Download math 2413 rt2su14 - HCC Learning Web

Survey
yes no Was this document useful for you?
   Thank you for your participation!

* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project

Document related concepts
no text concepts found
Transcript
Name: ________________________
06/06/2014
Math 2413 t2rsu14
1. Find the derivative of the following function using the limiting process.
f (x ) = −4x 2 + 5x
2. Find the derivative of the following function using the limiting process.
f (x ) =
2
x−3
1
____
3. The graph of the function f is given below. Select the graph of f ′.
2
a.
d.
b.
e.
c.
3
4. Find the derivative of the function.
f (x ) = x 4
5. Find the derivative of the function f (x ) = −7x 3 + 4x 2 + 1 .
6. Find the derivative of the function f (x ) = −4x 2 − 4cos (x ) .
7. Find the slope of the graph of the function at the given value.
f (x ) = −2x 2 +
6
when x = 5
x2
8. Find the derivative of the function f (x ) =
x5 − 9
.
x4
9. Determine all values of x , (if any), at which the graph of the function has a horizontal tangent.
y (x ) = x 3 + 12x 2 + 8
10. Suppose the position function for a free-falling object on a certain planet is given by s (t ) = −12t 2 + v 0 t + s 0 .
A silver coin is dropped from the top of a building that is 1372 feet tall. Find the instantaneous velocity of the
coin when t = 4.
4
11. The volume of a cube with sides of length s is given by V = s 3 . Find the rate of change of volume with
respect to s when s = 6 centimeters.
Ê
ˆÊ
ˆ
12. Find the derivative of the algebraic function H (v) = ÁÁÁ v 5 − 3 ˜˜˜ ÁÁÁ v 3 + 3 ˜˜˜ .
Ë
¯Ë
¯
5
13. Use the Product Rule to differentiate f (s) = s cos s .
8x
14. Use the Quotient Rule to differentiate the function f (x ) =
5
x +3
15. Use the Quotient Rule to differentiate the function f (x ) =
sin x
2
x +3
16. Find the derivative of the function.
f (s) = 9s sins + 5cos s .
5
9
17. Find the second derivative of the function f (x ) = 8x .
5
.
.
18. Find the second derivative of the function f (x ) =
3x 2 + 5x − 4
.
x
Ê
ˆ5
19. Find the derivative of the algebraic function f (x ) = ÁÁÁ x 6 + 4 ˜˜˜ .
Ë
¯
20. Find the derivative of the function.
f (x ) = x 7 (5 + 8x)
3
21. Find the derivative of the function y = 8cos 4x .
22. Find the derivative of the function.
Ê
ˆ
y = cos ÁÁÁ 2x 4 − 6 ˜˜˜
Ë
¯
23. Find an equation to the tangent line for the graph of f at the given point.
Ê
ˆ2
f (x ) = ÁÁÁ 5x 5 + 5 ˜˜˜ , ÊÁË 1,100 ˆ˜¯
Ë
¯
24. Find the second derivative of the function f (x ) = sin5x 6 .
6
25. Find
dy
by implicit differentiation.
dx
x 2 + y 2 = 25
26. Find
dy
by implicit differentiation.
dx
x 2 + 5x + 9xy − y 2 = 4
27. Evaluate
dy
for the equation 7xy = 21 at the given point ÊÁË −3,− 1 ˆ˜¯ . Round your answer to two decimal
dx
places.
2
x2 y
+
= 1 at ÊÁË 1,7 ˜ˆ¯ .
28. Use implicit differentiation to find an equation of the tangent line to the ellipse
2
98
29. Find
d2y
dx 2
in terms of x and y given that 3 − 7xy = 3x − 7y.
30. Assume that x and y are both differentiable functions of t . Find
y=
x.
7
dy
dx
when x = 49and
= 17 for the equation
dt
dt
31. A point is moving along the graph of the function y = sin6x such that
Find
dx
= 2 centimeters per second.
dt
dy
π
when x = .
dt
7
32. A spherical balloon is inflated with gas at the rate of 300 cubic centimeters per minute. How fast is the
radius of the balloon increasing at the instant the radius is 70 centimeters?
33. A conical tank (with vertex down) is 12 feet across the top and 18 feet deep. If water is flowing into the tank
at a rate of 18 cubic feet per minute, find the rate of change of the depth of the water when the water is 10
feet deep.
34. A ladder 20 feet long is leaning against the wall of a house (see figure). The base of the ladder is pulled away
from the wall at a rate of 2 feet per second. How fast is the top of the ladder moving down the wall when its
base is 13 feet from the wall? Round your answer to two decimal places.
8
35. A man 6 feet tall walks at a rate of 10 feet per second away from a light that is 15 feet above the ground (see
figure). When he is 13 feet from the base of the light, at what rate is the tip of his shadow moving?
9
ID: A
Math 2413 t2rsu14
Answer Section
1. ANS:
−8x + 5
2. ANS:
f ′ (x ) = −
2
(x − 3 )
2
3. ANS: A
4. ANS:
f ′ (x ) = 4x 3
5. ANS:
f ′ (x ) = −21x 2 + 8x
6. ANS:
f ′ (x ) = −8x + 4 sin (x )
7. ANS:
2512
f ′ (5) = −
125
8. ANS:
36
f ′ (x ) = 1 + 5
x
9. ANS:
x = 0 and x = −8
10. ANS:
–96 ft/sec
11. ANS:
108 cm2
12. ANS:
H ′ (s) = 8v 7 + 15v 4 − 9v 2
13. ANS:
5
4
f ′ (s) = −s sins + 5s cos s
14. ANS:
Ê
5ˆ
˜˜˜
Ë
¯
ÊÁ 5 ˆ˜ 2
ÁÁ x + 3 ˜˜
Ë
¯
8 ÁÁÁ −3 + 4x
f ′ (x ) = −
1
ID: A
15. ANS:
f ′ (x ) =
ÊÁ
ˆ
ÁÁ 3 + x 2 ˜˜˜ cos x − 2x sinx
Ë
¯
ÊÁ 2 ˆ˜ 2
ÁÁ x + 3 ˜˜
Ë
¯
16. ANS:
f ′ (s) = 9s cos s + 4 sins
17. ANS:
−13
18.
19.
20.
21.
22.
23.
24.
25.
−160 9
f ″ (x ) =
x
81
ANS:
8
f ″ (s) = − 3
x
ANS:
Ê
ˆ4
f ′ (x ) = 30x 5 ÁÁÁ x 6 + 4 ˜˜˜
Ë
¯
ANS:
2
f ′ (x ) = x 6 (5 + 8x) (35 + 80x)
ANS:
y ′ = −32sin4x
ANS:
Ê
ˆ
y ′ = −8x 3 sin ÁÁÁ 2x 4 − 6 ˜˜˜
Ë
¯
ANS:
y = 500x − 400
ANS:
f ″ (x ) = 150x 4 cos 5x 6 − 900x 10 sin 5x 6
ANS:
dy
x
=−
dx
y
26. ANS:
dy 2x + 5 + 9y
=
dx
2y − 9x
27. ANS:
dy
= −0.33
dx
28. ANS:
y = −2x + 9
29. ANS:
d2y
dx 2
=0
2
ID: A
30. ANS:
dy 17
=
dt 14
31. ANS:
dy
6π
= 12 cos
dt
7
32. ANS:
dr
3
=
cm/min
dt
196π
33. ANS:
81
ft/min
50π
34. ANS:
−1.71 ft/sec
35. ANS:
50
ft/sec
3
3