Calculus Equations Assignment please solve the questions attached. i want a detailed answer. In Class Review Sheet for Final Exam 1. Use the graph to deter

Calculus Equations Assignment please solve the questions attached. i want a detailed answer. In Class Review Sheet for Final Exam
1.
Use the graph to determine lim f ( x) . If the limit does not exist, you may write DNE.
x 0
y  f ( x)
2.
Use the graph below to determine each of the following. If a limit does not exist, you may write
DNE.
a. lim h( x)
x 1
b. h (1)
3.
Use the graph below to determine each of the following. If the limit does not exist, you may write
DNE.
a. lim f ( x)
x 1
b. lim f ( x)
x 1
c. lim f ( x)
x 1
1
4. Compute each of the following limits. If a limit does not exist, write DNE
a.
b.
x 3

x9
lim
x 9
x2  2 x  8

x 4
x 2  16
lim
5. Complete the following definition: A function f is continuous at the point where x  a if
______________________________________________________________________
6. Use f (x ) to answer the following questions. If a limit does not exist, you may write DNE.
2 x 2 if x  2
f ( x)   3
 x if x  2
a.
What is lim f ( x) ?
b.
What is lim f ( x) ?
c.
What is f ( 2) ?
d.
Is f (x ) continuous at the point x = 2? Explain your answer. You must use the
definition of continuous in your answer.
x2
x2
2
7. The following graph depicts the distance an object has travelled (in meters) at time t (in
seconds) . Use the graph to answer the questions below.
a. How far has the object travelled after 40 seconds?
b. Draw a line on the graph whose slope represents the average speed of the object between
10 and 35 seconds. Please label this line with an “A”.
c. Draw a line on the graph whose slope represents the instantaneous speed of the object at
20 seconds. Please label this line with a “B”.
d. What are the units associated with the slopes of the lines in parts a and b?
distance traveled
800
700
600
500
meters
400
300
200
100
0
0
10
20
30
seconds
3
40
50
60
8. Use the power rule to determine the derivative for each of the following functions. Your final
answers should contain no negative or fractional exponents.
a.
f ( x)  2 x5  3×3  17
b.
f ( x) 
c.
f ( x)  4 x 4  2 x 3 
2
 5 3 x 2  3x
3
x
1
x2
9. Below is a graph of f ( x ) , the derivative of some function f .
y
18
NOTE: This is
the graph of
f ( x ) , NOT
16
14
y  f ( x)
12
10
f (x )
8
6
4
2
-7
-6
-5
-4
-3
-2
-1
-2
x
1
2
3
4
5
6
7
-4
a. Mark the x values that are critical values of the original function f (x ) and
indicate whether each one corresponds to a local maximum, a local minimum, or
neither.
b. Find all intervals on the x-axis for which the original function f (x ) is increasing.
c. Find all intervals on the x-axis for which the original function f (x ) is decreasing.
4
d. Find all x values of the inflection points for the original function f (x ) .
e. Find all intervals on the x-axis for which the original function f (x ) is concave up.
f. Find all intervals on the x-axis for which the original function f (x ) is concave
down.
10. Consider the equation 2 x3  x 2 y  y 3  1  0 ,
dy
a. Calculate
assuming that the equation determines a function f such that
dx
y  f ( x) .
b. Find the slope of the tangent line to the graph of the equation at the point (2, 3) .
5
11. Gas is escaping from a spherical balloon at a rate of 10 cubic feet per hour. At what
rate is the radius changing when the volume is 400 cubic feet? (volume of a sphere is
4
given by V   r 3 ). Please round your answer to 3 decimal places and include units.
3
12. Calculate the derivative of each of the following functions
a.
y  e4 x tan x
b.
y  x3 ln(1  x)
c.
y  ( x  5)5 sin 2 x
d.
y
x cos x
( x 2  3)3
6
e.
f ( x)  arcsin(x 2  3x  1)
f.
f ( x)  x arctan( x)
13. If a farmer has 400 feet of fencing to use to enclose a pen for his pigs, what are the
dimensions (length and width) of the pen that maximize area, if one side of the pen
will be built against an existing barn and will not require fencing for that side?
14. Find the dimensions of the cylinder with minimal surface area that can hold 16 cm3.
Recall that the surface area of a cylinder is given by SA  2 r 2  2 rh and the volume
of a cylinder is given by V   r 2 h
7
15. Find the left hand Riemann sum for the function f ( x)  2  x 2 on the interval [0, 4]
for n  4 .
16. Use the following graph of the function y  f ( x) to answer the questions below.
a. Calculate

d
b. Calculate

e
c. Calculate

c
d. Calculate

b
e. Calculate

b
a
a
d
a
a
f ( x)dx
f ( x)dx
f ( x)dx
5 f ( x)dx
d
f ( x)dx   f ( x)dx
c
8
17. Suppose the function v(t ) models the rate at which helium is leaking out of a balloon
at time t in cubic centimeters per second. What does the definite integral

5
0
v(t )dt
represent with respect to the balloon? Please include units in your answer.
18. Suppose the function b (t ) models the rate at which Baby Bear’s porridge is cooling in
degrees Celsius per minute. Write a definite integral which represents the change in
temperature of Baby Bear’s porridge between time t  1 minute and time t  6
minutes.
Calculate the following definite integral using the Fundamental Theorem of Calculus. You must
show ALL your work (including an antiderivative) in order to receive any credit.
19.
1
 2x
5
0
 7 x  2 dx
20. Calculate the following indefinite integrals.
1
a.
 4sin( x)  x 
b.
1 x
2
2
x dx
dx
9
21. Use u-substitution to calculate each of the following integrals.
a.  x cos(3x 2 ) dx =
ln x
dx =
x
b.

c.
x
2
1  x 3 dx =
10
In Class Review Sheet for Final Exam
1.
Use the graph to determine lim f ( x) . If the limit does not exist, you may write DNE.
x 0
y  f ( x)
2.
Use the graph below to determine each of the following. If a limit does not exist, you may write
DNE.
a. lim h( x)
x 1
b. h (1)
3.
Use the graph below to determine each of the following. If the limit does not exist, you may write
DNE.
a. lim f ( x)
x 1
b. lim f ( x)
x 1
c. lim f ( x)
x 1
1
4. Compute each of the following limits. If a limit does not exist, write DNE
a.
b.
x 3

x9
lim
x 9
x2  2 x  8

x 4
x 2  16
lim
5. Complete the following definition: A function f is continuous at the point where x  a if
______________________________________________________________________
6. Use f (x ) to answer the following questions. If a limit does not exist, you may write DNE.
2 x 2 if x  2
f ( x)   3
 x if x  2
a.
What is lim f ( x) ?
b.
What is lim f ( x) ?
c.
What is f ( 2) ?
d.
Is f (x ) continuous at the point x = 2? Explain your answer. You must use the
definition of continuous in your answer.
x2
x2
2
7. The following graph depicts the distance an object has travelled (in meters) at time t (in
seconds) . Use the graph to answer the questions below.
a. How far has the object travelled after 40 seconds?
b. Draw a line on the graph whose slope represents the average speed of the object between
10 and 35 seconds. Please label this line with an “A”.
c. Draw a line on the graph whose slope represents the instantaneous speed of the object at
20 seconds. Please label this line with a “B”.
d. What are the units associated with the slopes of the lines in parts a and b?
distance traveled
800
700
600
500
meters
400
300
200
100
0
0
10
20
30
seconds
3
40
50
60
8. Use the power rule to determine the derivative for each of the following functions. Your final
answers should contain no negative or fractional exponents.
a.
f ( x)  2 x5  3×3  17
b.
f ( x) 
c.
f ( x)  4 x 4  2 x 3 
2
 5 3 x 2  3x
3
x
1
x2
9. Below is a graph of f ( x ) , the derivative of some function f .
y
18
NOTE: This is
the graph of
f ( x ) , NOT
16
14
y  f ( x)
12
10
f (x )
8
6
4
2
-7
-6
-5
-4
-3
-2
-1
-2
x
1
2
3
4
5
6
7
-4
a. Mark the x values that are critical values of the original function f (x ) and
indicate whether each one corresponds to a local maximum, a local minimum, or
neither.
b. Find all intervals on the x-axis for which the original function f (x ) is increasing.
c. Find all intervals on the x-axis for which the original function f (x ) is decreasing.
4
d. Find all x values of the inflection points for the original function f (x ) .
e. Find all intervals on the x-axis for which the original function f (x ) is concave up.
f. Find all intervals on the x-axis for which the original function f (x ) is concave
down.
10. Consider the equation 2 x3  x 2 y  y 3  1  0 ,
dy
a. Calculate
assuming that the equation determines a function f such that
dx
y  f ( x) .
b. Find the slope of the tangent line to the graph of the equation at the point (2, 3) .
5
11. Gas is escaping from a spherical balloon at a rate of 10 cubic feet per hour. At what
rate is the radius changing when the volume is 400 cubic feet? (volume of a sphere is
4
given by V   r 3 ). Please round your answer to 3 decimal places and include units.
3
12. Calculate the derivative of each of the following functions
a.
y  e4 x tan x
b.
y  x3 ln(1  x)
c.
y  ( x  5)5 sin 2 x
d.
y
x cos x
( x 2  3)3
6
e.
f ( x)  arcsin(x 2  3x  1)
f.
f ( x)  x arctan( x)
13. If a farmer has 400 feet of fencing to use to enclose a pen for his pigs, what are the
dimensions (length and width) of the pen that maximize area, if one side of the pen
will be built against an existing barn and will not require fencing for that side?
14. Find the dimensions of the cylinder with minimal surface area that can hold 16 cm3.
Recall that the surface area of a cylinder is given by SA  2 r 2  2 rh and the volume
of a cylinder is given by V   r 2 h
7
15. Find the left hand Riemann sum for the function f ( x)  2  x 2 on the interval [0, 4]
for n  4 .
16. Use the following graph of the function y  f ( x) to answer the questions below.
a. Calculate

d
b. Calculate

e
c. Calculate

c
d. Calculate

b
e. Calculate

b
a
a
d
a
a
f ( x)dx
f ( x)dx
f ( x)dx
5 f ( x)dx
d
f ( x)dx   f ( x)dx
c
8
17. Suppose the function v(t ) models the rate at which helium is leaking out of a balloon
at time t in cubic centimeters per second. What does the definite integral

5
0
v(t )dt
represent with respect to the balloon? Please include units in your answer.
18. Suppose the function b (t ) models the rate at which Baby Bear’s porridge is cooling in
degrees Celsius per minute. Write a definite integral which represents the change in
temperature of Baby Bear’s porridge between time t  1 minute and time t  6
minutes.
Calculate the following definite integral using the Fundamental Theorem of Calculus. You must
show ALL your work (including an antiderivative) in order to receive any credit.
19.
1
 2x
5
0
 7 x  2 dx
20. Calculate the following indefinite integrals.
1
a.
 4sin( x)  x 
b.
1 x
2
2
x dx
dx
9
21. Use u-substitution to calculate each of the following integrals.
a.  x cos(3x 2 ) dx =
ln x
dx =
x
b.

c.
x
2
1  x 3 dx =
10

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