1. The limit, as x tends to 2, of the function

exists, and is equal to 16. What is the value of the whole number coefficient b?

2. The first derivative of

is of the form

, where a is a positive number. What is the value of the whole number coefficient a?

3. For the ellipse

, the slope of the tangent at the point (10, -6) is b. Correct to the nearest hundredth, the value of b is ______________.

4. The number I is defined as

. The value of I, correct to the nearest hundredth, is ______________.

5. A spring has a motion that can be modelled by the differential equation

. One solution of this equation is

, where k is a positive number to be determined. What is the value of k, correct to two decimal places?

6. An LRT train travels for 120 s between two stations. It accelerates for 30 s, maintains a constant velocity for 70 s, and brakes to a stop in 20 s. The velocity function v(t), with velocity measured in m/s, is defined as follows:

a. Sketch a graph of the velocity as a function of time, assuming the velocity function to be continuous at t = 30 and t = 100. (Be sure to label your axes.)

b. Determine the total distance travelled by the train in the 120-second time interval.

4. Let the function f be defined as

.

a. Use a suitable numerical value of x to estimate the limit, if it exists, of f(x) as x approaches zero.

b. Use trigonometric identities and limit theorems to confirm the estimate made in part a above.