Math Analysis Exam Review
Polar Coordinates:
A point in polar coordinates is given. Find the corresponding rectangular coordinates.
𝜋
1. (2, π)
3. (√2, 2.36)
2. (4, − 3 )
The rectangular coordinates of a point are given. Find two polar coordinates (r, θ) for each point, one
with r > 0 and the other with r < 0.
4. (-3, 3)
5. (0, 2)
6. (√3, −1)
Convert the rectangular equation to polar form.
7. 6xy = 1
8. y2 = 2x
Convert each polar equation to rectangular form.
9. r = 6 sin θ
10. r2 cos 2θ = 2
Graph each polar equation. Label specific points.
11. r = 2 – 2 cos θ
12. r = 4 sin 3θ
6
6
6
6
Vectors: PC Sections 8.4-8.6
1.
Find each position vector.
Initial Point
Terminal Point
a) (-1, -5)
(2, 3)
b) (-6, -4)
(0, 1)
2. Given v  24, 7 .
a) Find a unit vector in
the direction of v .
b) Find || v ||.
3. If u  6, 8 & v  2,4 ,
what is 2u  3v ?
4. What is the angle between
u = 2i – j and v = 6i + 4j?
Parametric Equations:
5. A force of 45 pounds in the direction of 30° above the
horizontal is required to slide a table across a floor. The table is
dragged 20 feet. How much work was done?
PC Section 9.7
Graph the curve whose parametric equations are given and show its orientation. Then, find the
rectangular equation of each curve.
6. x = t2 – 4, y = 3t; -3 < t < 3
y
x
Rectangular Equation:
7. x t 2, y t 2,t 0
y
x
Rectangular Equation:
Sequences & Series:
PC Sections 11.1-11.5
Find each sum, if possible. If a series does not have a sum, write ‘no sum’.
8. 2 + 10 + 50 + 250 + …
9. 5 + 10 + 20 + … + 640
10. 22 + 30 + 38 + … + 102
11. 162 + 54 + 18 + 6 + …
12. A ball rebounds to two-thirds the height from which it is dropped. Use an infinite geometric series to
approximate the total distance the ball travels, after being dropped from 5 feet above the ground, until it
comes to rest.
13. Use the Principle of Mathematical Induction to show that the statement below is true for all natural
numbers n.
1 3  2  4  3  5  ...  n (n  2) 
14. Find the coefficient of x2 in the expansion of (2x – 3)6.
n(n  1)(2n  7)
6
Calculus: http://quizlet.com/23904400/final-exam-review-flash-cards/
Limits:
Calculus Chapter 1
15. Use the graph of f(x) to find the value of each limit, if it exists. Use ∞, -∞,  , or DNE when
appropriate.
a) lim f ( x ) =
b) lim f ( x) =
x
x 5
c) lim f ( x) =
d) lim f ( x) =
e) lim f ( x) =
f) f(-4) =
g) lim f ( x ) =
h) f(5) =
x  4
x  4
x  4
x6
i) lim f ( x) =
j) f(-10) =
f ( x) =
k) xlim
 10
l) f(-1) =
x  1
Find the value of each limit, if it exists.
 x  2  3x  2 
16. xlim
 1
3
x 2  3 x  10
19. lim
x 5
5 x
22. lim
x 0
| 4x |
x
17. lim
x2
x2
x  2 x 2  3x  6
3
sin 2 x
x 0
x
20. lim
23. lim
x 0
| 4x |
x
18. lim
x 4
21. lim
x 
24. lim
x 0
x  13  3
x4
8  3x
x2  6
| 4x |
x
Consider each function. Evaluate the limit, if it exists. If the limit does not exist, explain why.
 3x  2, if x  6

25. f ( x)  7, if x  6
 1 x  1, if x  6
2
 cos x, if x  0

26. g ( x )    sin x, if 0  x  
1  cos x, if x  

What is lim f ( x ) ?
What is lim g( x) ?
x0
x6
Differentiation:
Calculus Chapter 2
Use the limit of a derivative to differentiate each function.
27. f (x)  4x5
28. g(x) 
6
3x  2
Find the equation of the line that is tangent to the given point.
29. y = 5x3 – 2x + 9 at the point (-1, 6)
30. f(t) = 7 + 3 cos x at the point (π, 4)
Find the derivative of each function.
31. f ( x) 
2 x3  6 x  1
x2
32. g(t)  t 18t 2  3
33. h( x) 
5x
2
x  2x
34. j(θ) = cos5 (2θ)
35. k()  tan3 12sin4
d2y
Find
for each function.
dx 2
37. g(x) = sin2 (x – 5)
2
36. f ( x )  x  4 x
Given the graph of f ′(x), sketch the graph of f(x).
38.
f ′(x)
39.
f ′(x)
x
f (x)
x
f (x)
x
x
40. If h(x) = 4f 2 (x) + 5 g2 (x), f ′(x) = – g(x) and g ′(x) = f(x), what is h ′(x) in terms of f(x) and g(x)?
41. Consider the three conditions of continuity. If f(x) is continuous at the point x = a, what can you say
about limits as x approaches a from the right and from the left. What can you say about whether f(x) is
defined at a? What is the relationship between continuity at x = a and differentiability at x = a?
Find dy/dx .
42. x3 + x2y + 4y2 = 6
43. 4 cos x sin y = 1
44. Given x3 + y3 = 6xy.
a) Verify that y ' 
2 y  x2
y2  2x
b) At which point in the first quadrant is the tangent
line vertical to the graph of x3 + y3 = 6xy.
45. The position of a particle moving on the horizontal line y = 2 is given by x(t) = 4t3 – 16t2 +15t for t > 0.
At the instant when the acceleration becomes zero, what’s the velocity of the particle?


46. The position at time t > 0 of a particle moving along the s-axis is s(t )  10cos  x   . What is the
4

particle’s acceleration at two seconds?
47. Functions f and g and their first derivatives have the following values at x = -1 and x = 0.
x
f(x)
g(x)
f ′(x)
g ′(x)
-1
0
-1
2
1
0
-1
-3
-2
4
Find the value of each expression.
a) If M(x) = 3 f(x) – g(x), what is M ′(-1)?
b) If N(x) = f (g(x)), what is N ′(-1)?
48. Suppose oil spills from a ruptured tanker and spreads in a circular pattern. If the radius of the oil spill
increases at a constant rate of 1 meter per second, how fast is the area of the spill increasing when the radius
is 30 meters?
49. Gravel is being dumped from a conveyor belt at a rate of 30 ft3/min and its coarseness is such that it
forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the height
of the pile increasing when the pile is 10 ft high?
Applications of Differentiation:
Calculus Chapter 3
Find the relative extrema for each function.
50. y = -2x3 + 6x2 – 3
2
51. y  x 8  x
Find the value of C that is guaranteed by the Mean Value Theorem for each function on the interval…
52. f(x) = x3 – 5x + 2 on [-1, 3]
53. y  3x 6 on[5,14]
54. Analyze and sketch the graph of y = -x4 + 4x3 – 4x + 1. Label intercepts, local extrema, and points of
inflection. Then, list increasing and decreasing intervals. Lastly, list the intervals for which the function is
concave up or concave down.
Increasing:
Decreasing:
Concave up:
Concave down:
55. A particle is moving along a line, and its position function y = s(t) is shown in the graph below.
y
Distance from the origin
b) At what time is the particle’s
velocity equal to zero?
a) At what time is the particle’s
acceleration equal to zero?
5
10
t