BCC Project Calculus part 1 Spring 2014 1. Show that the equation has exactly 1 real root. ______________________________________________________________________________________ 2. Research the function and build the graph . On the same graph draw and compare their behavior. ______________________________________________________________________________________ 3. Find the point on the line , closest to the point (2,6) ______________________________________________________________________________________ 4. Research the function and sketch the graph 5. At 2:00 pm a car’s speedometer reads 30 mph. At 2:10 pm it reads 50 mph. Show that at some time between 2:00 and 2:10 the acceleration was exactly 120 mi/h2. ______________________________________________________________________________________ 6. Find the dimensions of the rectangle of the largest area that has its base on the x-axis and its other two vertices above the x-axis and lying on the parabola . ______________________________________________________________________________________ 7. Find , if ______________________________________________________________________________________ 8. Prove the identity ______________________________________________________________________________________ 9. Given tanh(x) = 0.8. Find other 5 values of hyperbolic functions: sinh(x), cosh(x), sech(x), csch(x), coth(x). ______________________________________________________________________________________ 10. Research and build the graph: h(x) = (x + 2)3 - 3x - 2 ______________________________________________________________________________________ 11. Find the number c that satisfies the conclusion of the Mean Value Theorem. f(x) = 5x2 + 3x + 6 x [-1, 1] ______________________________________________________________________________________ 12. Evaluate the limit: a) ; b) ______________________________________________________________________________________ 13. Sketch the curve. y = x/(x2+4) ______________________________________________________________________________________ 14. Find the area of the largest rectangle that can be inscribed in a right triangle with legs of lengths 5 cm and 6 cm if two sides of the rectangle lie along the legs. ________________________________________________________________________ 15. The altitude of a triangle is increasing at a rate of 1 cm/min while the area of the triangle is increasing at a rate of 2 cm sq. per minute. At what rate is the base of the triangle changing when the altitude is 10 cm and the area is 100 cm sq? ________________________________________________________________________ 16. Use a linear approximation or differentials to estimate the given number: tan440 ________________________________________________________________________ 17. Prove the identity: cosh 2x = cosh2x + sinh2x _______________ ...

1. BCC Project Calculus part 1 Spring 2014
1. Show that the equation has exactly 1 real root.
_____________________________________________________
_________________________________
2. Research the function and build the graph . On the same
graph draw and compare their behavior.
_____________________________________________________
_________________________________
3. Find the point on the line , closest to the point (2,6)
_____________________________________________________
_________________________________
4. Research the function and sketch the graph
5. At 2:00 pm a car’s speedometer reads 30 mph. At 2:10 pm it
reads 50 mph. Show that at some time between 2:00 and 2:10
the acceleration was exactly 120 mi/h2.
_____________________________________________________
_________________________________
6. Find the dimensions of the rectangle of the largest area that
has its base on the x-axis and its other two vertices above the
x-axis and lying on the parabola .
_____________________________________________________
_________________________________
7. Find , if
_____________________________________________________
_________________________________
8. Prove the identity
_____________________________________________________
_________________________________

2. 9. Given tanh(x) = 0.8. Find other 5 values of hyperbolic
functions: sinh(x), cosh(x), sech(x), csch(x), coth(x).
_____________________________________________________
_________________________________
10. Research and build the graph: h(x) = (x + 2)3 - 3x - 2
_____________________________________________________
_________________________________
11. Find the number c that satisfies the conclusion of the Mean
Value Theorem.
f(x) = 5x2 + 3x + 6
x [-1, 1]
_____________________________________________________
_________________________________
12. Evaluate the limit: a) ; b)
_____________________________________________________
_________________________________
13. Sketch the curve. y = x/(x2+4)
_____________________________________________________
_________________________________
14. Find the area of the largest rectangle that can be inscribed
in a right triangle with legs of lengths 5 cm and 6 cm if two
sides of the rectangle lie along the legs.
_____________________________________________________
___________________
15. The altitude of a triangle is increasing at a rate of 1 cm/min
while the area of the triangle is increasing at a rate of 2 cm sq.
per minute. At what rate is the base of the triangle changing
when the altitude is 10 cm and the area is 100 cm sq?
_____________________________________________________
___________________

3. 16. Use a linear approximation or differentials to estimate the
given number:
tan440
_____________________________________________________
___________________
17. Prove the identity: cosh 2x = cosh2x + sinh2x
_____________________________________________________
___________________
18. Prove that the formula for the derivative of tangent
hyperbolic inverse :
_____________________________________________________
___________________
19. Show that the equation has exactly one real root.
Find the intervals on which F(x) is increasing or decreasing.
Find local maximum and minimum of F(x). Find the intervals of
concavity and the inflection points.
_____________________________________________________
___________________
20. Suppose that for all values of x.
Show that .
_____________________________________________________
_________________________________
21. Suppose that the derivative of a function f(x) is :
On what interval is f (x) increasing?
_____________________________________________________
_________________________________
22. Explore and analyze the following three functions.
a) Find the vertical and horizontal asymptotes. I.

4. b) find the intervals of increase or decrease. II.
c) find local maximum and minimum values. III.
d) find the intervals of concavity and the inflection points.
e) use the information from parts a) to d) to sketch the graph
_____________________________________________________
_________________________________
23. A piece of wire 10 m long is cut into two pieces. One piece
is bent into a square and the other is bent into an equilateral
triangle. How should the wire be cut so that the total area
enclosed is a) maximum ? b) minimum ?
_____________________________________________________
_________________________________
24. Find the point on the line that is closest to the origin.
_____________________________________________________
_________________________________
25. Sketch the graph of
_____________________________________________________
_________________________________
26. Find f(x) if
27. Express the limit as a derivative and evaluate:
_____________________________________________________
________________________________________
28. The volume of the cube is increasing at the rate of 10
cm3/min. How fast is the surface area increasing when the
length of the edge is 30 cm.
_____________________________________________________
________________________________________
29. Evaluate dy, if , x = 2, dx = 0.2
_____________________________________________________

5. ________________________________________
30. Find the parabola that passes through point (1,4) and
whose tangent lines at x = 1 and x = 5 have slopes 6 and -2
respectively.
_____________________________________________________
________________________________________
31. Cobalt-60 has a half life of 5.24 years. A) Find the mass
that remains from a 00 mg sample after 20 years. B) How soon
will the mass of 100 mg decay to 1 mg?
_____________________________________________________
________________________________________
32. Find the points on the figure where the tangent line has
slope 1.
_____________________________________________________
________________________________________
1
33. Suppose that a population of bacteria triples every hour and
starts with 400 bacteria.
(a) Find an expression for the number n of bacteria after t hours.
_____________________________________________________
________________________________________
(b) Estimate the rate of growth of the bacteria population after
2.5 hours. (Round your answer to the nearest hundred.)
_____________________________________________________
________________________________________
34. Find the n-th derivative of the function y = xe-x
35. Find the equation of the line going through the point (3,5),
that cuts off the least area from the first quadrant.

6. 36. Research the function and sketch its graph. Find all
important points, intervals of increase and decrease
a) y =
b) y =
c)
37. Use Integration to find the area of a triangle with the given
vertices: (0,5), (2,-2), (5,1)
38. Find the volume of the largest circular cone that can be
inscribed into a sphere of radius R.
39. Find the point on the hyperbola xy = 8 that is the closest
to the point (3,0)
_____________________________________________________
____________________________________
40. For what values of the constants a and b the point (1,6) is
the point of inflection for the curve
41. If 1200 sq.cm of material is available to make a box with a
square base and an open top, find the largest possible volume of
the box.
42. Two cars start moving from the same point. One travels
south at 60 mi/h and the other travels west at 25 mi/h. At what
rate
is the distance between the cars increasing two hours
later?
43. A man starts walking north at 4 ft/s from a point . Five
minutes later a woman starts walking south at 5 ft/s from a
point
500 ft due east of . At what rate are the people moving apart 15
min after the woman starts walking?
44. At noon, ship A is 150 km west of ship B. Ship A is sailing
east at 35 km/h and ship B is sailing north at 25 km/h. How fast
is the distance between the ships changing at 4:00 PM?

7. 46. Two sides of a triangle have lengths 12 m and 15 m. The
angle between them is increasing at a rate of 2 degrees per
minute . How fast is the length of the third side increasing when
the angle between the sides of fixed length is 60 ?
47. Two people start from the same point. One walks east at 3
mi/h and the other walks northeast at 2 mi/h. How fast is
the distance between the people changing after 15 minutes?
48. The radius of a circular disk is given as 24 cm with a
maximum error in measurement of 0.2 cm.
Use differentials to estimate the maximum error in the
calculated area of the disk.
49. Use differentials to estimate the amount of paint needed to
apply a coat of paint 0.05 cm thick to a hemispherical dome
with diameter 50 m.
50. Use a linear approximation (or differentials) to estimate the
given number: 2.0015
51. Verify that the function satisfies the hypotheses of the Mean
Value Theorem on the given interval. Then find all numbers that
satisfy the conclusion of the Mean Value Theorem.
52. Does there exists a function f(x) such that for all x?
53. Suppose that f(x) and g(x) are continuous on [a,b] and
differentiable on (a,b). Suppose also that f(a)=g(a) and f’(x)<
g’(x) for a<x<b prove that f(b)<g(b). [Hint: apply the Mean
Value Theorem for the function h=f - g. ]

8. 54. Show that the equation has at most 2 real roots.
55 - 58. (a) Find the intervals on which is increasing or
decreasing.
(b) Find the local maximum and minimum values of f(x).
(c) Find the intervals of concavity and the inflection
points.
55.
56. for
57. for
58.
(a) Find the vertical and horizontal asymptotes.
(b) Find the intervals of increase or decrease.
(c) Find the local maximum and minimum values.
(d) Find the intervals of concavity and the inflection points.

9. (e) Use the information from parts (a)–(d) to sketch the graph
of f(x)
59.
60.
61.
62. , for
Find the limits:
63.
64.
65.

10. 66.
67. A stone is dropped from the upper observation deck of a
Tower, 450 meters above the ground.
(a) Find the distance of the stone above ground level at time t.
(b) How long does it take the stone to reach the ground?
(c) With what velocity does it strike the ground?
(d) If the stone is thrown downward with a speed of 5 m/s, how
long does it take to reach the ground?
68. What constant acceleration is required to increase the speed
of a car from 30 mi/h to 50 mi/h in 5 seconds?
A particle is moving with the given data. Find the position of
the particle.
69. a(t)= cos(t)+sin(t), s(0)=0, v(0)=5
70. ,
f (x) =
x3

11. 2x2 +1
f(x)=
x
3
2x
2
+1
y = 8− x2
y=8-x
2
f (t)
f(t)
′′f (t) = 2 −12t, f (0) = 9, f (2) =15
¢¢
f(t)=2-12t,f(0)=9,f(2)=15
tanh(lnx) =
x2 −1
x2 +1
tanh(lnx)=
x
2
-1

12. x
2
+1
lim
x→1
x15 −1
x −1
lim
x®1
x
15
-1
x-1
y = x3 −2x2 +1
y=x
3
-2x
2
+1
y = ax2 + bx + c
y=ax
2
+bx+c
x2 + 2y2 =1

13. x
2
+2y
2
=1
x
x2 + 4
x
x
2
+4
y = x3 − x −2
y=x
3
-x-2
y = x3 + ax2 + bx +1
y=x
3
+ax
2
+bx+1
f (x) = x3 + x −1,.............x ∈ [1,2]
f(x)=x

14. 3
+x-1,.............xÎ[1,2]
f (0) = −1, f (2) = 4, f '(x) ≤ 2
f(0)=-1,f(2)=4,f'(x)£2
x4 + 4x + c = 0
x
4
+4x+c=0
y = x4 −2x2 +3
y=x
4
-2x
2
+3
y = sinx + cosx
y=sinx+cosx
0 ≤ x ≤ 2π
0£x£2p
y = cos2 x −2sinx

15. y=cos
2
x-2sinx
0 ≤ x ≤ 2π
0£x£2p
y =
x2
x2 +3
y=
x
2
x
2
+3
y =
x2
x2 − 4
y=
x
2
x
2
-4

16. y = x2 +1− x
y=x
2
+1-x
y = e
−1
x+1
y=e
-1
x+1
1+ x + 2x3 +3x5 = 0
1+x+2x
3
+3x
5
=0
y = x tanx
y=xtanx
−
π
2
< x <

17. π
2
-
p
2
<x<
p
2
lim
x→0
(cscx −cotx)
lim
x®0
(cscx-cotx)
lim
x→∞
(
ln2 x
x
)
lim
x®¥
(
ln
2
x
x
)

18. lim
x→∞
(1+
a
x
)
x
b
lim
x®¥
(1+
a
x
)
x
b
y = x2 lnx
y=x
2
lnx
lim
x→0
(
5x − 4x
3x −2x
)

19. lim
x®0
(
5
x
-4
x
3
x
-2
x
)
v(t) =1.5 t
v(t)=1.5t
s(4) =10
s(4)=10
y = x2
y=x
2
3x + 4y =12
3x+4y=12