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Math 160. Practice test 2A.
Compute the derivatives:
f (x) = x^2 + 5
x^3 +^1 x^3
f (x) = (10x^2 + 3)
x
f (x) = 3 x (10x + 2)
f (x) = (x^2 + 2x)^1 /^2
f (x) =
x^2 +
1 + x
- Find dy/dx: x^3 + xy + y = 0
- Find dy/dx: x^5 y + y^2 = 1
- A large cylindrical tank with height 10 feet and radius 4 feet is filled with water. The water begins leaking out at a rate of 1 f t^3 /hr. How fast is the water level dropping when the tank is half-full? (remember that the volume of a cylinder is V = πr^2 h) 4
10
- A kite flying 100 f t above the ground moves horizontally away from its holder at a speed of 8 f t/sec. At what rate is the kite string being reeled out when 200 f t of the string have been reeled out?
200 f^ t 100 ft
8 ft/sec
- Cars A and B leave from the same point at the same time. Car A heads north at 45 miles per hour. Car B heads east at 65 miles per hour. How fast is the distance between them increasing after two hours?
- Air is pumped into a spherical balloon at a rate of 1 f t^3 /min. How fast is the radius of the balloon increasing when the volume of the balloon is 4π f t^3? (remember that the volume of a sphere is V = 43 πr^3 )
- The price at which x units can sold is given by the equation:
p(x) = −. 3 x^2 − 5 x + 300
What is the marginal revenue function? Use marginal analysis to estimate the revenue from the sale of the 10th^ unit.
- Let x be the distance car B has traveled and let y be the distance car A has traveled. Let D be the distance between them. Then
D^2 = x^2 + y^2 =⇒ 2 D · dD dt
= 2x · dx dt
After two hours, car A has traveled 90 miles and car B has traveled 130 miles. The total distance between them at that instant is: D =
902 + 130^2 ≈ 158. Plug in:
2 · 158 · dDdt = 2 · 130 · 65 + 2 · 90 · 45 =⇒ dDdt ≈ 79.
dV dt = 4πr
(^3) · dr dt When V = 4π,
4 π =^43 πr^3 =⇒ 3 = r^3 =⇒ r = 3
1 = 4π( 3
3)^2 ·
dr dt =⇒^
dr dt =^
4 π 32 /^3
≈. 038 f t/min
R(x) = x · p(x) = −. 3 x^3 − 5 x^2 + 300x =⇒ R′(x) = −. 9 x^2 − 10 x + 300 R′(9) = −.9(99)^2 − 10(99) + 300 = 137. 10
