Review Problems for Exam 2
Calculus 3
Problem 1. A curve C has parametric equations x = t2 − 2t + 2, y = 2(t − 1)3 + 3(t − 1) + 1.
(a) Find all points on C where the tangent is horizontal or vertical.
(b) Determine the intervals for t on which the curve is moving up/down and left/right.
(c) Find an equation for the line which is tangent to C at the point P where t = 1.
(d) Compute d2 y/dx2 , and then find the values of t for which C is concave up.
Problem 2. How does the curve with parametric equations x = t2 + 2t + 2, y = −2(t + 1)3 − 3(t + 1) + 1 compare
with the curve C described in the previous problem? Explain the difference. (Hint: replace t with −t.)
Problem 3. An object moves in the plane so that at time t it is located at the point with rectangular coordinates
(x, y) where x = 3t2 and y = t3 − 3t.
(a) Does the object pass through the point (3, 3)? Does it cross the vertical line x = 3 and if so where?
p
(b) Find the speed
x0 (t)2 + y 0 (t)2 at time t and determine any values of t for which the speed is minimal.
(c) Determine the arclength of the curve traversed by the object between times t = 0 and t = 1.
(d) Show that the curve traced by the object crosses itself in one point and find the arclencth of the loop of the
curve determined by the crossing point.
Problem 4. Let C be the curve segment described by the equations x = t2 cos t, y = t2 sin t, where −1 ≤ t ≤ 1.
Compute the arclength of C.
Problem 5. Find all of the points where the curve C with parametric equations x = t2 − 2t, y = (t − 1)3 − (t − 1)
crosses itself. Then determine the slope-intercept equations for the two lines tangent to C at that point.
Problem 6. A curve is described parametrically by the equations: x = t2 + 2t and y = t3 − t.
(a) Find all points where the curve has horizontal or vertical tangent lines.
(b) Determine for which intervals of t the curve is rising/falling moving right and moving left.
(c) Give a rough sketch of the curve using information from (b).
d2 y
.
(d) Find
dx2
Problem 7. Find parametric equations for the path of an object moving on the circle (x − 3)2 + (y + 2)2 = 9 as
described:
(a) Once around the circle counterclockwise starting at (6, −2).
(b) Four times around the circle counterclockwise starting at (6, −2).
(c) Half way around the circle counterclockwise starting at (3, 1).
(d) Once around the circle clockwise starting at (6, −2).
Problem 8. Compare the curves represented by the parametric equations. How do they differ?
(a) x = t2 , y = t3
(b) x = t4 , y = t6
(c) x = sin2 (t), y = sin3 (t)
Problem 9.
Consider the power series
∞
X
n=1
(−1)n √
1
xn .
n 5n
(a) Determine the radius of convergence of this series.
(b) Find the interval of convergence of this series.
∞ √
X
n n
(c) If f (x) =
x then show that the graph of f (x) is concave up for all x < 0 in the domain of f .
n
5
n=0
Problem 10. Find the intervals of convergence of the series:
∞
∞
X
X
1
1
(a)
(−1)n √ n (x − 5)n
(b)
(−1)n √ n (x − 5)4n
n
5
n
5
n=1
n=1
Problem 11. Let g(x) = √
1
. Use the definition of Maclaurin series to determine the first five terms of the
1−x
Maclaurin series of g(x).
∞
X
(−1)n n
x .
n2 6 n
n=1
(a) Find the radius of convergence and interval of convergence of the power series.
(b) Express f (x) as the sum of a power series whose indexing starts at n = 0 (using Σ notation).
(c) Find a power series representation for f 0 (x) and determine the interval of convergence of this power series.
∞
X
(−1)n 2n
x ?
(d) What is the radius and interval of convergence for the series
6n
n=1
Problem 12.
Let f (x) =
Problem 13. (a) Use the first five terms of the Maclaurin series for f (x) = e2x to approximate the value of e2 . Use
Taylor’s inequality to determine how close the approximation is to the actual value of e2 .
(b) Use the first five terms of the Maclaurin series for f (x) = e2x to approximate the value of e.1 . Use Taylor’s
inequality to determine how close the approximation is to the actual value of e.1 .
Problem 14. Find Maclaurin series expansions for each of the following:
3
3
d
(a) x5 ex
(b) dx
{x5 ex }
(c) (1 + 2x)−2 (hint: differentiate (1 + 2x)−1 )
Z
1
x2
1
x2
(f)
(g)
(h)
dx
(e)
1+x
1 + x5
1 + x5
1 + x5
Problem 15.
∞
X
3n−2
(a)
4n+1
n=0
Determine the sums of the following
∞
∞
X
X
3n−2
3n
(b)
(c)
n+1
4
n!
4n n!
n=0
n=3
(d)
∞
X
(−1)n
n=0
π 2n
n
4 (n)!
Problem 16. Briefly explain why the following statement is correct: The sequence
but still the Alternatiing Series Test can be used to explain that
∞
X
n=1
x2 arctan(x) dx
1 + x2
(i)
1 + x5
R
(d)
(e)
∞
X
(−1)n
n=0
n
n
n2 −2n+5
o∞
π 2n
4n (2n)!
is not decreasing
n=1
(−1)n n
converges.
− 2n + 5
n2
Problem 17. Determine whether the series converges absolutely, converges conditionally or diverges. Clearly
indicate which convergence test is used and how it is applied.
∞
∞
∞
∞
∞
n 2n
X
X
X
X
X
1
3n n2n
n 1
2 −n3
n 3 n
(b)
(−1)
(c)
n
e
(d)
(e)
(−1)
(a)
5n + n − 2
ln(n)
(3n)2n
(3n)2n
n=1
n=2
n=1
n=1
n=1