All work must be shown to be awarded full credit.
Provide exact solutions to all problems, unless otherwise stated.
allowed.
Student Name: __________________
Instructor: ____________________
Only a basic scientific calculator
Student ID:_______________________
Exam Score: ________
1. Find the limit analytically.
𝑥+2
lim ( 2
)
𝑥→ −2 3𝑥 + 5𝑥 − 2
x 2
 lim
x 2 3x  5x  2
x 2
lim
2
x 2
1
1
1
 lim


7


1  x 2 
1
1
3 x  2 x  
3  x   3  2  
3
3
3





2. Find the derivative
𝑓(𝑥) = 7𝑥 + sech(𝑥 2 − 5)
f (x )  7x  sech(x 2  5)

f '(x )  7  sech x
 
 5  tanh x

 5  2x

f '(x )  7  sech x 2  5 tanh x 2  5 x 2  5 '
2
2
3. Find the derivative
𝑓(𝑥) = 𝑥 3 ∙ arccsc(8𝑥)
 
f ' x   x
 
arccsc  8x   ' 
  x  ' arccsc  8x   x  arccsc  8x   ' 
 8x  ' 
 3x arccsc  8x   x
8x  8x   1
8
 3x arccsc  8x   x
8x  8x   1
f x  x 3 arccsc 8x
3
3
2
3
3
2
2
3
2
Math 1411 Final Exam – TIME Spring 2016
All work must be shown to be awarded full credit.
Provide exact solutions to all problems, unless otherwise stated.
allowed.
Only a basic scientific calculator
 
y  x 3 arccsc 8x
 

ln y  ln  x   ln  arccsc  8x  
ln y  3 ln x  ln  arccsc  8x  
ln y  ln x arccsc 8x
3
3

1
1
1
y'  3 
y
x arccsc 8x
 
arccsc  8x  '
1
1
1
y'  3 
y
x arccsc 8x
  8x

 1
1
y '  y 3 
 x arccsc 8x 8x

 
 8x  '
 8x 
2
1
8
 8x 
2



1

Math 1411 Final Exam – TIME Spring 2016
All work must be shown to be awarded full credit.
Provide exact solutions to all problems, unless otherwise stated.
allowed.
Only a basic scientific calculator
4. Find the derivative
𝑓(𝑥) = ln (√𝑥 3 + 2)
1
 3
 1
2
f x  ln  x  2   ln x 3  2

 2


1 1
1 1
f' x 
x3  2 ' 
3x 2
3
3
2x 2
2x 2

 


 

 
1
x
3
2

1
2


 
f ' x   ln  x 3  2
 
 




1 3
x 2
2
1
2

1
' 


x3  2

1
1
2
 x
3

2 ' 

1
2
 3
 x 2



1
x
3
2

1
2

1
2

' 



1 3
x 2
2

1
1
2
3x 2
5. Find the derivative
𝑔(𝑥) =
e 2x
e 2x  e 2x
 e 2x
g ' x   2x
2x
e  e
 
𝑒 2𝑥
𝑒 2𝑥 + 𝑒 −2𝑥
g x 
 


 
  2e
e  e 
2e 2x e 2x  e 2x  e 2x
2x
  




e 2x ' e 2x  e 2x  e 2x e 2x  e 2x '


' 
2
2x
2 x

e e
2x
2x
 2e 2x
2
Math 1411 Final Exam – TIME Spring 2016
All work must be shown to be awarded full credit.
Provide exact solutions to all problems, unless otherwise stated.
allowed.
e 2x
e 2x  e 2x
e 2x
ln y  ln 2x
e  e 2x
ln y  ln e 2x  ln e 2x  e 2x
Only a basic scientific calculator
y


ln y  2x ln e  ln e

ln y  2x  ln e

2x
e
2x
e
2x

2x




1
1
y '  2  2x
e 2x  e 2x '
2x
y
e e


1
y '  y  2  2x
2e 2x  2e 2x 
2x
e e




6. Use implicit differentiation to find 𝑑𝑦/𝑑𝑥
𝑥𝑦 2 − 𝑒 𝑦 = −5
xy 2  e y  5
xy  e  '   5  '
xy  ' e  '  0
 x  ' y  x y  ' e y '  0
y
2
y
2
2
2
y
y 2  x 2yy ' e yy '  0
2xy  e  y '  y
y
y' 
2
y 2
2xy  e y
7. Consider the function 𝑓(𝑥) = 2𝑥 3 − 7𝑥 2 − 40𝑥 + 5
a. Find any critical numbers if possible
b. Find the intervals in which 𝑓(𝑥) is increasing and decreasing
c. Use the First Derivative test to find the relative minimum and maximum.
Math 1411 Final Exam – TIME Spring 2016
All work must be shown to be awarded full credit.
Provide exact solutions to all problems, unless otherwise stated.
Only a basic scientific calculator
allowed.
8. A rectangular solid, with square base, has a surface area of 455 square centimeters. Find the
dimensions that will result in a solid with maximum area.
9. Find the indefinite integral
∫
𝑑𝑥
√9 − 𝑥 2
10. Find the indefinite integral
∫ 2𝑥 2 √𝑥 3 + 3 𝑑𝑥
11. Find the indefinite integral
∫
sinh 𝑥
𝑑𝑥
cosh(𝑥) + 5
12. Find the area of the region bounded by the functions 𝑓(𝑥) = −6𝑥 2 + 8𝑥 − 4, 𝑥 = 1, 𝑥 = 2 and
𝑦 = 0.
Math 1411 Final Exam – TIME Spring 2016