## CSETMathGuru: THE Site for Single Subject Math

**Are you prepared for Subtest III: Calculus / Trigonometry?**

Then, have a stab at the following Qs!

Assess yourself: You should do OK on the Test if you can solve about 75% of the Qs below, taking about 20 minutes/Q!

1. Find all values (or intervals) of

*x*such that a) y > 1/2 b) y = 1/2 and c) y < 1/2 if y = -sin x defined over [-2pi, 2pi].

2. Solve the trigonometric equation: 2cos2x sin x + 3 cos x sin x = -sin x in the interval [-2pi, 2pi].

3. Using Riemann Sums, show that the Area of the region bounded by the graph f(x) = 3x - x2 and the x-axis between x = 1 and x = 2 is 13/6.

4. A sector shaped like a piece of a pie is cut from a circle of radius r. The outer circular arc of the sector has length s. If the total perimeter of the sector (ie. 2r + s) equals 100 cm, what values of r and s shall maximize the area of the sector?

5. The height of a rock thrown vertically upward on the moon with a velocity of 24 m/sec is given by h(t) = 24t - 0.8t², t being the time elapsed in seconds.

a) What is the rock's velocity and acceleration after t seconds?

b) How long does it take to reach the highest point? How high does the rock go?

c) How long does the rock take to reach half its maximum height? What is its velocity and acceleration at that moment?

d) How long is the rock aloft for its entire journey?

e) With what velocity does it strike the surface of the moon? What is its acceleration then?

6. Prove the Law of Cosines: If a, b and c are sides of triangle ABC and x is the angle opposite c, then

c² = a² + b² - 2ab cos x

7. A ladder 20 ft long leans against a vertical building. If the bottom of the ladder slides away from the building horizontally at a rate of 3ft/sec, how fast is the ladder sliding down the building when the top of the ladder is 8 feet from the ground.

8. A water bucket containing 10 gallons of water develops a leak. The volume v of water in the bucket t seconds later is given by v(t) = 10(1- (t/100))^2 until the bucket is empty 100 seconds later.

a) At what rate is the water leaking after exactly 1 minute?

b) When is the instantaneous rate of change of volume equal to the average rate of change of volume from t = 0 to t =10?

c) At the instant described in part "b", how much water is in the tank?