# MATH 165 Test #3 Derivatives Name: ______________________________________

```MATH 165 Test #3 Derivatives Name: ______________________________________
No notes or books.
Each problem is worth 6 points. Show your work for partial credit.
1. Find
dy
of y = 3x 2 − 2x
dx
2. Find
dy
of y = tan x 2 + sec x
dx
3. Find
dy
of y =
dx
4. Find
dy
of sin ( xy ) = x 2 − y
dx
x3
x−2
5. Find an equation of the tangent line at (1,1) on the curve y 2 = x 3 ( 2 − x )
6. Find a second degree polynomial such that f (1) = 4, f ′(1) = 0, and f ′′(1) = 2.
7. Find the derivative using the definition. y = x − 2
8. A balloon is rising at a constant rate of 4 ft/s. A boy is cycling along a straight road at
a speed of 10 ft/s. When he passes under the balloon, it is 40 ft above him. How fast
is the distance between the boy and the balloon increasing 3 s later?
9.
The radius of a sphere is measured to be 10 inches with a maximum possible error of
0.25 inches. Find the estimated maximum error and relative error in the calculated
volume of the sphere ( V = 43 π ⋅ r 3 ).
sin 4x
x→0 sin 6x
10. Find the limit. lim
(For problems 11, 12, and 13) A particle moves on a vertical line so that its coordinate at
time t is y = t 3 − 12t + 3, t ≥ 0 .
11a. Find the velocity function.
11b. Find the acceleration function.
12. When is the particle moving upward?
13a. Find the distance that the particle travels in the time interval 0 ≤ t ≤ 3 .
13b. When is the particle speeding up?
14. A lighthouse is 4 km from the nearest point P on a straight shoreline. The light
revolves 3 revolutions per minute. When the light is shining on the shore 1 km from
point P, what is the rate of change (km/min) of the light along the shoreline?
15. The ideal gas law at absolute temperature T (in Kelvin), pressure P (in
atmospheres), and volume V (in liters) is PV = nRT , where n is the number of
moles of the gas and R = 0.0821 is the gas constant. At a certain instant, P = 7 atm
and is increasing at a rate of 0.10 atm/min, and V = 10 liters and is decreasing at a
rate of 0.15 liters/min. Find the rate of change of T with respect to time at that
instant if n = 10 moles.
16. One side of a right triangle is known to be 20 cm long and the opposite angle is
measured as 30 degrees, with a possible error of ±1 .
a)
Find the estimated maximum error in computing the length of the hypotenuse.
b)
What is the percentage error?
BONUS (additional 6 points): On a military watch (the hour hand goes around once in
24 hours), the minute hand is 8 mm long and the hour hand is 4 mm long. How fast is the
distance between the tips of the hands on a military watch decreasing at 3 o'clock?
```