1. (1 point) Differentiate y = varctan(x). AS y'= 2. (1 point) If f(x) = 5 sin(4x) sin (4x), find f'(x). f' (x ) = 3. (1 point) Find dy/ dx by implicit differentiation. tan (x - y) = 1+.x2 (b) dy dx [?/Increasing/Decreasing] 4. (1 point) The accompanying figures show the position ver- S sus time curves of four different particles moving on a straight line. For each particle, determine whether its instantaneous ve- locity is increasing or decreasing with time. S' (c) [?/Increasing/Decreasing] (a) [?/Increasing/Decreasing](d) [?/Increasing/Decreasing] 5. (1 point) The gure below shows three graphs: A (in blue), B (in red), and C (in green). One is the position function of a car, one is the velocity of the car, and one is its acceleration. Identify each curve. 1. Position Function: _ 2. Acceleration Function: _ Velocity Function: _ 3. Note: You only have two attempts at this problem. Note: You can click on the graph to enlarge the images. 6. (1 point) A curve passes through the point (0, 5) and has the property that the slope of every point P of the curve is 4 times the y- coordinate of P. What is the equation of the curve? N) = 7. (1 point) If a stone is thrown vertically upward from the surface of the moon with a velocity of 10 mfs, its height (in meters) after! sec onds is 11(1) : 101 0.8312. (3) What is the velocity of the stone after 3 seconds? (is) What is the velocity of the stone after it has risen 25 m? (a) v(3) : m/s (b) v : m/s 8. (1 point) If a ball is thrown vertically upward with a velocity of 80 ft/s, then its height after 1' seconds is s(t) = 80t * 16:2. (a) What is the maximum height reached by the ball? ('0) What is the velocity of the ball when it is 96 ft above the ground on its way up? (c) What is the velocity of the ball when it is 96 ft above the ground on its way down? (a) height = ft (b) velocity = ft/s (c) velocity 2 ft/s 9. (1 point) (21) Find the average rate of change of the area of the circle with respect to its radius r as r changes from: (i) 2 to 3, (ii) 2 to 25, (iii) 2 to 2.1 (b) Find the instantaneous rate of change when r = 2. [Do you notice anything interesting about the formula of dA/dr?] **Please leave all your answers in terms of "pi\".** (a) (i) (ii) (iii) (b)A'(2) : 10. (1 point) Newton's Law of Gravitation says that the magnitude F of the force exerted by a body of mass m on a body of mass M is GmM F = r2 distance between the bodies. (a) Find dF/dr. [What does the minus sign mean?] (1)) Suppose it is known that Earth attracts an object with a force that decreases at a rate of 2 N/km when r = 20000 km. How fast does this force change when r : 10000 km? (a) dF/dr : (b) N/km 11. (1 point) Human hair from a grave in Africa proved to have only 60% of the carbon 14 of living tissue. When was the body buried? The half life of carbon 14 is 5730 years. The body was buried about , where G is the gravitational constant and r is the years ago. Hint: The half-life of Carbon-14 is 5730 years. Use this and the the information about 60 % to help you nd r. Then nd I. 12. (1 point) Newton's Law of Cooling states that the rate at which an object cools is proportional to the difference in temperature be tween the object and the surrounding medium. Thus, if an ob ject is taken from an oven at 304F and left to cool in a room at 72F, its temperature T after 1' hours will satisfy the differential equation dT E _ k(T 72). If the temperature fell to 198DF in 0.7 hour(s), what will it be after 2 hour(s)? After 2 hour(s), the temperature will be degree F. Hint: Use a substitution. 13. (1 point) A particle moves according to a law that s = 0.012'4 7 0.041'3 where t is in seconds and s is in feet. For this problem, enter INF for innity, if needed. 3) Find the velocity at time t. v(t) : b) What is the velocity after 3 s? C) The particle is at rest at two times. Give these times in in- creasing order. , 11) Over what interval is the particle moving in a positive direc tion? ( , ) e) Find the total distance traveled during the rst 8 5. Find the acceleration at time t. a(t) : g) There is only one interval where the particle is slowing down. What is the interval? ( . ) 14. (1 point) The frequency of vibrations of a vibrating violin string is 1 T given by f = E E, where L is the length of the string, T is the tension, and p is its linear density. Find the rate of change of the frequency with respect to: (a) the length (when T and p are constant) (b) the tension (when L and p are constant) (c) the linear density (when L and T are constant) The pitch of a note is determined by the frequency f. (The higher the frequency, the higher the pitch.) Use the signs of the derivatives in (a) through (c) to determine what happens to the pitch of a note: (d) when the effective length of a string is decreased by plac- ing a nger on the string so a shorter portion of the string vi- brates. (e) when the tension is increased by tuming a timing peg. (i) when the linear density is increased by switching to another string. *For parts (a) through (c), use \"p\" for \"p\". *For parts (d) through (i), enter \"h\" for higher note, or \"i\" for lower note. (a) (b) (c) (d) (e) (f) 15. (1 point) In a sh farm, a population of sh is introduced into a pond and is harvested regularly. A model for the rate of change dP of the sh population is given by the equation 3 : r9(1 P I [E ))P(t) * 0PM, where m is the birth rate of the sh, PC is C the maximum population that the pond can sustain, and [i is the percentage of the population that can be harvested. (3) What value of dP/dt corresponds to a stable population? (b) If the pond can sustain 10,000 sh, the birth rate is 5 percent and the harvesting rare is 4 percent, nd the stable population level. (a) (b)