be minimize if the fence <strong>is</strong> built <strong>to</strong> be √ 200 by 2 √ 200 where √ 200 <strong>is</strong><br />
the length <strong>of</strong> the 4 parallel sections.<br />
V. A Fr<strong>is</strong>bee has landed in the ocean 150 from the shore. A dog <strong>is</strong> standing<br />
200 feet down the shore from where the Fr<strong>is</strong>bee landed. The dog can run<br />
5 feet per second and swim 2 feet per second. How far should the dog run<br />
down the shore before he get jumps in the water <strong>to</strong> get the Fr<strong>is</strong>bee in the<br />
least amount <strong>of</strong> time?<br />
Answer:<br />
T ime = 1 1<br />
(x) + 1502 + (200 − x) 2<br />
5 2<br />
6
T (x) = 1<br />
5<br />
T ′ (x) = 1<br />
5 +<br />
1<br />
(x) + 1502 + (200 − x) 2<br />
2<br />
−400 + 2x<br />
4 1502 + (200 − x) 2<br />
0 = 1<br />
5 +<br />
−400 + 2x<br />
4 1502 + (200 − x) 2<br />
1<br />
5 =<br />
400 − 2x<br />
4 1502 + (200 − x) 2<br />
( 1<br />
5 )2 400 − 2x<br />
= (<br />
4 1502 )2<br />
+ (200 − x) 2<br />
1<br />
25 =<br />
4(200 − x) 2<br />
16(1502 + (200 − x) 2 )<br />
1<br />
25 =<br />
(200 − x) 2<br />
4(1502 + (200 − x) 2 )<br />
4(150 2 + (200 − x) 2 ) = 25(200 − x) 2<br />
300 2 + 4(200 − x) 2 = 25(200 − x) 2<br />
21(200 − x) 2 = 300 2<br />
(200 − x) 2 = 30000<br />
<br />
7<br />
30000<br />
200 − x = ±<br />
7<br />
<br />
30000<br />
x = 200 ±<br />
7<br />
x = 134.535, 265.465<br />
<br />
30000<br />
T (x) < 0 when x < 200 −<br />
7<br />
<br />
30000<br />
T (x) > 0 when x > 200 −<br />
7<br />
There <strong>is</strong> a minimum at x = 134.535<br />
The dog should run 134.535 feet before jumping in the water.<br />
7
VI. Find the limit.<br />
1.<br />
2.<br />
3.<br />
Answer:<br />
Answer:<br />
e<br />
lim<br />
x→∞<br />
x<br />
x3 L.H.<br />
= lim<br />
x→∞<br />
L.H.<br />
= lim<br />
x→∞<br />
L.H.<br />
= lim<br />
x→∞<br />
= 1<br />
6<br />
e<br />
lim<br />
x→∞<br />
x<br />
x3 <br />
∞<br />
<br />
∞<br />
ex 3x2 <br />
∞<br />
<br />
∞<br />
ex <br />
∞<br />
<br />
6x ∞<br />
ex 6<br />
sin(3x)<br />
lim<br />
x→0 cos(4x)<br />
sin(3x)<br />
lim<br />
x→0 cos(4x)<br />
= 0<br />
1<br />
= 0<br />
lim<br />
x→0 +<br />
ln( √ x)<br />
x2 8
4.<br />
5.<br />
Answer:<br />
Answer:<br />
Answer:<br />
lim<br />
x→0 +<br />
ln( √ x)<br />
x2 = lim<br />
x→0 +<br />
(1/2) ln(x)<br />
= lim<br />
x→0 +<br />
= lim<br />
x→0 +<br />
= −∞<br />
5<br />
lim<br />
x→0<br />
x − 7x x 2<br />
ln(x)<br />
2x2 1 1<br />
(ln(x) − ) (∞ · −∞)<br />
x 2x<br />
5<br />
lim<br />
x→0<br />
x − 7x x<br />
x<br />
L.H.<br />
= lim<br />
x→0<br />
<br />
0<br />
0<br />
= ln(5) − ln(7)<br />
ln(5)5 x − ln(7)7 x<br />
1<br />
sin(x) − x<br />
lim<br />
x→0 x<br />
<br />
sin(x) − x 0<br />
lim<br />
x→0 x 0<br />
L.H. cos(x) − 1<br />
= lim<br />
x→0 1<br />
= 0<br />
9
6.<br />
7.<br />
8.<br />
Answer:<br />
Answer:<br />
Answer:<br />
lim<br />
x→0<br />
lim<br />
x→0<br />
sin(x<br />
lim<br />
x→0<br />
2 )<br />
x<br />
sin(x<br />
lim<br />
x→0<br />
2 )<br />
x<br />
L.H.<br />
= lim<br />
x→0<br />
= 0<br />
<br />
0<br />
0<br />
2x cos(x2 )<br />
1<br />
lim<br />
x→0 +<br />
cos(x)<br />
x<br />
lim<br />
x→0 +<br />
<br />
cos(x) 1<br />
x 0 +<br />
<br />
= ∞<br />
√ 1 + 2x − √ 1 − 4x<br />
x<br />
√ 1 + 2x − √ 1 − 4x<br />
L.H.<br />
= lim<br />
=<br />
2<br />
2<br />
x→0<br />
+ 4<br />
2<br />
1<br />
= 1 + 2<br />
= 3<br />
x<br />
2<br />
2 √ −4 − 1+2x 2 √ 1−4x<br />
1<br />
10<br />
<br />
0<br />
0
9.<br />
10.<br />
11.<br />
Answer:<br />
Answer:<br />
2x − sin(x)<br />
lim<br />
x→0 2x − tan(x)<br />
2x − sin(x)<br />
lim<br />
x→0 2x − tan(x)<br />
= lim<br />
x→0<br />
= 1<br />
1<br />
= 1<br />
<br />
0<br />
0<br />
2 − cos(x)<br />
2 − sec2 (x)<br />
lim<br />
x→0 +<br />
<br />
1 1<br />
−<br />
x ex <br />
− 1<br />
lim<br />
x→0 +<br />
<br />
1 1<br />
−<br />
x ex <br />
− 1<br />
= lim<br />
x→0 +<br />
x e − 1 − x<br />
x(ex <br />
0<br />
− 1) 0<br />
L.H.<br />
= lim<br />
x→0 +<br />
ex − 1<br />
ex − 1 + xex <br />
0<br />
0<br />
L.H.<br />
= lim<br />
x→0 +<br />
= 1<br />
2<br />
e x<br />
e x + e x + xe x<br />
lim (sin(x) ln(x))<br />
x→0 +<br />
11
12.<br />
13.<br />
Answer:<br />
Answer:<br />
Answer:<br />
lim (sin(x) ln(x))<br />
x→0 +<br />
= lim<br />
x→0 +<br />
ln(x)<br />
csc(x)<br />
L.H.<br />
= lim<br />
x→0 +<br />
1/x<br />
− csc(x) cot(x)<br />
= lim<br />
x→0 +<br />
− sin2 <br />
(x) 0<br />
x cos(x) 0<br />
L.H.<br />
= lim<br />
x→0 +<br />
−2 sin(x) cos(x)<br />
cos(x) − x sin(x)<br />
= 0<br />
1<br />
= 0<br />
lim<br />
x→0 +<br />
2<br />
x − ln(x )<br />
lim<br />
x→0 +<br />
2<br />
x − ln(x )<br />
= lim (x − 2 ln(x))<br />
x→0 +<br />
= 0 − (−∞)<br />
= ∞<br />
lim (tan(3x))6x<br />
x→0 +<br />
12
14.<br />
Answer:<br />
ln(y) = ln( lim<br />
x→0 + (tan(3x))6x )<br />
= lim 6x ln(tan(3x)) (0(−∞))<br />
x→0 +<br />
= lim<br />
x→0 +<br />
6 ln(tan(3x))<br />
x −1<br />
−∞<br />
∞<br />
<br />
L.H.<br />
= lim<br />
x→0 +<br />
18 sec2 (3x)<br />
tan(3x)<br />
−x−2 = lim<br />
x→0 +<br />
−18x2 sec2 (3x)<br />
tan(3x)<br />
= lim<br />
x→0 +<br />
−18x2 cos(3x)<br />
sin(3x) cos2 (3x)<br />
= lim<br />
x→0 +<br />
−18x2 <br />
0<br />
sin(3x) cos(3x) 0<br />
L.H.<br />
= lim<br />
x→0 +<br />
−36x<br />
cos2 (3x) − sin2 (3x)<br />
= 0<br />
1<br />
= 0<br />
y = e ln(y)<br />
= e 0<br />
= 1<br />
lim (x)2/(1−x)<br />
x→1 +<br />
13
15.<br />
Answer:<br />
ln(y) = ln( lim<br />
x→1 + (x)2/(1−x) )<br />
= lim<br />
x→1 +<br />
2<br />
1 − x ln(x)<br />
= lim<br />
x→1 +<br />
<br />
2 ln(x) 0<br />
1 − x 0<br />
L.H.<br />
= lim<br />
x→1 +<br />
2/x<br />
−1<br />
= −2<br />
y = e ln(y)<br />
= e −2<br />
x 2<br />
lim e + x<br />
x→∞<br />
1/(1−x) 14
x 2<br />
ln(y) = ln( lim e + x<br />
x→∞<br />
1/(1−x) )<br />
= lim<br />
x→∞<br />
= lim<br />
x→∞<br />
1<br />
1 − x ln e x + x 2<br />
ln (ex + x2 <br />
) ∞<br />
1 − x −∞<br />
e<br />
L.H.<br />
= lim<br />
x→∞<br />
x +2x<br />
ex +x2 −1<br />
−e<br />
= lim<br />
x→∞<br />
x − 2x<br />
ex + x2 <br />
−∞<br />
∞<br />
L.H. −e<br />
= lim<br />
x→∞<br />
x − 2<br />
ex <br />
−∞<br />
+ 2x ∞<br />
−ex ex <br />
−∞<br />
+ 2 ∞<br />
L.H.<br />
= lim<br />
x→∞<br />
L.H.<br />
= lim<br />
x→∞<br />
= lim<br />
x→∞ −1<br />
= −1<br />
y = e ln(y)<br />
= e −1<br />
15<br />
−e x<br />
e x