1
151 WebCalc Fall 2002-copyright Joe Kahlig
In Class Questions
MATH 151-Fall 02
October 10
1. Find dy/dx
(a)
y4
= x2 + 1
x−y
There are two methods for this problem. do it as it is written or try and simplify first.
Method 1:
Method 2:
(x − y) ∗ 4y 3 y 0 − y 4 (1 − y 0 )
= 2x
(x − y)2
y 4 = (x2 + 1)(x − y)
(x − y) ∗ 4y 3 y 0 − y 4 (1 − y 0 ) = 2x(x − y)2
y 4 = x3 + x − x 2 y − y
4xy 3 y 0 − 4y 4 y 0 − y 4 + y 4 y 0 = 2x(x − y)2
4y 3 y 0 = 3x2 + 1 − [2xy + x2 y 0 ] − y 0
4xy 3 y 0 − 4y 4 y 0 + y 4 y 0 = 2x(x − y)2 + y 4
4y 3 y 0 = 3x2 + 1 − 2xy − x2 y 0 − y 0
(4xy 3 − 4y 4 + y 4 )y 0 = 2x(x − y)2 + y 4
(4y 3 + x2 + 1)y 0 = 3x2 + 1
2x(x − y)2 + y 4
y =
4xy 3 − 4y 4 + y 4
y0 =
0
3x2 + 1
4y 3 + x2 + 1
(b) cos(x − y) = y sec(x2 )
− sin(x − y) ∗ (1 − y 0 ) = y 0 sec(x2 ) + 2xy sec(x2 ) tan(x2 )
− sin(x − y) + sin(x − y) ∗ y 0 = y 0 sec(x2 ) + 2xy sec(x2 ) tan(x2 )
(sin(x − y) − sec(x2 ))y 0 = sin(x − y) + 2xy sec(x2 ) tan(x2 )
sin(x − y) + 2xy sec(x2 ) tan(x2 )
sin(x − y) − sec(x2 )
y0 =
x2
x+4
Take the first derivative and simplify it. Then take a second derivative.
2. find y 00 for y =
Answer: y 00 =
32
(x+4)3
3. Find the formula for f (n) (x) for f (x) =
f (n) (x) =
1
(1 − x)2
(n + 1)!
(1 − x)n+2
4. Compute D 78 cos(3x) = −378 cos(3x)
5. Find a tangent vector of unit length at t = 2 for r(t) =< t, 25t − 5t2 >.
→0
−
→
−
r (t) =< 1, 25 − 10t > and r 0 (2) =< 1, 5 >.
Answer:
√1
26
D
< 1, 5 >=
√1 , √5
26
26
E
→
6. If −
r (t) =< t, 25t − 5t2 > is the position vector of an object, what is the speed of the object
at t = 2?
√
→
−
| r 0 (2)| = | < 1, 5 > | = 26