MATH 1500 Fall 2014 Quiz 3D Solutions
Solve each of the following questions. Show all work.
[8] 1. Suppose y is a function of x defined inplicitly by
tan y + x2 y 3 = exy .
Determine dy/dx.
Solution: Taking derivatives of both sides gives
d
dy
2 dy
3
2
sec y
= exy (xy)
+ 2xy + x 3y
dx
dx
dx
dy
3
2
xy
2 dy
2 dy
+ 2xy + x 3y
=e
⇒ sec y
y+x
dx
dx
dx
dy
dy
dy
⇒ sec2 y
+ 2xy 3 + x2 3y 2
= yexy + xexy
dx
dx
dx
2
Moving all the dy/dx terms to one side, factoring and then dividing yields
sec2 y
dy
⇒
dx
dy
dy
dy
+ 3x2 y 2
− x exy = yexy − 2xy 3
dx
dx
dx
sec2 y + 3x2 y 2 − xexy
⇒
= yexy − 2xy 3
yexy − 2xy 3
dy
=
dx
sec2 y + 3x2 y 2 − xexy
[7] 2. Determine the derivative of f (x) =
sin(x3 − 2) + e5x
√
. Do not simplify.
3
x − csc x
Solution:
√
d √
− 2) + e5x )( 3 x − csc x) − dx
( 3 x − csc x)(sin(x3 − 2) + e5x )
√
f (x) =
( 3 x − csc x)2
√
1 −2/3
2
3
5x
3
+ csc x cot x (sin(x3 − 2) + e5x )
3x cos(x − 2) + 5e ( x − csc x) − 3 x
√
=
( 3 x − csc x)2
0
d
(sin(x3
dx
[5] 3. Evaluate
tan 2x
.
x→0 sin 9x
lim
Solution:
tan 2x
sin 2x
= lim
x→0 sin 9x
x→0 cos 2x sin 9x
sin 2x
2x
2x
= lim
sin 9x
x→0
9x cos 2x
9x
sin 2x
2
2x
= lim
sin 9x
x→0
9 cos 2x
9x
2(1)
=
9(cos 0)(1)
2
= .
9
lim