Mat 270 Final Exam Review Sheet Fall 2012 (Final on December
13th, 7:10 PM - 9:00 PM in PSH 153)
1. Find the slope of the secant line to the graph of y  f ( x) between the points
f (b)  f (a)
Answer:
(a, f (a)), and (b, f (b)) .
ba
2. Find the equation of the secant line to the graph of y  f ( x)  2 x 2  3 between the points
at x  1 and x  2 .
Answer: y  6 x  1
3. Find the equation of the tangent line to the graph of y  f ( x)  2 x 2  3 at the points x  1 .
Answer: y  4 x  1
4. Find the equation of the normal line to the graph of y  f ( x)  2 x 2  3 at the points x  1 .
Answer: x  4 y  21
x 9
5. Use conjecture method to evaluate the limit f ( x) 
at x = 9.
x 3
Answer: 6
6. Evaluate the following limits using algebraic process:
x 9
a) lim
Answer: 6
x 9
x 3
b) lim(1  2 x)1/ x
Answer: e 2
x 0
x  16  4
x 0
x
2 
 1
d) lim 
 2

x 2 x  2
x  2x 

c) lim
e)
f)
cos x5
x 
x
lim  tan x
lim
x ( / 2)
3  2x  4x2
x 
x2  9
cos x
h) lim
x ( / 2) x   / 2
i) lim( 4 ln x  tan 1 x)
g) lim
x 1
j)
lim
x 
11x3  42
5 x3  64 x 6  1500
Answer: 1/ 8
Answer: 1/ 2
Answer: DNE
Answer: DNE
Answer: 4
Answer: 1
Answer: 0.7853
Answer: 11/13
x 2  3x  2
7. Find vertical and horizontal asymptote(s) if any: f ( x)  10
x  x9
Answer: VA: x  0 and HA: y  0
x2  5x  6
x2  2 x
Answer: VA: x  0 and horizontal asymptote HA is y  1.
9. Write the limit definition of derivative for the function y  f ( x) at x = a
8. Find vertical and horizontal asymptote(s) if any: f ( x) 
10. Use limit definition of derivative to find f ( x) of the function f ( x)  3x  1
1
11. Find the equation of the tangent line to y  at the point (1, 1)
x
12. Find the equation of the tangent line to y  e x at the point x  ln 3
Answer: y  3x  3  3ln 3
13. Where is the slope of the tangent line to the curve y  2 x 2  14 x equal to –24?
At x  2.5
14. Find the derivative y  f ( x) :
1) f ( x)  10 x 4  e x  13x  13
Answer: f ( x)  40 x3  e x  13
2)
3)
f ( x)  x 2 e 2 x
sin x  cos x
f ( x) 
sin 2 x
Answer: f ( x) 
sin 2 x(cos x  sin x)  2cos 2 x(sin x  cos x)
sin 2 2 x
5)
sin x  cos x
sin x  cos x
f ( x)  cos2 x  sin 2 x
6)
f ( x)  (ln | x |)2  ln( x 2 )
7)
f ( x)  (2sin x  x3  9)1/ 2
8)
f ( x) | x |  | 2 x  3|
Answer:
9)
f ( x)  x  x
Answer:
4)
f ( x) 
10) f ( x) 
4
Answer: f ( x)  0
2ln | x | 2
Answer: f ( x) 

x
x
x 2(2 x  3)

| x | | 2x  3 |
2 x 1
4 x x x
2x
4x  3
11) f ( x)  cos 1 x 
3
 tan 1 (2 x)
x
12) f ( x)  x 2 x
13) x3/ 2  y 2/ 3  15
15. Find the equation of the tangent line to y  g ( x) at x = 3 where
f (3)  1, f (3)  4, g ( x)  x2  f ( x)
Answer: y  10 x  20
27
16. Find the equation of the tangent line to y  2
at x = 2
x 9
cos x
at x   / 3
1  cos x
cos x
18. Find the equation of the normal line to y 
at x   / 3
1  cos x
4
5
19. The slope of y  f 1 ( x) at (4, 7) is , find f (7) .
Answer:
5
4
1
2
20. The function f ( x)  x  x  1 is one-to-one for x  0 . Find ( f )(3)
21. Find ( f 1 )(3) , the derivative of the inverse function of f if f ( x)  x3  x  7 .
1
22. A cost function of the form C ( x)  x 2 reflects diminishing returns to sale. Find the and
2
graph the cost, average cost and marginal cost functions.
17. Find the equation of the tangent line to y 
23. An 8 foot ladder is leaning against a wall. How fast is the top of the ladder sliding down
the wall if the bottom of the ladder is sliding directly away from the wall at 2 ft/sec and
the foot of the ladder is 2 ft away from the wall?
Answer: 2 / 15
24. A spherical snowball melts at a rate proportional to its surface area. Show that the rate of
change of the radius is constant. Given the surface area S  4 r 2 . Hint: Volume of a
4
dV
dr
dV
sphere is V   r 3 
and
 4 r 2
 4k r 2 where k is a constant. Now find
3
dt
dt
dt
dr
that
 k , shows that rate of change of radius is a constant.
dt
25. Baseball runners stand at first and second base in a baseball game. At the moment a ball
is hit, the runner at first base runs to second base at 18ft/sec; simultaneously on the
second base runs to the third base at 20ft/sec. How fast is the distance between the
runners changing 1 second after the ball is hit? The distance between consecutive bases is
90 ft and the bases lie at the corner of a square. Hints: Draw a rough diagram and choose
dD
dx
dy
D2  (90  x)2  y 2 , find
 ?, given
 18,
 20 . Answer: distance is decreasing
dt
dt
dt
at a rate of 11.99 ft/s.
26. Find all relative/absolute max/min, critical point(s), interval of increase/decrease,
inflection point(s) interval of concavity:
1)
2)
3)
f ( x)  x ( x 2 / 5  4)
f ( x)  10 x2 ln | 6 x |
4)
5)
f ( x)  x ( x 2 / 5  4), [0, 4]
f ( x)  x 2/ 3 (4  x 2 ), [3, 4]
f ( x)  10 x 2 ln(6 x)
27. Determine the open interval where the function f ( x)  6 x 2 ln(3x) is concave up.
28. An 8 ft tall fence runs parallel to the side of a house 3 ft away. What is the length of the
shortest ladder that clears the fence and reaches the house? Assume that the vertical wall
of the house and the horizontal ground have infinite extend. Answer: 15 ft. (see example
4 at page number 260)
29. Find the dimensions of the right circular cylinder of maximum volume that can be placed
inside the sphere of radius R.
30. A 5324 cubic foot tank with a square base and an open top is to be constructed of a sheet
of steel of a given thickness. Find the length of a side of the square base of the tank with
minimum surface area.
Answer: Length of side of square base is 22 ft.
31. Linear Approximation:
1) Approximate 1.03 using the linear approximation of f ( x)  (1  x)m at x = 0
0.03
Answer: (1  x)m  1  mx, 1.03  1 
 1.015
2
2) Approximate 1.03 using the linear approximation of f ( x)  x at x = 1
Answer: x  0.5  0.5x, 1.03  0.5  0.5(1.03)  1.015
32. Verify the conditions of Mean Value Theorem or Rolle’s Theorem and find the value of c
in the open interval given.
1)
2)
f ( x)  2 x3  3x  1, [2, 2]
f ( x)  2 x( x  1)3 , [0,1]
33. Evaluate the following integrals:
1)
2)
 (20 x
 (20 x
4
 sin x  cos x  e x ) dx
3
 sin 2 x  cos 2 x  e4 x ) dx
1
 2  x dx b)
4 x19  5 x8
dx
4) 
x9
6.  sec2 x dx
3) a)
1  tan 
d
sec 
x
dx
9. 
2 x
8.
x
 2  x2 dx
c)
1
 2  x2 dx
d)
5.  (cos(4 x)  sin( x / 4)) dx
7.  | ax  b | dx

2
10.

0
1  sin 2x
2x
 2  2x dx e)
2  x2
 1  x2 dx
3.5
11.


1
 x 2  2 dx
2
x

1
 x 2  1 dx
2
4x
1
4
13.
1
3
15.
4
1  sin 2x dx
1
17.  cos 2 x dx
 tan x dx
20.  sec x dx
21.  csc x dx
12.

1
 x 2  2 dx
x2

1
 x 2  1 dx
2
4x
1
3
14.
1
16.  sin 2 x dx
18.  (sin 2 x  cos 2 x) dx
Answer: ln | sec x | C
19.
22.
Answer: ln | csc x  cot x | C
sin( x10 )
C
10
(cos x)11
C
Answer: 
11
9
10
 x cos( x ) dx
Answer:
23.  sin x(cos x)10 dx
4
24.

16  x 2 dx
0
1
25.
 2xdx
1
26.
 2 | x | dx
1
1
4
e2
x
dx
27. 
1  x2
1
28.

1
ln x
dx
x
34. Evaluate the following:
x
1)
2
d
2et 3dt

dx 1
Answer: 2e x
2
3
10
2)
2
d
2et 3dt

dx x
Answer: 2e x
2
3
x2
d
2t 3dt
3)

dx 1
Answer: 4x 7
x2
d
2t 3dt
4)

dx 2 x
Answer: 4 x7  32 x3
35. Given f (t )  3t 2  1, f (0)  10 , find f (2 x)
36. Find LHS, RHS and Midpoint sum (MPS) use Riemann sum:
1)
f ( x)  x 2  1, [2, 4], n  4
2)
3)
4)
f ( x)  cos x, [2, 4], n  4
f ( x)  x 2  1, [2, 4], n  6
f ( x)  cos 2 x, [0,  / 4], n  6
37. Evaluate the sum:
40
 (n
2
 3n  0.2n  1)
n 1
38. Evaluate the sum:
40
[(n  2)
2
 0.3n  2]
n 1
39. If two resistors R1 and R2 are connected in parallel, as shown in the figure, then the total
1 1 1
resistance R measured in ohms (Ω), is given by   .
R R1 R2
If R1 and R2 are increasing at a rate of 0.3 Ω/s and 0.2 Ω/s respectively, how fast is R
changing when R1  80  and R2  100  ?
40. Two people start from the same point. One walks east at 3 miles/hour and the other walks
northwest at 2 miles/hour. How fast is the distance between the people changing after 15
minutes?
41. The minute hand on a watch is 8 mm long and the hour hand is 4 mm long. How fast is
the distance between the tips of the hands changing at one o’clock?
42. Find (in two decimal places) the coordinates of the point on the curve y  tan x that is
closest to the point (1, 1).