Midterm Exam, Honors Calculus Math 221, Fall 2012
1. A sequence cfw_an is dened by a1 = 1 and
an+1 = 3 + 2an ,
n 1.
Prove by mathematical induction that
an = 2n+1 3
for n 1.
Solution: We want to prove the statement Pn : an = 2n+1 3.
P1 is true: 1 = 4 3.
Honors Calculus Math 221
Homework 10, due December 5, 2012
1
1. Find L3 (f ) and U3 (f ) for the function f (x) = x+1 and the interval
[a, b] = [0, 3].
2. Evaluate
d
dx
0
1 + t4 dt.
x
x sin x dx.
3. Evaluate
0
4. Evaluate the improper integral
0
2
xex dx.
Honors Calculus Math 221
Homework 9
1. A projectile is red with a speed of 800 ft/s at an angle of /6 above the
horizontal. Find its maximum height during the ight.
Solution: We know that x(t) = v0 cos t and y (t) = 1 gt2 + v0 sin t. The
2
maximal height
Honors Calculus Math 221
Homework 8
1. Let f (x) =
+
12x2 . Find the critical points, the intervals on
which f is increasing or decreasing, the intervals on which f is concave up
or concave down and any inection points. Plot the function.
Solution: The c
Honors Calculus Math 221
Homework 7
2
1. Solve the dierential equation y + x y = 4x with initial condition y (1) = 3.
Solution: The integrating factor is exp(2 ln(x) = x2 . Multiplying by the
integrating factor, we get
x2 y + 2xy = 4x3
which is the same a
Honors Calculus Math 221
Homework 6
1. Let f 1 : R R be the inverse function of f (x) = x5 + 2x + 2. Find
(f 1 ) (5).
Solution: We note that f (x) = 5x4 + 2 and f (1) = 5. Therefore, by
Theorem 4.5.1,
1
1
(f 1 ) (5) =
=.
f (1)
7
2. A function is dened by
Honors Calculus Math 221
Homework 5
1. Show that the function f (x) = 1 + x2 with domain D = [0, ) is
one-to-one. Find the inverse function f 1 (y ) and the domain of the inverse
function.
Solution: If 0 x1 < x2 then f (x1 ) = 1 + x2 1 + x2 f (x2 ), so th
Honors Calculus Math 221
Homework 4, due October 10, 2012
1. Use the denition of limit to show that lim x = 2.
x4
x4 5x3 + 9 x2 7x + 2
.
x1
x2 2x + 1
2. Using the limit rules nd the limit lim
4. Find all points c at which the function
x2 + 2x if
2
if
f (x
Honors Calculus Math 221
Homework 3
(1)k1
1. Determine whether the series
k=1
3k 2
1
converges abso+ 2k + 1
lutely, converges conditionally or diverges.
Solution: To test for absolute convergence we consider the series
(1)
k=1
3k 2
1
.
+ 2k + 1
Since
1
1
Honors Calculus Math 221
Homework 2
n+1
is non-decreasing, non3n + 2
increasing or not monotone. Is the sequence bounded? Does the sequence
converge?
Solution: We look at the dierence of an+1 and an
n+2
n+1
1
an+1 an =
=
.
3n + 5 3n + 2
(3n + 2)(3n + 5)
1
Honors Calculus Math 221
Homework 1
1. Use mathematical induction to prove that, for all positive integers n,
n
12
3
2
k=1 k = 4 n (n + 1) .
Solution: The given equation is the statement Pn . P1 is true since 13 =
12
2
4 1 (1 + 1) . Suppose Pn is true. Th
Review problems, Honors Calculus Math 221
Chapter 4
1
(1) Use the denition to nd the derivative of f (x) = x at x = 3.
1
(2) Find a point a > 0 such that the tangent line of f (x) = x at x = a
passes through the point (0, 1).
(3) Find the second derivativ
Review problems for Chapters 2, 3 and 4, Honors Calculus
Chapter 2
(1) Prove by mathematical induction that
1 + 5 + 9 + + (4n 3) = n(2n 1).
(2) Find the limits of the following sequences
3n2 + 1
, bn = (n2 + 1)1/n .
an =
n+2
(3) Prove that the sequence
n