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?