865 words - 3 pages

Math 1431 Exam 2 Review 1. Find the following limits (if they exist): a. = >− x
x x 5
4sinlim 0
e. =⎟⎟⎠⎞⎜⎜⎝⎛ ⎟⎠⎞⎜⎝⎛ −>− xxx 12lim0 b. =−+ >− x
x x
24lim 0
f. x
xx x 2
cossin2lim 0>−
= c. =⎟⎠⎞⎜⎝⎛ −+
>− x x
x
1 1
1
lim 0
g. )9tan(
5lim 0 x
x x >−
= d. =
− −+
−>− 9 6lim 2
2
3 x xx
x h.
x x
x 6 )sin(lim
2
0>− = 2. Determine if the following are continuous. If the function is not continuous, state the type of discontinuity.
a. ⎪ ⎩
⎪ ⎨

> = <+
= 1 18 11
)( 3
2
xx x xx
xf b.
⎪ ⎩
⎪ ⎨

> = <
= 2 28 22
)( 3
2
xx x xx
xf c.
⎪⎩
⎪ ⎨ ⎧
−>− −= −<−
= 25
27 25
)( 2 xx
x xx
xf d. 2 1( )
2 1
⎧ < − ⎪= ⎨ ⎪ + ≥ −⎩
x if x f x
x if x
3. Let 1 3( )
3
+ ≠⎧ ⎪= ⎨ ⎪ =⎩
x if x f x
k if x
For what value of k would f(x) be continuous at x=3?

4. Find A and B so that f (x) is continuous: ⎪ ⎩
⎪ ⎨

−>+ −= −<−
= 13
1 116
)(
2
xBx xA xx
xf 5. Given
⎩ ⎨ ⎧
≤− >
= 33 32
)( 2 xx xx
xf , find )3('f if it exists. 6. Find the derivative of the following: a. ( )4123)( −= xxf b. )2(sec3 xy = c. x
xxf 53)( += d. x
xf 1)( = e. 2 2)( x xxxf += f. 342 )1()2()( −+= xxxxf g.
2 )1)(1()(
+ +−=
x xxxf h. )5( 3 xxxy += i. x xxf
cos1 cos1)(
− += j. )164(sin)( 24 +−= xxxf k.
2
cot x
xy = l. θθθ tansec)( −=f 7. Find dx dy using implicit differentiation.
a. 73422 =+−+ yxyx b. 02cossin =−− yx c. 133 =+− yxyx d. 16=− yxxy e. 10=xy f. 12sin =yx g. 53/23/2 =+ yx

h. xyyx 4)cos( =+ 8. Use the definition of derivative to find the derivative of the following: a. 23)( 2 +−= xxxf b. 2( ) 5
f x x
= +
c. ( ) 1f x x= + 9. Find ⎟ ⎠ ⎞⎜
⎝ ⎛ −+− 102 4 3 34
3
3
xxx dx d
10. Find dx dy at 2−=x for 3)1)(14( xxy −+= 11. Find the second derivative at the point (-­‐2,1) for 322 =− yx 12. Use interval notation to give the solution set to: a. 0)43)(12(...

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