Math 2414 Homework assignments
page 1 of 6
Each section’s problems are due the class period after the section is covered in class and will not be accepted late.
5.1 a. Find y ′ for y =
x2 + 1
x(2 x3 − 1) 2
(Use logarithmic differentiation.)
b. Find an equation for the tangent line to the graph of =
f ( x) ln( x 2 − 3) at the point where x = 2.
5.2 a. Integrate:
∫
x+5
dx
x −1
∫
b. Evaluate the integral:
5.3 a. Show that f=
( x)
3
9 x2 − 9 x + 9
dx
x2 + 1
∫
c. Evaluate the integral:
cos 2 x − sin x
cos3 x
dx
3 x3 − 1 is one-to-one, and calculate f −1 ( x) .
b. Prove that f ( x) = x3 + 2 x − 1 has an inverse function (use Thm 5.7)
Then, use Thm 5.9 to find the slope of the tangent line to f −1 ( x) at the point (2, 1).
1
5.4 a. Find y ′ for y = ecot x
dy
b. Find
: x2e y + 2 =
xy 2
dx
2
5.5 a. Find the derivative: y =
x3
3x
c. Integrate:
∫
e x +1
dx
( x + 1) 2
b. Find the derivative:
=
y x 2 log 3 x3 + 1
3
e x − 2 x3 − 1
5.6 Evaluate the limits: a. lim
x →0
5 x3
tan x
b. lim (1 + cos 3 x )
x→
π
2
−
dy
for y = x arctan x
dx
dy
1
 x
b. Find
for=
y
ln( x 2 − 9) + arcsin   Simplify completely.
2
dx
3
5.7 a. Find
5.8 a. Evaluate the definite integral:
∫
3
x+3
2
x +9
0
dx
5.9 a. Differentiate: f ( x) = ln(sech x5 )
c. Evaluate the definite integral:
∫
b. Integrate:
b. Integrate:
1
0
1
dx
9 − 4 x2
∫
∫
dx
8 + 2 x − x2
x cosh 3 x 2 sinh x 2 dx
Math 2414 Homework assignments
8.1 Integrate:
8.2 Integrate:
8.3 Integrate
8.4 Integrate:
8.5 Integrate:
a.
a.
a.
a.
a.
∫
∫
2
2
page 2 of 6
3
x sec ( x ) dx
b.
x2 / 3 + 3
dx
x1/ 3
1
x
e cos   dx
3
∫
∫
8
2x
2
b.
∫
2
b.
25 − 9x 2
dx
x
b.
3x 2 + 7 x + 6
dx
x3 + 2 x 2 − 3x
1/ 2
x 2 ln x dx
b.
∫
∫
∫
6
sec x dx
3/ 2
3/2
8.7 Integrate using the table:
8.8 Evaluate the definite integrals:
4
0
a.
π /4
2 cos3 x dx
0
16 x 2 − 9
dx
x3
x 2 − 16 x − 70
dx
( x + 5) 2 ( x − 2)
∫
2
1
Calculate the maximum error in this approximation.
∫
∫
c.
8.6 a. Use the trapezoidal rule with n = 4 to approximate the value of:
b. Approximate the value of
∫
x
dx
2x + 3
1
sin (4 x) cos (4 x) dx
∫
∫
e
c.
3
c.
∫
x 2 + 3x + 5
dx
x3 + x
2
dx
( x + 1) 2
1+ x 3 dx with Simpson’s rule, n = 4.
∫
a.
1
x (1 − tan x )
∫
∞
1
ln x
dx
x4
b.
dx
b.
∫
4
0
∫(
ex
e
4
dx
x2 − 4
2x
2
− 9)
dx
Math 2414 Homework assignments
page 3 of 6
 (n − 1)!
an = (−1) n+1  2

 n +1 
9.1 a. Write the first five terms of the sequence (begin with n = 1):
b. Simplify the expression for a n and decide whether the sequence converges:
an =
n n 2n+1 (3n)!
n n−1 (3n + 1)! 2n
7 − . . . in sigma form and decide if it converges or diverges.
9.2 a. Write the geometric series: −7 + 73 − 79 + 27
Decide if the following series converge or diverge; show the use of one of the tests from this section.
∞
∑
b.
n =1
2n − 1
5n + 3
∞
5
∑ 2n
c.
d.
n =0
∞
2n
∑ n3
n =1
9.3 Decide if the following series converge or diverge; show the use of a test from this section.
∞
∑
a.
n =1
5
n
b.
∞
∑
arctan n
c.
2
n +1
n=2
∞
∑ ne−n
n =1
9.4 Determine the convergence or divergence of the series using a Comparison Test:
∞
∑
a.
n=2
8n 2 + 6
b.
n3 − n
∞
2n − 6
∑ 5n + 3
n =1
9.5 Determine if each the following series converges absolutely, converges conditionally, or diverges .
a.
∞
∑
n =1
(−1) n (n − 1)
n n
b.
∞
∑
n=2
sin
( (2n2−1)π )
n3 − 5
9.6 Determine if the following series converge or diverge (show work):
a.
∞
3n−1
∑ n 2n
n =1
b.
∞
∑
n =1
 n 


 2n + 1 
n
c.
∞
∑
n =1
(−1) n (3n + 4)
4n − 9
page 4 of 6
9.7 a. Find the sixth degree MacLaurin polynomial for the function =
f ( x) sin(3 x) − 4 cos x.
b. Find the fourth degree Taylor polynomial centered at x = 1 for the function f ( x) = x 7 / 5
Use your polynomial to approximate the value of f (1.2) . Give answer to six decimal places.
Use Taylor’s Theorem to obtain an upper bound for the error when P4 is used to approximate f (1.2) .
9.8 Determine the interval of convergence for each power series in a, b, and c:
∞
∑
a.
2n ! ( x + 1)
n
n =0
d.
∞
b.
∑
n =0
∞
2 xn
(n + 1)!
∞
Find f ′( x) and ∫ f ( x)dx for f ( x) = ∑
n =1
c.
∑
n =1
2n ( x − 4) n
n
( −1)n ( x − 2) n
n
Write the sigma form of each, and write out the first 5 non-constant terms of each.
It not necessary to find the interval of convergence
9.9/9.10 Find power series representations for each of the following functions. Use the series in the chart
in the text as “given”. In giving your answer, write summation notation, then write out
at least the first five terms of the requested series.
a. f ( x) = xe − x
b. f ( x) =
2
d 
2
 x arctan( x ) 
dx
c. f ( x) = 2sin x3
Math 2414 Homework assignments
page 5 of 6
7.1 a. Find the area of the region bounded by: y = x , x = 1 , and x + y =
6.
b. Find the area of the region bounded by: x= 2 − 3 y and y 3 + x =
0.
7.2 a. Use disks or washers to find the volume of the solid formed by revolving the region bounded by:
y = 1, y = 3, x = 0, and y = 1x around the y-axis.
b. Use disks or washers to find the volume of the solid formed by revolving the region bounded by:
y = x 2 and y= 4 − x 2 around the x-axis.
c. Use disks or washers to find the volume of the solid formed by revolving the region bounded by:
=
x + y 3 and y=
+ x 2 3 around the line y = 2.
7.3 a. Use cylindrical shells to find the volume of the solid formed by revolving the region bounded by:
y = x3 + 1, x + 2 y = 2, and x = 1, around the y-axis.
b. Use cylindrical shells to find the volume of the solid formed by revolving the region bounded by:
x = y 2 and y= x − 2 , around the line y = 3.
c. Use cylindrical shells to find the volume of the solid formed by revolving the QI portion of the
region bounded by: x =
6 − y2 , y =
0, and x =
0 , around the line x = -2.
7.4 a. Find the arc length of the graph of: ( y + 1) 2 =( x − 4)3 , from (5, 0) to (8, 7).
b. Find the area of the surface generated by revolving: 8 y =
2 x 4 + x −2 , x ∈ [1, 2] around the y-axis.
7.6 a. Find the center of mass of the region bounded by =
y x 2 − 2 x and y = x .
b. Find the center of mass of the region bounded =
by y e − x /2 =
and y 0, with 0 ≤ x ≤ 4.
Math 2414 Homework assignments
page 6 of 6
10.1 Write each conic in standard form, then graph.
a. 3 x 2 − 18 x − 12 y − 21 =
0
b. 16 x 2 + 9 y 2 − 160 x − 36 y + 292 =
0
10.2 Find the corresponding rectangular equation by eliminating the parameter.
State any restrictions on the variables. Graph the curve. Indicate orientation.
 x = t − 1 
a. 

y
t + 2 
=
 x= 2 + 3cos θ 
b. 

 y = 1 − 4sin θ 
c. Write two different sets of parametric equations for the line passing through (4, 3) and (-2, 7).
Use the form a =
at + xo , y =
bt + yo , where a and b are integers.
10.3 a. Find
dy
dx
and
d2y
dx 2
=
x 2t 3 + 2 
if 
.
2
y
=
−
t
+
t


b. Find the equation of the tangent line to the curve: x =
2t − 10 and y =
6t 2 / 3 − 12
at the point where t = 8.
c. Find all points (x, y) where the curve given by: x =2t 3 − 9t 2 − 24t + 182 and y =t 3 − 48t − 120
has horizontal or vertical tangent lines.
 x = 8 t 3/ 2 
3
d. Calculate the length of the arc of the curve: 
 between t = 1 and t = 3.
2
 y= 2t − t 
10.4
(
a. Convert the following from polar to rectangular, then plot: −3,
4π
3
)
b. Convert from polar to rectangular: r 2 sin(2θ ) = 2 Identify and graph the curve.
c. Convert the following from rectangular to polar and plot the point:
(2
3, − 2 )
d. Convert from rectangular to polar: ( x 2 + y 2 + 4 y ) 2 = 4 x 2 + 4 y 2
Isolate r in your answer and identify the curve.
10.5 a. Graph r= 5sin θ and r= 2 + sin θ on the same graph and find the points of intersection.
b. Sketch the curve r = 5cos(2θ ) and find the area inside the entire curve.
c. Calculate the surface area generated if the part of the curve =
r eθ , 0 ≤ θ ≤
is revolved about the y-axis.
10.6 Identify and sketch the conics:
a.
r=
8
2 − cos θ
b. r =
π
2
4
1 + sin θ