Name:
Date:
Resources:
http://www.calcchat.com/book/Calculus-9e/
http://apcentral.collegeboard.com/apc/public/courses/teachers_corner/2178.html
http://www.calculus.org/
http://cow.math.temple.edu/
http://www.mathsisfun.com/calculus/
http://www.wolframalpha.com/widgets/view.jsp?id=dc816cd78d306d7bda61f6facf5f17f7
http://www.wolframalpha.com/widgets/view.jsp?id=c44e503833b64e9f27197a484f4257c0
PCTI Mathematics Department
Sal F. Gambino, Supervisor
Summer Packet Grading
• On the first day of school, the teacher will check for
completion/effort of the packet.
This will be weighted at 50%.
• Teacher will then review the packet with the students.
Upon completion of the review, the students will be given
an assessment based on the summer packet.
The assessment will be weighted at 50%.
• The two weighted scores combined will count as one
project grade. Therefore, the grade for the summer packet
will be placed under the “project” category.
1. Use the definition of a derivative to find y’ of 𝑦 =
2
𝑡
Related Rates
#2
#3
2. A swimming pool is 11 meters long, 6 meters wide, 1 meter deep at the shallow end, and 3
1
meters deep at the deep end (see the figure above). Water is being pumped into the pool at 4
cubic meter per minute, and there is 1 meter of water in the deep end.
a.) What percent of the pool is filled?
b.) At what rate is the water level rising?
3. A trough is 12 feet long and 3 feet across the top (see figure). Its ends are isosceles triangles
with altitude of 3 feet.
a.) If water is being pumped into the trough at 2 cubic feet per minute, how fast is the water
level rising when the depth h is 1 foot?
3
b.) If the water is rising at a rate of 8 inch per minute when h = 2, determine the rate at which
water is being pumped into the trough.
Solve
1
4. ∫−1(5𝑥 3 − 7𝑥)𝑑𝑑
5. ∫ (sec(2𝑥) tan(2𝑥))𝑑𝑑
12
6. ∫3 (𝑥 − 3)1/2 𝑑𝑑
2
7. ∫1 (𝑥 − 1)√2 − 𝑥𝑑𝑑
8. ∫( 2x − 9 sinx)dx
9. ∫ 𝑥𝑥𝑥𝑥3𝑥 2 dx
Sketch region bounded by graph and find area of regions.
10. 𝑦 = 𝑥 2 − 1, 𝑦 = −𝑥 + 2
11. 𝑓(𝑦) = 𝑦 2 , 𝑔(𝑦) = 𝑦 + 2
4
12. 𝑔(𝑥) = 2−𝑥, 𝑦 = 4, 𝑥 = 0
13. Define𝑓(𝑥) =
|𝑥−1|
𝑥
(a) Show that 𝑓(𝑥) is continuous at 𝑥 = 2.
(b) Where on the interval [-2,2] is 𝑓 discontinuous? Show the work that leads to your
conclusion.
(c) Classify the discontinuities in part (b) as removable or nonremovable.
Find the limit (if it exists). If the limit does not exist, explain why.
(𝑥 + 𝑦)3 − 𝑥 3
𝟏𝟏. lim
𝑦→0
𝑦
𝟏𝟏. lim(𝑥 − 2)(𝑥 − 2)2
𝑥→6
𝟏𝟏. lim
𝑥→0
1 − 𝑐𝑐𝑐𝑐
𝑠𝑠𝑠𝑠
2𝑥 2
𝑥→∞ 3𝑥 2 + 5
𝟏𝟏. lim
1
𝟏𝟏. lim �8 + �
𝑥→∞
𝑥
Free-Falling Object, for 19 and 20 use the position function 𝑠(𝑡) = −4.9𝑡 2 + 250 which gives
the height (in meters) of an object that has fallen from a height of 250 meters.
19. Find the velocity of the object when 𝑡 = 4 .
20. At what velocity will the object impact the ground?
Find the one sided limit (if it exists)
𝟐𝟐. lim∓
𝑥→0
𝟐𝟐. lim−
𝑥→0
𝟐𝟐. lim−
𝑥→3
𝑠𝑠𝑠4𝑥
5𝑥
𝑐𝑐𝑐 2 𝑥
𝑥
|𝑥 − 3|
𝑥−3
24. Find the equation of the tangent to the graph of 𝑓(𝑥) = 3𝑥 − 5𝑐𝑐𝑐2𝑥 𝑎𝑎 𝑥 = 0.
25. Suppose 𝑔(0) = 4, 𝑔′ (0) = 8, 𝑎𝑎𝑎 𝑔′′ (0) = −12. 𝐼𝐼 ℎ(𝑥) = �𝑔(𝑥) . 𝑊ℎ𝑎𝑎 𝑖𝑖 ℎ′′ (0)?
26-27. Determine relative max or min, and state on which intervals function is increasing or
decreasing
26.𝑓(𝑥) =
𝑥 4 +1
𝑥2
𝟐𝟐. 𝑓(𝑥) = sin(𝑥) + 1, 0 < 𝑥 < 2𝜋
28. The concentration C of a chemical in the bloodstream t hours after injection into muscle
3𝑡
tissue is 𝐶(𝑡) = 27 − 𝑡 3 , 𝑡 ≥ 0. Determine the time when the concentration is greatest.
29. R is a region bounded by f(y) = 𝑦 2 − 3 𝑎𝑎𝑎 𝑔(𝑦) = 3𝑦 + 1. Write the equation that gives
the area of R.
30. The functions 𝑓(𝑥) = 2 − 𝑥 2 and 𝑔(𝑥) = −𝑥 are given.
a. Find the area of the region bounded by the graphs of 𝑓(𝑥) and 𝑔(𝑥).
b. Find the area of the region bounded by the graph of 𝑓(𝑥) and the 𝑥-axis.
c. Find the area of the region bounded by the graph of 𝑔(𝑥), the 𝑥-axis, and the line 𝑥=2.
Determine whether the Mean Value Theorem can be applied. If so, find all values of c on the
open interval (a,b) such that 𝑓′(𝑐) =
31. 𝑓(𝑥) = 𝑥 2⁄3 , [1, 8]
𝑓(𝑏)−𝑓(𝑎)
𝑏−𝑎
. If not, explain why not.
𝜋 𝜋
𝟑𝟑. 𝑓(𝑥) = 𝑥 − cos 𝑥 , �− , �
2 2
Use the First Derivative Test to find any relative extrema of the function.
33. 𝑓(𝑥) = 4𝑥 3 − 5𝑥
3
𝜋𝜋
34. 𝑔(𝑥) = 2 sin � 2 − 1� , [0, 4]
35. Determine the points of inflection and discuss the concavity of the graph of
𝑓(𝑥) = (𝑥 + 2)2 (𝑥 − 4)
36. Find the particular solution of the differential equation 𝑓 ′′ (𝑥) = 6(𝑥 − 1) whose graph
passes through the point (2, 1) and is tangent to the line 3𝑥 − 𝑦 − 5 = 0.
Use properties of summation and evaluate the sum
20
𝟑𝟑. �(𝑖 + 1)2
38.
𝑖=1
25
�(2𝑖 − 𝑖 2 )
𝑖=1
39. Use left and right-hand sums to approximate the areas of the region using the indicated
number of subintervals of equal width
10
𝑦 = 𝑥 2 +1
40. Use the limit process to find the area of the region between the graph of the function and
the x-axis over the given interval.
𝑦 = 5 − 𝑥 2 [-2,1]
41. 𝑓(𝑥) = 2𝑥 − 8 Set up a definite integral that yields area of the region.
8
8
42. Properties of definite integrals: Given ∫4 𝑓(𝑥)𝑑𝑑 = 12 and ∫4 𝑔(𝑥)𝑑𝑑 = 5
Evaluate
8
(a) ∫4 [𝑓(𝑥) + 𝑔(𝑥)]𝑑𝑑
8
(b) ∫4 [𝑓(𝑥) − 𝑔(𝑥)]𝑑𝑑
8
(c) ∫4 [2𝑓(𝑥) − 3𝑔(𝑥)]𝑑𝑑
8
(d) ∫4 7𝑓(𝑥)𝑑𝑑
9
43. Use the Fundamental Theorem of Calculus to evaluate the definite integral ∫4 (𝑥 √𝑥) 𝑑𝑑
44. Sketch the region bounded by the graphs of the functions. Then find its area.
4
𝑦 = 𝑥, 𝑦 = 0, 𝑥 = 1, 𝑥 = 9
45. Use the Second Fundamental Theorem of Calculus to find F’(x)
𝑥
F(x) = ∫−3(𝑡 2 + 3𝑡 + 2)𝑑𝑑
46. Find the average value of the function over the given interval. Find the values of x at which
the function assumes its average value.
𝐹(𝑥) = 𝑥 3 , [0,2]
1
47. Define 𝑓(𝑥) = 𝑥,
2
a.) Use the Trapezoidal Rule with 𝑛 = 5 to estimate ∫1 𝑓(𝑥)𝑑𝑑.
b.) ∫ 𝑓(𝑥) = 𝑙𝑙|𝑥| + 𝐶. Use the Fundamental Theorem of Calculus to calculate the exact value
2
of ∫1 𝑓(𝑥)𝑑𝑑.
1
c.) Explain why the Trapezoidal Rule cannot be used to estimate ∫−1 𝑓(𝑥)𝑑𝑑.
Multiple-choice questions
48. Let 𝑓(𝑥) = (3 + 2𝑥 − 𝑥 2 )3 be defined for the closed interval −2 ≤ 𝑥 ≤ 3. If M is the ycoordinate of the absolute maximum and m is the y-coordinate of the absolute minimum, what
is the absolute value of M+m?
a) 189
b) 125
c) 64
d) 61
e) none of these
49. A square is inscribed in a circle. How fast is the area of the square changing when the area
of the circle is increasing at the rate of 1 𝑖𝑖2 /min?
𝑖𝑖2
a) 1 𝑚𝑚𝑚
b)
1 𝑖𝑖2
2 𝑚𝑚𝑚
2 𝑖𝑖2
c) 𝜋 𝑚𝑚𝑚
𝑖𝑖2
d) 1 𝑚𝑚𝑚
𝜋 𝑖𝑖2
e) 2 𝑚𝑚𝑚
50. What is the slope of the line tangent to the curve 𝑦 3 + 𝑥 2 𝑦 2 − 3𝑥 3 = 9 at the point
(1,2)?
1
a) 16
1
b) 8
1
c) 4
1
d) − 4
1
e) − 8
2
8
51. Find ∫0 3𝑥 2 𝑓(𝑥 3 )𝑑𝑑 𝑖𝑖 ∫0 𝑓(𝑡)𝑑𝑑 = 𝑘
a) 𝑘 3
b) 9k
c) 3k
d) k
e)
𝑘
3
𝑥2
52. If 𝐹(𝑥) =∫0 √𝑡 + 3𝑑𝑑, what is 𝐹’(𝑥)
a) √𝑥 2 + 3
b)
1
2√𝑋 2 +3
c) 2𝑥(√𝑥 2 + 3)
d)
3
2(𝑋 2 +3)2
3
e) None of these
53. Let R be the region between the function 𝑓(𝑥) = 𝑥 3 + 6𝑥 2 + 10𝑥+4, the x-axis, and the
lines x = 0 and x = 4. Using Trapezoidal Rule, compute the area when there are 4 equal
subdivisions.
a) 196
b) 288
c) 296
d) 396
e) None of these
54. Which of the following statements about the function given by 𝑓(𝑥) = 𝑥 4 − 2𝑥 3
is true?
a) The function has no relative extremum.
b) The graph of the function has one point of inflection and the function has two relative
extrema.
c) The graph of the function has two points of inflection and the function has one relative
extremum.
d) The graph of the function has two points of inflection and the function has two relative
extrema.
e) The graph of the function has two points of inflection and the function has three relative
extrema.
55. The area of the region in the first quadrant between the graph of 𝑦 = 𝑥√4 − 𝑥 2 and the xaxis is:
a)
2√2
3
8
b) 3
c) 2√2
d) 2√3
16
e) 3