Math 3A HW #6
Answers must be submitted via Moodle before 10AM on Wednesday April 6th, 2016.
Good luck!
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Use implicit differentiation to find dy/dx.
1) x3 + 3x 2y + y 3 = 8
2
A) - x + 3xy
x2 + y 2
2) y cos
A)
C)
1)
x2 + 3xy
C)
x2 + y 2
2
B) - x + 2xy
x2 + y 2
1
= 10x + 10y
y
2)
10 - y sin
10
1
1
sin
+ y cos
- 10
y
y
B)
cos
10y 2
1
sin
- 10y 2
y
D)
Use implicit differentiation to find dy/dx and d2y/dx2.
3) xy - x + y = 2
dy = y + 1 ; d2y = 2y + 2
A)
dx
x + 1 dx2
(x + 1)2
C)
x2 + 2xy
D)
x2 + y 2
1
y
1
- 10
y
10y
1
1
sin
+ y cos
- 10y
y
y
3)
dy = - 1 + y ; d2y = y + 1
B)
dx
x + 1 dx2
(x + 1)2
dy = 1 - y ; d2y = 2y - 2
dx 1 + x dx2
(x + 1)2
D)
dy = - 1 + y ; d2y = 2y - 2
dx
x + 1 dx2
(x + 1)2
At the given point, find the slope of the curve or the line that is tangent to the curve, as requested.
4) y 5 + x3 = y 2 + 9x, tangent at (0, 1)
A) y = -
9
x- 1
5
B) y = -
9
x
7
C) y =
9
x+1
5
D) y = 3x + 1
Use logarithmic differentiation to find the derivative of y with respect to the independent variable.
5) y = x5 sin x
sin x
A) x sin x cos x ln x +
x
B) 5 cos x ln x + sin x
x
sin x
C) 5 x5 sin x cos x ln x +
x
D) 5 sin x ln x
1
4)
5)
6) y = (ln x)ln x
(ln x)ln x
A)
x
6)
B)
C) ln x ln (ln x)
D)
ln (ln x) + 1
x
ln x
ln (ln x) + 1
(ln x)
x
Find the derivative of y with respect to x.
2x + 3
7) y = sec-1
7
A)
7)
14
(2x + 3)2 - 3
(2x + 3)
-14
C)
(2x + 3)
14
B)
D)
(2x + 3)2 - 1
(2x + 3)2 - 49
-14
1 + (2x + 3)2
Solve the problem.
8) Water is falling on a surface, wetting a circular area that is expanding at a rate of 2 mm2/s. How
fast is the radius of the wetted area expanding when the radius is 113 mm? (Round your answer
to four decimal places.)
A) 0.0056 mm/s
B) 0.0028 mm/s
C) 0.0177 mm/s
D) 354.9997 mm/s
8)
9) Assume that the profit generated by a product is given by P(x) = 3 x, where x is the number of
units sold. If the profit keeps changing at a rate of $500 per month, then how fast are the sales
changing when the number of units sold is 1900? (Round your answer to the nearest dollar per
month.)
A) $130,767/month
B) $8/month
C) $14,530/month
D) $7265/month
9)
Solve the problem. Round your answer, if appropriate.
10) Water is being drained from a container which has the shape of an inverted right circular cone.
The container has a radius of 4.00 inches at the top and a height of 8.00 inches. At the instant
when the water in the container is 6.00 inches deep, the surface level is falling at a rate of 0.9
in./sec. Find the rate at which water is being drained from the container.
A) 25.4 in.3/s
B) 36.8 in.3s
C) 42.4 in.3/s
D) 24.3 in.3/s
2
10)